수학노트 - 사용자 기여 [ko]

임시

2023-12-13T07:33:55Z

2022-09-24T05:28:13Z

Pythagoras0: /* 위키데이터 */

== 노트 ==

===말뭉치===
# p5.js is a JavaScript library for creative coding, with a focus on making coding accessible and inclusive for artists, designers, educators, beginners, and anyone else!<ref name="ref_4012356a">[https://p5js.org/ p5.js]</ref>
# JavaScript is a cross-platform, object-oriented scripting language.<ref name="ref_646c500f">[https://www.w3resource.com/javascript-exercises/ JavaScript Exercises, Practice, Solution]</ref>
# JavaScript contains a standard library of objects, such as Array, Date, and Math, and a core set of language elements such as operators, control structures, and statements.<ref name="ref_646c500f" />
# We have started this section for those (beginner to intermediate) who are familiar with JavaScript.<ref name="ref_646c500f" />
# Hope, these exercises help you to improve your JavaScript coding skills.<ref name="ref_646c500f" />
# Interpreted Language - JavaScript is an interpreted programming language.<ref name="ref_6e246136">[https://www.programiz.com/javascript Learn JavaScript Programming]</ref>
# Platform Independence - JavaScript codes are run on browsers.<ref name="ref_6e246136" />
# JavaScript uses the just-in-time compilation technique.<ref name="ref_6e246136" />
# Since the compilation is handled at run time, JavaScript is considered an interpreted language.<ref name="ref_6e246136" />
# This approach is commonplace today, and libraries like Backbone.js, Ember.js, and Angular.js have made it easier for developers to build these rich JavaScript apps.<ref name="ref_2fc1fa5b">[https://medium.com/airbnb-engineering/isomorphic-javascript-the-future-of-web-apps-10882b7a2ebc Isomorphic JavaScript: The Future of Web Apps]</ref>
# It wasn’t long before developers started to build out entire applications in the browser using JavaScript, taking advantage of these new capabilities.<ref name="ref_2fc1fa5b" />
# With Node.js, a fast, stable server-side JavaScript runtime, we can now make this dream a reality.<ref name="ref_2fc1fa5b" />
# This idea isn’t new — Nodejitsu wrote a great description of isomorphic JavaScript architecture in 2011 — but it’s been slow to adopt.<ref name="ref_2fc1fa5b" />
# JavaScript is a lightweight, cross-platform and interpreted scripting language.<ref name="ref_f2fe6bcb">[https://www.geeksforgeeks.org/javascript-tutorial/ JavaScript Tutorials]</ref>
# With advances in browser technology and JavaScript having moved into the server with Node.js and other frameworks, JavaScript is capable of so much more.<ref name="ref_f2fe6bcb" />
# JavaScript was created in the first place for DOM manipulation.<ref name="ref_f2fe6bcb" />
# You need to enable JavaScript to run this app.<ref name="ref_c3091bdb">[https://babeljs.io/en/repl Babel · The compiler for next generation JavaScript]</ref>
# In the JavaScript community, engineers share hundreds of thousands of pieces of code so we can avoid rewriting basic components, libraries, or frameworks of our own.<ref name="ref_e2cf5940">[https://engineering.fb.com/2016/10/11/web/yarn-a-new-package-manager-for-javascript/ Yarn: A new package manager for JavaScript]</ref>
# The most popular JavaScript package manager is the npm client, which provides access to more than 300,000 packages in the npm registry.<ref name="ref_e2cf5940" />
# In the days before package managers, it was commonplace for JavaScript engineers to rely on a small number of dependencies stored directly in their projects or served by a CDN.<ref name="ref_e2cf5940" />
# The first major JavaScript package manager, npm, was built shortly after Node.js was introduced, and it quickly became one of the most popular package managers in the world.<ref name="ref_e2cf5940" />
# Our JavaScript Tutorial is designed for beginners and professionals both.<ref name="ref_027160c0">[https://www.javatpoint.com/javascript-tutorial Learn JavaScript Tutorial]</ref>
# JavaScript is not a compiled language, but it is a translated language.<ref name="ref_027160c0" />
# With JavaScript, users can build modern web applications to interact directly without reloading the page every time.<ref name="ref_027160c0" />
# All popular web browsers support JavaScript as they provide built-in execution environments.<ref name="ref_027160c0" />
# Kotlin/JS provides the ability to transpile your Kotlin code, the Kotlin standard library, and any compatible dependencies to JavaScript.<ref name="ref_2768fc8b">[https://kotlinlang.org/docs/reference/js-overview.html Kotlin Programming Language]</ref>
# The recommended way to use Kotlin/JS is via the kotlin.js and kotlin.multiplatform Gradle plugins.<ref name="ref_2768fc8b" />
# They provide a central and convenient way to set up and control Kotlin projects targeting JavaScript.<ref name="ref_2768fc8b" />
# This includes essential functionality such as controlling the bundling of your application, adding JavaScript dependencies directly from npm, and more.<ref name="ref_2768fc8b" />
# In browsers, JavaScript shares a thread with a load of other stuff that differs from browser to browser.<ref name="ref_8be67c0e">[https://web.dev/promises/ JavaScript Promises: An introduction]</ref>
# But typically JavaScript is in the same queue as painting, updating styles, and handling user actions (such as highlighting text and interacting with form controls).<ref name="ref_8be67c0e" />
# Promises arrive in JavaScript!<ref name="ref_8be67c0e" />
# The above and JavaScript promises share a common, standardized behaviour called Promises/A+.<ref name="ref_8be67c0e" />
# This page summarizes the JavaScript features that VS Code ships with.<ref name="ref_3218ae9f">[https://code.visualstudio.com/docs/languages/javascript JavaScript Programming with Visual Studio Code]</ref>
# A jsconfig.json file defines a JavaScript project in VS Code.<ref name="ref_3218ae9f" />
# If not all JavaScript files in your workspace should be considered part of a single JavaScript project.<ref name="ref_3218ae9f" />
# To ensure that a subset of JavaScript files in your workspace is treated as a single project.<ref name="ref_3218ae9f" />
# JavaScript is an object-oriented programming language employed by most websites along with HTML and CSS to create robust, dynamic and interactive user experiences.<ref name="ref_86546202">[https://www.edx.org/learn/javascript Learn JavaScript with Online Courses and Classes]</ref>
# The JavaScript programming language was introduced in 1995 and has since become one of the most popular with support by all major web browsers.<ref name="ref_86546202" />
# JavaScript programs are used both client-side and server-side to add functionality to web pages.<ref name="ref_86546202" />
# Javascript is one of the main programming languages used in web development.<ref name="ref_86546202" />
# This document serves as the complete definition of Google’s coding standards for source code in the JavaScript programming language.<ref name="ref_cbf24346">[https://google.github.io/styleguide/jsguide.html Google JavaScript Style Guide]</ref>
# The JavaScript community has invested effort to make sure clang-format does the right thing on JavaScript files.<ref name="ref_cbf24346" />
# JavaScript includes many dubious (and even dangerous) features.<ref name="ref_cbf24346" />
# JSDoc serves multiple purposes in JavaScript.<ref name="ref_cbf24346" />
# In this JavaScript aticle, we will go over event handlers, event listeners, and event objects.<ref name="ref_b8b0af3f">[https://www.digitalocean.com/community/tutorial_series/how-to-code-in-javascript How To Code in JavaScript]</ref>
# Using the API's JavaScript functions, you can queue videos for playback; play, pause, or stop those videos; adjust the player volume; or retrieve information about the video being played.<ref name="ref_76d43eb7">[https://developers.google.com/youtube/iframe_api_reference YouTube Player API Reference for iframe Embeds]</ref>
# – The API will call this function when the page has finished downloading the JavaScript for the player API, which enables you to then use the API on your page.<ref name="ref_76d43eb7" />
# The code in this section loads the IFrame Player API JavaScript code.<ref name="ref_76d43eb7" />
# After the API's JavaScript code loads, the API will call the onYouTubeIframeAPIReady function, at which point you can construct a YT.Player object to insert a video player on your page.<ref name="ref_76d43eb7" />
# We encourage any developers working at the intersection of machine learning and web/JS applications to join and participate in the activities of the SIG.<ref name="ref_0e37525e">[https://www.tensorflow.org/js Machine Learning for Javascript Developers]</ref>
# JavaScript is used mainly for enhancing the interaction of a user with the webpage.<ref name="ref_26a4a969">[https://www.guru99.com/interactive-javascript-tutorials.html JavaScript Tutorial for Beginners: Learn Javascript in 5 Days]</ref>
# JavaScript Engines are complicated.<ref name="ref_26a4a969" />
# Here, JavaScript engine applies optimizations at each step of the process.<ref name="ref_26a4a969" />
# It reads a compiled script and analyzes the data that passes in JavaScript engine.<ref name="ref_26a4a969" />
# Data attributes You can use all Bootstrap plugins purely through the markup API without writing a single line of JavaScript.<ref name="ref_f9c0473a">[https://getbootstrap.com/docs/3.4/javascript/ JavaScript · Bootstrap]</ref>
# We also believe you should be able to use all Bootstrap plugins purely through the JavaScript API.<ref name="ref_f9c0473a" />
# > "3.4.1" No special fallbacks when JavaScript is disabled Bootstrap's plugins don't fall back particularly gracefully when JavaScript is disabled.<ref name="ref_f9c0473a" />
# If you care about the user experience in this case, use <noscript> to explain the situation (and how to re-enable JavaScript) to your users, and/or add your own custom fallbacks.<ref name="ref_f9c0473a" />
# JavaScript (JS) is a lightweight, interpreted, or just-in-time compiled programming language with first-class functions.<ref name="ref_76748401">[https://developer.mozilla.org/en-US/docs/Web/JavaScript JavaScript]</ref>
# JavaScript is a prototype-based, multi-paradigm, single-threaded, dynamic language, supporting object-oriented, imperative, and declarative (e.g. functional programming) styles.<ref name="ref_76748401" />
# Do not confuse JavaScript with the Java programming language.<ref name="ref_76748401" />
# Both "Java" and "JavaScript" are trademarks or registered trademarks of Oracle in the U.S. and other countries.<ref name="ref_76748401" />
# JavaScript is very easy to implement because it is integrated with HTML.<ref name="ref_5a21c5f4">[https://www.tutorialspoint.com/javascript/index.htm Javascript Tutorial]</ref>
# Javascript is a MUST for students and working professionals to become a great Software Engineer specially when they are working in Web Development Domain.<ref name="ref_5a21c5f4" />
# Javascript is the most popular programming language in the world and that makes it a programmer’s great choice.<ref name="ref_5a21c5f4" />
# Once you learnt Javascript, it helps you developing great front-end as well as back-end softwares using different Javascript based frameworks like jQuery, Node.<ref name="ref_5a21c5f4" />
# This tutorial will teach you JavaScript from basic to advanced.<ref name="ref_95a957d0">[https://www.w3schools.com/js/DEFAULT.asp JavaScript Tutorial]</ref>
# If you try all the examples, you will learn a lot about JavaScript, in a very short time!<ref name="ref_95a957d0" />
# Join Cassidy as she does a live build to celebrate JavaScript's 25th birthday.<ref name="ref_4afd0de1">[https://www.javascript.com/ JavaScript.com]</ref>
# Measure your JS skills for free in just 10 minutes.<ref name="ref_4afd0de1" />
# You can’t get very far in tech without running smack into JavaScript.<ref name="ref_bde19c99">[https://www.skillcrush.com/blog/javascript/ What Is JavaScript? A Guide for Total Beginners]</ref>
# The results of JavaScript may seem simple, but there’s a reason why we teach an entire segment on JavaScript in both our Front End Web Developer and Break into Tech Blueprints.<ref name="ref_bde19c99" />
# From there, they move on to JavaScript.<ref name="ref_bde19c99" />
# Meanwhile, because JavaScript is such an integral part of web functionality, all major web browsers come with built-in engines that can render JavaScript.<ref name="ref_bde19c99" />
# JavaScript enables interactive web pages and is an essential part of web applications.<ref name="ref_ee2655a4">[https://en.wikipedia.org/wiki/JavaScript JavaScript]</ref>
# As a multi-paradigm language, JavaScript supports event-driven, functional, and imperative programming styles.<ref name="ref_ee2655a4" />
# JavaScript engines were originally used only in web browsers, but they are now embedded in some servers, usually via Node.js.<ref name="ref_ee2655a4" />
# The choice of the JavaScript name has caused confusion, sometimes giving the impression that it is a spin-off of Java.<ref name="ref_ee2655a4" />
===소스===
<references />

==메타데이터==
===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q2005 Q2005]
===Spacy 패턴 목록===
* [{'LEMMA': 'JavaScript'}]
* [{'LEMMA': 'JS'}]

Discrete differential geometry

2022-09-20T02:59:05Z

Pythagoras0:

==메모==
* [http://euclid.colorado.edu/%7Eiveyt/BSintro.pdf http://euclid.colorado.edu/~iveyt/BSintro.pdf]

==related items==
* [[Discrete conformal transform]]

==articles==
* Bobenko, Alexander I., and Felix Günther. “Discrete Riemann Surfaces Based on Quadrilateral Cellular Decompositions.” arXiv:1511.00652 [math], November 2, 2015. http://arxiv.org/abs/1511.00652.
* Bobenko, Alexander I., Yuri B. Suris, and Jan Techter. “On a Discretization of Confocal Quadrics.” arXiv:1511.01777 [nlin], November 5, 2015. http://arxiv.org/abs/1511.01777.
* Knöppel, Felix, and Ulrich Pinkall. ‘Complex Line Bundles over Simplicial Complexes and Their Applications’. arXiv:1506.07853 [cs, Math], 25 June 2015. http://arxiv.org/abs/1506.07853.

==메타데이터==
===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q958922 Q958922]
===Spacy 패턴 목록===
* [{'LOWER': 'discrete'}, {'LOWER': 'differential'}, {'LEMMA': 'geometry'}]

== 노트 ==

===말뭉치===
# Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact.<ref name="ref_3fe5d594">[https://link.springer.com/book/10.1007/978-3-7643-8621-4 Discrete Differential Geometry]</ref>
# It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry.<ref name="ref_37059fcb">[https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ Discrete Differential Geometry]</ref>
# Acknowledgements These notes grew out of a course on discrete differential geometry (DDG) taught annually starting in 2011, rst at Caltech and now at CMU.<ref name="ref_4bd7573c">[https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf Discrete differential geometry:]</ref>
# We in- troduce discrete differential geometry in the context of discrete curves and curvature (Chapter 1).<ref name="ref_c199dfae">[http://geometry.caltech.edu/pubs/GSD06.pdf Discrete differential geometry:]</ref>
# Part IV focuses on applications of discrete differential geometry.<ref name="ref_0fcf3c73">[https://nvbogachev.netlify.app/teaching/gcs20f/Alexander_I._Bobenko.pdf Oberwolfach seminars]</ref>
# Our work in discrete differential geometry has also been supported by the Deutsche Forschungsgemeinschaft (DFG), as well as other funding agencies.<ref name="ref_0fcf3c73" />
# The DFG Research Center MATHEON in Berlin, through its Application Area F Visualization, has supported work on the applications of discrete differential geometry.<ref name="ref_0fcf3c73" />
# Problems of discrete differential geometry also naturally appear in (and are relevant for) other areas of mathematics.<ref name="ref_6631a9ec">[https://ems.press/journals/owr/articles/1194 Discrete Differential Geometry]</ref>
# This is the first book on a newly emerging field of discrete differential geometry providing an excellent way to access this exciting area.<ref name="ref_769b6ee3">[https://www.goodreads.com/book/show/1969822.Discrete_Differential_Geometry Discrete Differential Geometry]</ref>
# I basically set about doing the homework from Keenan Crane's discrete differential geometry course(s).<ref name="ref_d6cd45fa">[https://github.com/LukeMcCulloch/Python-discrete-differential-geometry LukeMcCulloch/Python-discrete-differential-geometry: Discrete differential geometry in Python. Derived from CMU and Caltech Codebases and Published papers]</ref>
# See: here and pdf discrete differential geometry book for starters.<ref name="ref_d6cd45fa" />
# We consider the problem of constructing a discrete differential geometry defined on nonplanar quadrilateral meshes.<ref name="ref_8b3d1610">[https://link.aps.org/doi/10.1103/PhysRevE.85.066708 Discrete differential geometry: The nonplanar quadrilateral mesh]</ref>
# The lectures aimed at giving an introduction to the main ideas of (an integrable part of) discrete differential geometry.<ref name="ref_39ddbaaf">[https://www.cambridge.org/core/books/symmetries-and-integrability-of-difference-equations/lectures-on-discrete-differential-geometry/196221C1C0DE5061038969F068D96CF4 Lectures on Discrete Differential Geometry (Chapter 10)]</ref>
# Discrete differential geometry (DDG) is a new field presently emerging on the border between differential and discrete geometry.<ref name="ref_39ddbaaf" />
===소스===
<references />

[[분류:integrable systems]]
[[분류:math and physics]]
[[분류:migrate]]

Discrete differential geometry

2022-09-20T02:55:37Z

2022-09-19T02:59:57Z

Pythagoras0:

== 노트 ==

===말뭉치===
# Elliptic Curve Digital Signature Algorithm or ECDSA is a cryptographic algorithm used by Bitcoin to ensure the effective and secure control of ownership of funds.<ref name="ref_5a293cd7">[https://wiki.bitcoinsv.io/index.php/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm]</ref>
# Elliptic Curve Digital Signature Algorithm or ECDSA is a cryptographic algorithm used by Bitcoin to ensure that funds can only be spent by their rightful owners.<ref name="ref_862451f6">[https://en.bitcoin.it/wiki/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm]</ref>
# The ECDSA signing and verification algorithms make use of a few fundamental variables which are used to obtain a signature and the reverse process of getting a message from a signature.<ref name="ref_862451f6" />
# ECDSA is also used for Transport Layer Security (TLS), the successor to Secure Sockets Layer (SSL), by encrypting connections between web browsers and a web application.<ref name="ref_6b86fa80">[https://www.hypr.com/elliptic-curve-digital-signature-algorithm/ What is the Elliptic Curve Digital Signature Algorithm (ECDSA)?]</ref>
# The encrypted connection of an HTTPS website, illustrated by an image of a physical padlock shown in the browser, is made through signed certificates using ECDSA.<ref name="ref_6b86fa80" />
# Here is where ECDSA offers the required flexibility.<ref name="ref_154afd47">[https://www.maximintegrated.com/en/design/technical-documents/tutorials/5/5767.html Elliptic Curve Digital Signature Algorithm Explained]</ref>
# This article introduces the ECDSA concept, its mathematical background, and shows how the method can be successfully deployed in practice.<ref name="ref_154afd47" />
# This article discusses the concept of the Elliptic Curve Digital Signature Algorithm (ECDSA) and shows how the method can be used in practice.<ref name="ref_154afd47" />
# Computations needed for ECDSA authentication are the generation of a key pair (private key, public key), the computation of a signature, and the verification of a signature.<ref name="ref_154afd47" />
# The ECDSA (Elliptic Curve Digital Signature Algorithm) is a cryptographically secure digital signature scheme, based on the elliptic-curve cryptography (ECC).<ref name="ref_dfba95b2">[https://cryptobook.nakov.com/digital-signatures/ecdsa-sign-verify-messages ECDSA: Elliptic Curve Signatures]</ref>
# ECDSA relies on the math of the cyclic groups of elliptic curves over finite fields and on the difficulty of the ECDLP problem (elliptic-curve discrete logarithm problem).<ref name="ref_dfba95b2" />
# The ECDSA sign / verify algorithm relies on EC point multiplication and works as described below.<ref name="ref_dfba95b2" />
# A 256-bit ECDSA signature has the same security strength like 3072-bit RSA signature.<ref name="ref_dfba95b2" />
# In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.<ref name="ref_70efc61b">[https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm]</ref>
# ECDSA does the same thing as any other digital signing signature, but more efficiently.<ref name="ref_55c5f655">[https://www.encryptionconsulting.com/education-center/what-is-ecdsa/ Elliptic Curve Digital Signature Algorithm (ECDSA)]</ref>
# This is due to ECDSA’s use of smaller keys to create the same level of security as any other digital signature algorithm.<ref name="ref_55c5f655" />
# ECDSA is used to create ECDSA certificates, which is a type of electronic document used for authentication of the owner of the certificate.<ref name="ref_55c5f655" />
# The way ECDSA works is an elliptic curve is that an elliptic curve is analyzed, and a point on the curve is selected.<ref name="ref_55c5f655" />
# Firms do no longer have to incur the wrath of data loss and manipulation, through Elliptic Curve Digital Signature Algorithm (ECDSA), data is now safe.<ref name="ref_124458f8">[https://www.1kosmos.com/identity-management/the-elliptic-curve-digital-signature-algorithm/ The Elliptic Curve Digital Signature Algorithm (ECDSA)]</ref>
# ECDSA adopts various concepts in its operation.<ref name="ref_124458f8" />
# Everyone has probably heard of ECDSA in one form or another.<ref name="ref_124458f8" />
# If you want to see how Elliptic Curve Digital Signature Algorithm functions, it’s difficult to make sense of it on the grounds that most reference reports online are lacking.<ref name="ref_124458f8" />
# An Elliptic Curve Digital Signature Algorithm (ECDSA) uses ECC keys to ensure each user is unique and every transaction is secure.<ref name="ref_090e87d6">[https://avinetworks.com/glossary/elliptic-curve-cryptography/ What is Elliptic Curve Cryptography? Definition & FAQs]</ref>
# Both Bitcoin and Ethereum apply the Elliptic Curve Digital Signature Algorithm (ECDSA) specifically in signing transactions.<ref name="ref_090e87d6" />
# The ECDSA algorithm uses elliptic curve cryptography (an encryption system based on the properties of elliptic curves) to provide a variant of the Digital Signature Algorithm.<ref name="ref_fa291530">[https://www.ibm.com/docs/SSLTBW_2.4.0/com.ibm.zos.v2r4.csfb300/csfb300_Elliptic_Curve_Digital_Signature_Algorithm__ECDSA_.htm Elliptic Curve Digital Signature Algorithm (ECDSA)]</ref>
# The most widely used digital signature in broadcast authentication is ECDSA, as described in Section 3.<ref name="ref_d07faf15">[https://www.sciencedirect.com/topics/computer-science/elliptic-curve-digital-signature-algorithm Elliptic Curve Digital Signature Algorithm - an overview]</ref>
# In this section, we will study a few of the digital signatures computed from public keys, including ECDSA versions.<ref name="ref_d07faf15" />
# The first block is only authenticated using digital signature ECDSA.<ref name="ref_d07faf15" />
# Next, when they rebroadcast verified legitimate packets, they also include partial results of the ECDSA verification process.<ref name="ref_d07faf15" />
# If you’re into SSL certificates or cryptocurrencies, you’d likely come across the much-discussed topic of “ECDSA vs RSA” (or RSA vs ECC).<ref name="ref_621714c4">[https://sectigostore.com/blog/ecdsa-vs-rsa-everything-you-need-to-know/ ECDSA vs RSA: Everything You Need to Know]</ref>
# ECDSA and RSA are two of the world’s most widely adopted asymmetric algorithms.<ref name="ref_621714c4" />
# It’s an extremely well-studied and audited algorithm as compared to modern algorithms such as ECDSA.<ref name="ref_621714c4" />
# ECDSA was born when two mathematicians named Neal Koblitz and Victor S. Miller proposed the use of elliptical curves in cryptography.<ref name="ref_621714c4" />
# Let's discuss now how and why the ECDSA signatures that Sony used in the Playstation 3 were faulty and how it allowed hackers to gain access to the PS3's ECDSA private key.<ref name="ref_312e7735">[https://www.instructables.com/Understanding-how-ECDSA-protects-your-data/ Understanding How ECDSA Protects Your Data.]</ref>
# The ECDSA algorithm is very secure for which it is impossible to find the private key...<ref name="ref_312e7735" />
# As with elliptic-curve cryptography in general, the bit size of the public key believed to be needed for ECDSA is about twice the size of the security level, in bits.<ref name="ref_7c8129e9">[https://www.cardlogix.com/glossary/ecdsa-elliptic-curve-digital-signature-algorithm/ ECDSA (Elliptic Curve Digital Signature Algorithm)]</ref>
# the size of an ECDSA public key would be 160 bits, whereas the size of a DSA public key is at least 1024 bits.<ref name="ref_7c8129e9" />
# On the other hand, the signature size is the same for both DSA and ECDSA: approximately bits, where is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.<ref name="ref_7c8129e9" />
# The elliptic curve digital signature algorithm (ECDSA) is a common digital signature scheme that we see in many of our code reviews.<ref name="ref_d3e0a981">[https://blog.trailofbits.com/2020/06/11/ecdsa-handle-with-care/ ECDSA: Handle with Care]</ref>
# You’re probably familiar with attacks against ECDSA.<ref name="ref_d3e0a981" />
# When DSA is used with the elliptic curve group as this mathematical group, we call this ECDSA.<ref name="ref_d3e0a981" />
# ECDSA works the same way as DSA, except with a different group.<ref name="ref_d3e0a981" />
# Elliptic Curve Digital Signature Algorithm (ECDSA) is a cryptographic algorithm used by Bitcoin to ensure that funds can only be spent by their rightful owners.<ref name="ref_c92ef94b">[https://en.bitcoinwiki.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm – BitcoinWiki]</ref>
# In December 2010, a group calling itself fail0verflow announced recovery of the ECDSA private key used by Sony to sign software for the PlayStation 3 game console.<ref name="ref_c92ef94b" />
# One characteristic of DSA and ECDSA is that they need to produce, for each signature generation, a fresh random value (hereafter designated as k).<ref name="ref_9063c683">[https://tools.ietf.org/html/rfc6979 rfc6979]</ref>
# The randomized nature of DSA and ECDSA also makes implementations harder to test.<ref name="ref_9063c683" />
# Deterministic DSA and ECDSA only deal with the need for randomness at the time of signature generation.<ref name="ref_9063c683" />
# It is used in the specification of the encoding of an ECDSA private key (x) within an ASN.1-based structure.<ref name="ref_9063c683" />
# The Elliptic Curve Digital Signature Algorithm (ECDSA) variant is described, an analogue of the Digital Signature Algorithm (DSA).<ref name="ref_a8997298">[https://link.springer.com/10.1007/978-1-4419-5906-5_251 Elliptic Curve Signature Schemes]</ref>
# The Elliptic Curve Digital Signature Algorithm (ECDSA) is a variant of the Digital Signature Algorithm (DSA) which uses Elliptic curve cryptography.<ref name="ref_a8da8e72">[https://cryptography.fandom.com/wiki/Elliptic_Curve_DSA Elliptic Curve DSA]</ref>
# On the other hand, the signature size is the same for both DSA and ECDSA: bits, where is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.<ref name="ref_a8da8e72" />
# Provides an abstract base class that encapsulates the Elliptic Curve Digital Signature Algorithm (ECDSA).<ref name="ref_7d5e9289">[https://docs.microsoft.com/en-us/dotnet/api/system.security.cryptography.ecdsa ECDsa Class (System.Security.Cryptography)]</ref>
# Initializes a new instance of the ECDsa class.<ref name="ref_7d5e9289" />
# Create(ECCurve) Creates a new instance of the default implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA) with a newly generated key over the specified curve.<ref name="ref_7d5e9289" />
# Create(ECParameters) Creates a new instance of the default implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA) using the specified parameters as the key.<ref name="ref_7d5e9289" />
# These are all prerequisites to apply Elliptic Curve Digital Signature Algorithm (ECDSA).<ref name="ref_1ca63053">[https://sefiks.com/2018/02/16/elegant-signatures-with-elliptic-curve-cryptography/ Elegant Signatures with Elliptic Curve Cryptography]</ref>
# ECDSA is highly adopted in IOT devices because of their low power consumption.<ref name="ref_1ca63053" />
# Moreover, Bitcoin transactions are signed with ECDSA, too.<ref name="ref_1ca63053" />
# To get started, ECDSA bases its operation on the basis of a mathematical equation that draws a curve.<ref name="ref_ad530292">[https://academy.bit2me.com/en/what-is-ecdsa-elliptic-curve/ What is the ECDSA signature algorithm?]</ref>
# Under this operating scheme, ECDSA guarantees in the first instance the following: Unique and unrepeatable signatures for each generation set private keys and public.<ref name="ref_ad530292" />
# Thanks to these two characteristics, ECDSA is considered a safe standard for deploying digital signature systems.<ref name="ref_ad530292" />
# For example, the security certificate infrastructure SSL y TLS Internet makes heavy use of ECDSA.<ref name="ref_ad530292" />
# This means one template argument to ECDSA will include ECP .<ref name="ref_3aa88b4b">[https://www.cryptopp.com/wiki/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm]</ref>
# Elliptic Curve Digital Signature Algorithm, or ECDSA, is one of three digital signature schemes specified in FIPS-186.<ref name="ref_3aa88b4b" />
# The key formats are ignorant to the objects which use them (such as ECDSA).<ref name="ref_3aa88b4b" />
# In Fireware v12.3 U1 or higher, the Firebox supports Elliptic Curve Digital Signature Algorithm (ECDSA) certificates.<ref name="ref_792a001d">[https://www.watchguard.com/help/docs/help-center/en-US/Content/en-US/Fireware/certificates/cert_ecdsa.html About Elliptic Curve Digital Signature Algorithm (ECDSA) certificates]</ref>
# Compared to RSA, ECDSA certificates have equivalent security, smaller keys, and increased efficiency.<ref name="ref_792a001d" />
# In some countries, governments require ECDSA certificates for regulation compliance.<ref name="ref_792a001d" />
# In Fireware v12.6.2 or higher, the Firebox supports creating a Certificate Signing Request (CSR) with ECDSA.<ref name="ref_792a001d" />
# The Elliptic Curve Digital Signature Algorithm or ECDSA is a cryptographic scheme for producing digital signatures using public and private keys.<ref name="ref_ade6a467">[https://river.com/learn/terms/e/ecdsa/ River Financial]</ref>
# All Bitcoin keys and signatures are currently generated using ECDSA.<ref name="ref_ade6a467" />
# ECDSA signatures are used to sign all Bitcoin transactions thanks to these strong security features.<ref name="ref_ade6a467" />
# Critically, point division is incalculable, meaning a public key cannot currently be used to derive a private key, giving the ECDSA scheme its security.<ref name="ref_ade6a467" />
# This document describes how to specify Elliptic Curve Digital Signature Algorithm (DSA) keys and signatures in DNS Security (DNSSEC).<ref name="ref_24833376">[https://www.rfc-editor.org/rfc/rfc6605.html RFC 6605: Elliptic Curve Digital Signature Algorithm (DSA) for DNSSEC]</ref>
# This document defines the DNSKEY and RRSIG resource records (RRs) of two new signing algorithms: ECDSA (Elliptic Curve DSA) with curve P-256 and SHA-256, and ECDSA with curve P-384 and SHA-384.<ref name="ref_24833376" />
# Current estimates are that ECDSA with curve P-256 has an approximate equivalent strength to RSA with 3072-bit keys.<ref name="ref_24833376" />
# Using ECDSA with curve P-256 in DNSSEC has some advantages and disadvantages relative to using RSA with SHA-256 and with 3072-bit keys.<ref name="ref_24833376" />
# One modern ap- plication of the ECDSA is found in the Bitcoin protocol, which has seen a surge in popularity as an open source, digital currency.<ref name="ref_a31ee04e">[http://koclab.cs.ucsb.edu/teaching/ecc/project/2015Projects/Malvik+Witzoee.pdf Elliptic curve digital signature algorithm and its]</ref>
# In this paper we will present the ECDSA, covering signature generation and verication.<ref name="ref_a31ee04e" />
# We will then discuss the consequences the choice of elliptic curves has on the performance and security of the ECDSA.<ref name="ref_a31ee04e" />
# The implications this choice has on ECDSA will then be discussed.<ref name="ref_a31ee04e" />
# The task is to write a toy version of the ECDSA, quasi the equal of a real-world implementation, but utilizing parameters that fit into standard arithmetic types.<ref name="ref_dff662a0">[https://rosettacode.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm Elliptic Curve Digital Signature Algorithm]</ref>
# It provides step by step examples to generate and verify ECDSA for differing key sizes.<ref name="ref_55c19d07">[https://www.ijser.org/paper/Make-a-Secure-Connection-Using-Elliptic-Curve-Digital-Signature.html Make a Secure Connection Using Elliptic Curve Digital Signature]</ref>
# The Elliptic Curve Digital Signature Algorithm (ECDSA) is a Digital Signature Algorithm (DSA) which uses keys derived from elliptic curve cryptography (ECC).<ref name="ref_ff7f5974">[https://www.hypr.com/security-encyclopedia/elliptic-curve-digital-signature-algorithm What is the Elliptic Curve Digital Signature Algorithm (ECDSA)?]</ref>
# A main feature of ECDSA versus another popular algorithm, RSA, is that ECDSA provides a higher degree of security with shorter key lengths.<ref name="ref_ff7f5974" />
# How does ECDSA work in Bitcoin ECDSA (‘Elliptical Curve Digital Signature Algorithm’) is the cryptography behind private and public keys used in Bitcoin.<ref name="ref_67739b8a">[https://medium.com/@blairlmarshall/how-does-ecdsa-work-in-bitcoin-7819d201a3ec How does ECDSA work in Bitcoin]</ref>
# bits2octets is not used in standard DSA or ECDSA.<ref name="ref_d135f032">[https://datatracker.ietf.org/doc/html/rfc6979 Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)]</ref>
# The obtained value of k is used in DSA or ECDSA.<ref name="ref_d135f032" />
# This offers a property that ECDSA lacks: Exclusive Ownership.<ref name="ref_fba3795d">[https://soatok.blog/2022/05/19/guidance-for-choosing-an-elliptic-curve-signature-algorithm-in-2022/ Guidance for Choosing an Elliptic Curve Signature Algorithm in 2022]</ref>
# NIST P-256 is the go-to curve to use with ECDSA in the modern era.<ref name="ref_fba3795d" />
# If you’re running old software, you may still be vulnerable to timing attacks that can recover your ECDSA secret key.<ref name="ref_fba3795d" />
# ECDSA requires a secure randomness source to sign data.<ref name="ref_fba3795d" />
# This paper describes the ANSI X9.62 ECDSA, and discusses related security, implementation, and interoperability issues.<ref name="ref_66f98fbb">[https://link.springer.com/article/10.1007/s102070100002 The Elliptic Curve Digital Signature Algorithm (ECDSA)]</ref>
# It’s mathematically simple to compute a key in one direction with ECDSA, but it’s very difficult to reverse the process.<ref name="ref_1768896e">[https://www.okta.com/identity-101/ecdsa/ Elliptic Curve Digital Signature Algorithm (ECDSA) Defined]</ref>
# Breaking the ECDSA curve means solving something called the elliptic curve discrete logarithm problem, and that’s notoriously hard to do.<ref name="ref_1768896e" />
# ANSI accepted ECDSA as a standard in 1999, and IEEE and NIST accepted it as a standard in 2000.<ref name="ref_1768896e" />
# It’s mathematically challenging to crack an ECDSA code, although hackers will certainly try to do so.<ref name="ref_1768896e" />
# As with elliptic curve cryptography in general, the bit size of the public key believed to be needed for ECDSA is about twice the size of the security level, in bits.<ref name="ref_9a7eb019">[https://www.vocal.com/cryptography/ecdsa-elliptic-curve-digital-signature-algorithm/ Elliptic Curve Digital Signature Algorithm]</ref>
# For an example showing the verification procedure of ECDSA, see Test Example.<ref name="ref_3ec7456b">[https://infocenter.nordicsemi.com/topic/sdk_nrf5_v17.0.2/lib_crypto_ecdsa.html Elliptic Curve Digital Signature Algorithm]</ref>
# In section 2, we summarize existing elliptic curve digital signature algorithm (ECDSA).<ref name="ref_2089358e">[https://arxiv.org/pdf/1808.02988 A Secure Multiple Elliptic Curves Digital Signature Algorithm for Blockchain]</ref>
# The signer can obviously operate the ECDSA times (t-ECDSA), and get the signa- ture (1, 1, 2, 2, , , , ) in elliptic curves, but this will make the length of the sig- nature long.<ref name="ref_2089358e" />
# So this ECDSA is like mentioned once again nothing more than numbers (very important ones though!).<ref name="ref_b47123cd">[https://uploads-ssl.webflow.com/5d25da7a03e410dc1f3b7f36/5e66557f4fda928d5d163b71_elliptic%20curve.pdf 2.2.1 elliptic curve digital signature algorithm (ecdsa)! 1/2]</ref>
# Just as the hash is used with PoW, the hash in the ECDSA is used to once again change a huuuuuuuuuuuge number into a readable output (which is still alphanumeric).<ref name="ref_b47123cd" />
# But lets get back to the basics of ECDSA.<ref name="ref_b47123cd" />
# The private key encrypted via ECDSA leads to the public key.<ref name="ref_b47123cd" />
# In this paper, we analyse the Junru's ECDSA and improve his scheme by using two random numbers for signature generation.<ref name="ref_6630c46b">[https://www.inderscienceonline.com/doi/abs/10.1504/IJITST.2016.080406 An improvement of a elliptic curve digital signature algorithm]</ref>
# Therefore, the improved scheme can enhance the security of the Junru's ECDSA.<ref name="ref_6630c46b" />
# So please read on to find the beauty of the Elliptic Curve Digital Signature Algorithm beast.<ref name="ref_3180bd72">[https://trustica.cz/en/2018/06/07/elliptic-curve-digital-signature-algorithm/ Elliptic Curve Digital Signature Algorithm]</ref>
# The ECDSA provides advantages of elliptic curve cryptography to the function of the digital signature algorithm to authenticate and protect transmissions between involved parties.<ref name="ref_1c7aa13b">[https://libres.uncg.edu/ir/ecsu/f/Thomas_Johnson_Thesis-Final.pdf Elliptic curve digital signature algorithm]</ref>
# Implementing ECDSA 47 3.1 An Example of Implementing ECDSA . . . . . . . . . . . . .<ref name="ref_1c7aa13b" />
# In this blog, I would like to introduce some background concept on the ECDSA, ECDH and AES128 first.<ref name="ref_d4bb3f7d">[https://jimmywongiot.com/2019/06/10/how-to-protect-the-ble-connection-with-encryption-at-application-layer-instead-of-link-layer/ Background Information on the ECDSA / ECDH / AES128]</ref>
# Section 2 present a modular reduction used for accelerating one of those protocols RSA or ECDSA.<ref name="ref_3d77ede9">[https://arxiv.org/pdf/1508.00184 International Journal of Embedded systems and Applications(IJESA) Vol.5, No.2, June 2015 COMPARISON AND EVALUATION OF DIGITAL]</ref>
# Section 3 describes the simulation process used to clarify and illustrate the differences between RSA and ECDSA.<ref name="ref_3d77ede9" />
# ECDSA schemes provide the same functionality as RSA schemes including sign and/or verify signed packets.<ref name="ref_3d77ede9" />
# The claim is that a 192 bit ECDSA key is similar to a 1024 bit RSA key in terms of the security that it offers.<ref name="ref_3d77ede9" />
# The Elliptic Curve Digital Signature Algo- rithm (ECDSA) is the most commonly used cryptographic scheme in permissioned blockchains.<ref name="ref_b5bba0e2">[https://arxiv.org/pdf/2112.02229 Efficient FPGA-based ECDSA Verification Engine for Permissioned Blockchains]</ref>
# Based on these optimized modular and point arithmetic modules, we propose an ECDSA verification engine that can be used by any application for fast verification of ECDSA signatures.<ref name="ref_b5bba0e2" />
# By default, Fabric uses 256-bit ECDSA scheme for signature generation and verification.<ref name="ref_b5bba0e2" />
# All the compute-intensive operations of validation were of- floaded to the FPGA accelerator, including verification of ECDSA signatures.<ref name="ref_b5bba0e2" />
# tocol compatible with ECDSA in which one of the users plays the role of recovery party: a user involved only once in a preliminary set-up prior even to the key-generation step of the protocol.<ref name="ref_c73d17f9">[https://arxiv.org/pdf/2009.01631 Springer Nature 2021 LATEX template A Provably-Unforgeable Threshold EdDSA]</ref>
# For ex- ample, ECDSA provides integrity, authentication, and non-repudiation.<ref name="ref_2e9e92f8">[https://arxiv.org/pdf/1902.10313 Eﬃcient and Secure ECDSA Algorithm and its Applications: A Survey]</ref>
# On one hand, several approaches have been developed to improve the eciency of the ECDSA algo- rithm to reduce the cost of computation, energy, memory, and consumption of processor capabilities.<ref name="ref_2e9e92f8" />
# The opera- tion that consumes more time in ECC/ECDSA is the point multiplication (PM) or scalar multiplication (SM).<ref name="ref_2e9e92f8" />
# Many researchers have made improvements to the PM to increase the per- formance of the ECC/ECDSA as we will see in Section 4.<ref name="ref_2e9e92f8" />
# We show how this information allows an attacker to apply lattice techniques to recover 256-bit private keys for ECDSA and ECSchnorr sig- natures.<ref name="ref_65cc1578">[https://arxiv.org/pdf/1911.05673 TPM-FAIL: TPM meets Timing and Lattice Attacks Daniel Moghimi1, Berk Sunar1, Thomas Eisenbarth1, 2, and Nadia Heninger3]</ref>
# Similarly, we extract the private ECDSA key from a hardware TPM manu- factured by STMicroelectronics, which is certied at Common Criteria (CC) EAL 4+, after fewer than 40,000 observations.<ref name="ref_65cc1578" />
# The discovery of previously unknown vulnerabilities in TPM implementations of ECDSA and ECSchnorr sig- nature schemes, and the pairing-friendly BN-256 curve used by the ECDAA signature scheme.<ref name="ref_65cc1578" />

===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q915079 Q915079]
===Spacy 패턴 목록===
* [{'LOWER': 'elliptic'}, {'LOWER': 'curve'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LOWER': 'algorithm'}]
* [{'LOWER': 'ecdsa'}]
* [{'LOWER': 'elliptic'}, {'LOWER': 'curve'}, {'LOWER': 'dsa'}]

플레이페어 암호

2022-09-18T06:00:38Z

Pythagoras0: /* 메타데이터 */ 새 문단

== 노트 ==

===말뭉치===
# The first recorded description of the Playfair cipher was in a document signed by Wheatstone on 26 March 1854.<ref name="ref_b2ed2f31">[https://en.wikipedia.org/wiki/Playfair_cipher Playfair cipher]</ref>
# The Playfair cipher uses a 5 by 5 table containing a key word or phrase.<ref name="ref_b2ed2f31" />
# A different approach to tackling a Playfair cipher is the shotgun hill climbing method.<ref name="ref_b2ed2f31" />
# The Playfair cipher is a written code or symmetric encryption technique that was the first substitution cipher used for the encryption of data.<ref name="ref_2d06685f">[https://www.techopedia.com/definition/29770/playfair-cipher#:~:text=Techopedia%20Explains%20Playfair%20Cipher,-Created%20by%20Sir&text=The%20Playfair%20cipher's%20essential%20method,message%20into%20two%2Dletter%20bits. What is the Playfair Cipher?]</ref>
# The Playfair cipher's essential method involves creating key tables that arrange the letters of the alphabet into a square grid.<ref name="ref_2d06685f" />
# As an example of the Playfair cipher, begin with the following message text: HELLO WORLD.<ref name="ref_2d06685f" />
# The Playfair cipher was invented in 1854 by Charles Wheatstone, but named after lord Playfair who heavily promoted the use of the cipher.<ref name="ref_951dd169">[https://www.boxentriq.com/code-breaking/playfair-cipher Playfair Cipher (online tool)]</ref>
# Instead of encrypting single letters, the Playfair cipher encrypts pairs of letter (digrams or bigrams).<ref name="ref_951dd169" />
# next → ← prev Playfair Cipher Program in Java Playfair cipher is proposed by Charles Whetstone in 1889.<ref name="ref_df7d395b">[https://www.javatpoint.com/playfair-cipher-program-in-java#:~:text=In%20comparison%20to%20monoalphabetic%20cipher,262%20(676)%20digraphs. Playfair Cipher Program in Java]</ref>
# In this section, we will discuss Playfair cipher along with its implementation in a Java program.<ref name="ref_df7d395b" />
# Playfair Cipher Playfair cipher is an encryption algorithm to encrypt or encode a message.<ref name="ref_df7d395b" />
# Since Playfair cipher encrypts the message digraph by digraph.<ref name="ref_df7d395b" />
# In playfair cipher, initially a key table is created.<ref name="ref_2dabac93">[https://www.tutorialspoint.com/cryptography/traditional_ciphers.htm Traditional Ciphers]</ref>
# The Play-Fair algorithm uses a 5x5 matrix of letters called Playfair square or Wheatston-square, constructed using keyword.<ref name="ref_f0e8e277">[https://www.educative.io/answers/encryption-using-playfair-cipher Encryption using Playfair Cipher]</ref>
# Below is an example of a Playfair cipher, solved by Lord Peter Wimsey in Dorothy L. Sayers’s Have His Carcase (1932).<ref name="ref_2e588872">[https://www.britannica.com/topic/Playfair-cipher Playfair cipher | data encryption]</ref>
# The loss of a significant part of the plaintext frequency distribution, however, makes a Playfair cipher harder to cryptanalyze than a monoalphabetic cipher.<ref name="ref_2e588872" />
# The coastwatchers regularly used the Playfair system.<ref name="ref_b38730e1">[http://www.practicalcryptography.com/ciphers/playfair-cipher/ Practical Cryptography]</ref>
# Cryptanalysis of the playfair cipher is much more difficult than normal simple substitution ciphers, because digraphs (pairs of letters) are being substituted instead of monographs (single letters).<ref name="ref_b38730e1" />
# 12 Example: Playfair Cipher Program le for this chapter: playfair This project investigates a cipher that is somewhat more complicated than the simple substitution cipher of Chapter 11.<ref name="ref_1b93c6c2">[https://people.eecs.berkeley.edu/~bh/pdf/v1ch12.pdf 12]</ref>
# Advantages to Digraphic Ciphers Kahn, in his discussion of the Playfair Cipher, points out several advantages of the digraphic (enciphering letter pairs) ciphers: 1.<ref name="ref_5a23f982">[https://www.nku.edu/~christensen/1402%20Playfair%20cipher.pdf Spring 2015]</ref>
# The Playfair cipher encrypts pairs of letters, called bigrams.<ref name="ref_2400a7c4">[https://privacycanada.net/playfair-cipher/ Detailed Guide 2022]</ref>
# In this study, the authors proposed an enhanced key security of Playfair cipher algorithm using Playfair cipher 16x16 matrix, XOR, two's complement, and bit swapping.<ref name="ref_e7ef0553">[https://dl.acm.org/doi/abs/10.1145/3316615.3316689 An Enhanced Key Security of Playfair Cipher Algorithm]</ref>
# Also, the proposed method is faster than the existing key security of Playfair cipher algorithm.<ref name="ref_e7ef0553" />
# The Playfair Cipher is a simple pencil-and-paper cipher which doesn't suffer from some of the common weaknesses with simple substitution ciphers.<ref name="ref_83ca1967">[https://chrome.google.com/webstore/detail/playfair-cipher/igmiiphijligfkpgioddalhdipioegca?hl=en Playfair Cipher]</ref>
# In the Playfair cipher, there are many letters that will stand for an E, making it much harder to crack!<ref name="ref_c79cac58">[https://www.popularmechanics.com/science/math/a40706789/playfair-cipher-puzzle/ Can You Solve This Playfair Cipher Puzzle?]</ref>
# Throughout my upbringing, I often heard of detectives and spies using the Playfair cipher as a way to encode/decode messages’ meanings.<ref name="ref_89dcf64c">[https://hankeringforhistory.com/history-of-the-playfair-cipher/ History of the Playfair Cipher]</ref>
# The Playfair cipher is notable because it is one of the first ciphers that paired letters (also known as a digraph) instead of using a single letter cipher.<ref name="ref_89dcf64c" />
# The Playfair cipher was originally thought by the British Foreign Office as too complex; they feared that using this cipher would take too much time and would be ineffective in the field.<ref name="ref_89dcf64c" />
# The Playfair cipher was predominantly used by British forces during the Second Boer War (1899-1902) and World War I (1914-1918).<ref name="ref_89dcf64c" />
# The first block cipher we will discuss is the Playfair Cipher which dates back to the 1800's.<ref name="ref_d06879c6">[http://facweb1.redlands.edu/fac/Tamara_Veenstra/cryptobook/playfair.html Playfair Cipher]</ref>
# Usually a Playfair cipher uses a keyword to arrange the alphabet in a five-by-five grid.<ref name="ref_d06879c6" />
# To encipher a message using the Playfair square we first break our message into blocks of two letters.<ref name="ref_d06879c6" />
# For the second block in the plaintext we find E and T in the Playfair square.<ref name="ref_d06879c6" />
# This is a C Program to implement Playfair cipher.<ref name="ref_8fa496d0">[https://www.sanfoundry.com/c-program-encode-message-using-playfair-cipher/ Playfair Cipher Encryption Program in C]</ref>
# Here is the source code of the C Program to Encode a Message Using Playfair Cipher.<ref name="ref_8fa496d0" />
# The encryption key for a Playfair cipher is a word, i.e., a nite sequence of characters taken from the set of plaintext characters.<ref name="ref_0fad4328">[https://math.asu.edu/sites/default/files/playfair.pdf Playfair ciphers]</ref>
# and we want to encrypt the message rocky mountain meadow using a Playfair cipher with this keyword.<ref name="ref_0fad4328" />
# Playfair cipher is the first and best-known digraph substitution cipher, which uses the technique of symmetry encryption.<ref name="ref_8e3f376f">[https://www.linkedin.com/pulse/playfair-cipher-jahnavi-redrouthu?trk=pulse-article_more-articles_related-content-card PLAYFAIR CIPHER]</ref>
# The Playfair cipher is relatively fast and doesn’t require special equipment.<ref name="ref_8e3f376f" />
# The Playfair Cipher uses digraphs to encrypt and decrypt from a 5x5 matrix constructed from a sequence key of 25 unique letters.<ref name="ref_ba0fa829">[https://github.com/damiannolan/simulated-annealing-playfair-cipher-breaker/blob/master/README.md simulated-annealing-playfair-cipher-breaker/README.md at master · damiannolan/simulated-annealing-playfair-cipher-breaker]</ref>
# Playfair cipher have made significant progress on improving its security, but little attention was given to optimize or maintain the original time and space efficiency.<ref name="ref_d461cd70">[https://www.spiedigitallibrary.org/conference-proceedings-of-spie/12167/121671M/Hybrid-Playfair--a-modified-Playfair-cipher-combining-2D-and/10.1117/12.2628650.full Hybrid Playfair: a modified Playfair cipher combining 2D and 3D Playfair]</ref>
# To close the gap, this paper proposes Hybrid Playfair, a modified Playfair cipher using a three-dimensional key and 2-character message pairs for encryption and decryption.<ref name="ref_d461cd70" />
# This paper exhibits a variation in the implementation of a classical cipher technique, the Playfair cipher.<ref name="ref_466b8022">[https://people.cis.fiu.edu/gubbisadashiva/wp-content/uploads/sites/9/2019/11/A-Variation-in-the-Working-of-Playfair-Cipher.pdf A variation in the working of playfair cipher]</ref>
# Sections II and III discuss the classical Playfair cipher technique and other related works in this domain.<ref name="ref_466b8022" />
# Matrix formed with key phrase HELLO WORLD as per Playfair cipher letters from the alphabet (Fig. 2).<ref name="ref_466b8022" />
# Plaintext enciphered to ciphertext as per Playfair cipher with assumed inputs 3.<ref name="ref_466b8022" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q829915 Q829915]
===Spacy 패턴 목록===
* [{'LOWER': 'playfair'}, {'LEMMA': 'square'}]
* [{'LOWER': 'playfair'}, {'LOWER': 'cipher'}]
* [{'LOWER': 'playfair'}, {'LEMMA': 'system'}]

플레이페어 암호

2022-09-18T06:00:35Z

Pythagoras0: /* 노트 */ 새 문단

정보 엔트로피

2022-09-16T11:04:06Z

Pythagoras0:

== 노트 ==

===말뭉치===
# He captured it in a formula that calculates the minimum number of bits — a threshold later called the Shannon entropy — required to communicate a message.<ref name="ref_8c92b5e4">[https://www.quantamagazine.org/how-claude-shannons-concept-of-entropy-quantifies-information-20220906/ Quanta Magazine]</ref>
# The term “entropy” is borrowed from physics, where entropy is a measure of disorder.<ref name="ref_8c92b5e4" />
# A cloud has higher entropy than an ice cube, since a cloud allows for many more ways to arrange water molecules than a cube’s crystalline structure does.<ref name="ref_8c92b5e4" />
# In an analogous way, a random message has a high Shannon entropy — there are so many possibilities for how its information can be arranged — whereas one that obeys a strict pattern has low entropy.<ref name="ref_8c92b5e4" />
# Shannon entropy allows to estimate the average minimum number of bits needed to encode a string of symbols based on the alphabet size and the frequency of the symbols.<ref name="ref_16c8e33c">[https://www.shannonentropy.netmark.pl/ Shannon entropy calculator — Real example how to calculate and interpret information entropy]</ref>
# The concepts of entropy, as used in information theory and in various scientific disciplines, are now countless (Shannon, 1948).<ref name="ref_26d5c7c6">[https://www.sciencedirect.com/topics/engineering/shannon-entropy Shannon Entropy - an overview]</ref>
# Based on this fact, Shannon proposed measuring the average of this flux of information called entropy.<ref name="ref_26d5c7c6" />
# (7.79) can be a log 2 or an ln, with the entropy units in bits (binary units) or nats (natural units), respectively.<ref name="ref_26d5c7c6" />
# In general, the probability distribution for a given stochastic process is not known, and, in most situations, only small datasets from which to infer the entropy are available.<ref name="ref_26d5c7c6" />
# Shannon entropy, also known as information entropy or the Shannon entropy index, is a measure of the degree of randomness in a set of data.<ref name="ref_3feca1bb">[https://www.omnicalculator.com/statistics/shannon-entropy Shannon Entropy Calculator]</ref>
# The more characters there are, or the more proportional are the frequencies of occurrence, the harder it will be to predict what will come next - resulting in an increased entropy.<ref name="ref_3feca1bb" />
# Apart from information theory, Shannon entropy is used in many fields.<ref name="ref_3feca1bb" />
# In the Shannon entropy equation, p i is the probability of a given symbol.<ref name="ref_8a5af6e0">[http://bearcave.com/misl/misl_tech/wavelets/compression/shannon.html Shannon Entropy]</ref>
# The number of bits per character can be calculated from this frequency set using the Shannon entropy equation.<ref name="ref_8a5af6e0" />
# Shannon entropy provides a lower bound for the compression that can be achieved by the data representation (coding) compression step.<ref name="ref_8a5af6e0" />
# Shannon entropy makes no statement about the compression efficiency that can be achieved by predictive compression.<ref name="ref_8a5af6e0" />
# For anyone who wants to be fluent in Machine Learning, understanding Shannon’s entropy is crucial.<ref name="ref_90b7575b">[https://towardsdatascience.com/the-intuition-behind-shannons-entropy-e74820fe9800 The intuition behind Shannon’s Entropy]</ref>
# A little more formally, the entropy of a variable is the “amount of information” contained in the variable.<ref name="ref_90b7575b" />
# The analogy results when the values of the random variable designate energies of microstates, so Gibbs formula for the entropy is formally identical to Shannon's formula.<ref name="ref_6dcdd18d">[https://en.wikipedia.org/wiki/Entropy_(information_theory) Entropy (information theory)]</ref>
# Entropy has relevance to other areas of mathematics such as combinatorics and machine learning.<ref name="ref_6dcdd18d" />
# The definition can be derived from a set of axioms establishing that entropy should be a measure of how "surprising" the average outcome of a variable is.<ref name="ref_6dcdd18d" />
# In this case a coin flip has an entropy of one bit.<ref name="ref_6dcdd18d" />
# Some details: We treat Shannon (discrete) and differential (continuous) entropy separately.<ref name="ref_4977c7cb">[https://stats.stackexchange.com/questions/305148/what-are-the-units-of-entropy-of-a-normal-distribution What are the units of entropy of a normal distribution?]</ref>
# For example, you wouldn’t calculate nutrition in the same way you calculate entropy in thermodynamics.<ref name="ref_fdaef900">[https://www.statisticshowto.com/shannon-entropy/ Shannon Entropy]</ref>
# Shannon entropy for imprecise and under-defined or over-defined information.<ref name="ref_fdaef900" />
# In the general case Arithmetic coding results in a near optimal encoding of messages that is very close to the number obtained from the Shannon entropy equation.<ref name="ref_cae75a49">[https://heliosphan.org/shannon-entropy.html Shannon Entropy]</ref>
# Statistical entropy was introduced by Shannon as a basic concept in information theory measuring the average missing information in a random source.<ref name="ref_401e9b4d">[https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/shannon-entropy-a-rigorous-notion-at-the-crossroads-between-probability-information-theory-dynamical-systems-and-statistical-physics/4A4B7B069BCF64CC595635D865317C83 Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics]</ref>
# Extended into an entropy rate, it gives bounds in coding and compression theorems.<ref name="ref_401e9b4d" />
# The relevance of entropy beyond the realm of physics, in particular for living systems and ecosystems, is yet to be demonstrated.<ref name="ref_401e9b4d" />
# My goal is to really understand the concept of entropy, and I always try to explain complicated concepts using fun games, so that’s what I do in this post.<ref name="ref_983e8028">[https://medium.com/udacity/shannon-entropy-information-gain-and-picking-balls-from-buckets-5810d35d54b4 Shannon Entropy, Information Gain, and Picking Balls from Buckets]</ref>
# In colloquial terms, if the particles inside a system have many possible positions to move around, then the system has high entropy, and if they have to stay rigid, then the system has low entropy.<ref name="ref_983e8028" />
# The molecules in ice have to stay in a lattice, as it is a rigid system, so ice has low entropy.<ref name="ref_983e8028" />
# The molecules in water have more positions to move around, so water in liquid state has medium entropy.<ref name="ref_983e8028" />
# In physics, the word entropy has important physical implications as the amount of "disorder" of a system.<ref name="ref_f0d9e29e">[https://mathworld.wolfram.com/Entropy.html Entropy -- from Wolfram MathWorld]</ref>
# But what properties single out Shannon entropy as special?<ref name="ref_e22fb276">[https://math.ucr.edu/home/baez/entropy/ Shannon Entropy from Category Theory]</ref>
# Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the "information loss", or change in entropy, associated with a measure-preserving function.<ref name="ref_e22fb276" />
# Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous.<ref name="ref_e22fb276" />
# The entropy H is correspondingly reduced in the posterior relative to the prior distribution.<ref name="ref_80dfd193">[https://journals.ametsoc.org/view/journals/atot/35/5/jtech-d-17-0056.1.xml On Some Shortcomings of Shannon Entropy as a Measure of Information Content in Indirect Measurements of Continuous Variables]</ref>
# In this case a measurement yields a posterior PDF with no change in the expected value but a significant increase in the spread and in the Shannon entropy.<ref name="ref_80dfd193" />
# 2.6 Entropy Rate Revisited . . . . . . . . . . . . . . . . . . . . . . .<ref name="ref_dbc05de0">[https://ee.stanford.edu/~gray/it.pdf Entropy and]</ref>
# 105 5.4 Limiting Entropy Densities 5.5 Information for General Alphabets . . . . . . . . . . . . . . . . .<ref name="ref_dbc05de0" />
# Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function.<ref name="ref_4031275c">[https://golem.ph.utexas.edu/category/2022/05/shannon_entropy_from_category.html The n-Category Café]</ref>
# Thus, whenever you pick something out, you have absolutely no doubt (entropy is zero for Container 3) that it will be a circle.<ref name="ref_c753903f">[https://stats.stackexchange.com/questions/87182/what-is-the-role-of-the-logarithm-in-shannons-entropy What is the role of the logarithm in Shannon's entropy?]</ref>
# As you can see, entropy (doubt, surprise, uncertainty) for Container 1 is less (56%), but more for Container 2 (99%).<ref name="ref_c753903f" />
# For Container 3, there is no (0%) entropy - you have 100% chance to pick a circle.<ref name="ref_c753903f" />
# The electroencephalographic Shannon entropy increased continuously over the observed concentration range of desflurane.<ref name="ref_b3894916">[https://pubs.asahq.org/anesthesiology/article/95/1/30/39031/Shannon-Entropy-Applied-to-the-Measurement-of-the Shannon Entropy Applied to the Measurement of the Electroencephalographic Effects of Desflurane]</ref>
# Recently, approximate entropy, a measure of the “amount of order” of the electroencephalographic signal has been shown to correlate well with the concentration of desflurane at the effect site.<ref name="ref_b3894916" />
# The Shannon entropy 2 is a standard measure for the order state of sequences and has been applied previously to DNA sequences.<ref name="ref_b3894916" />
# In this investigation, we applied the Shannon entropy to electroencephalographic data from anesthetized patients and correlated the concentration of anesthetic agent and entropy value.<ref name="ref_b3894916" />
# KEYWORDS: Monte Carlo, keff, convergence, Shannon entropy, MCNP 1.<ref name="ref_b481b7f2">[https://www.oecd-nea.org/science/wpncs/sccsa/documents/brown-physor-2006.pdf Physor-2006, ans topical meeting on reactor physics]</ref>
# Line-plots of Shannon entropy vs. batch are easier to interpret and assess than are 2D or 3D plots of the source distribution vs. batch.<ref name="ref_b481b7f2" />
# When running criticality calculations with MCNP5, it is essential that users examine the convergence of both keff and the fission source distribution (using Shannon entropy).<ref name="ref_b481b7f2" />
# A number of theoretical approaches based on, e.g., conditional Shannon entropy and Fisher information have been developed, along with some experimental validations.<ref name="ref_3f6b5e67">[https://spj.sciencemag.org/journals/research/2021/9780760/ Quantifying Information via Shannon Entropy in Spatially Structured Optical Beams]</ref>
# Shannon’s concept of entropy can now be taken up.<ref name="ref_8f62e39e">[https://www.britannica.com/science/information-theory/Entropy information theory - Entropy]</ref>
# Thus, the bound computed using entropy cannot be attained with simple encodings.<ref name="ref_8f62e39e" />
# This is better than the 2.0 obtained earlier, although still not equal to the entropy.<ref name="ref_8f62e39e" />
# Because the entropy is not exactly equal to any fraction, no code exists whose average length is exactly equal to the entropy.<ref name="ref_8f62e39e" />
# Pintacuda, N.: Shannon entropy: A more general derivation.<ref name="ref_001ffaad">[https://link.springer.com/article/10.1007/BF00532728 On Shannon's entropy, directed divergence and inaccuracy]</ref>
# In this Letter, we report a comparative analysis of the Shannon entropy and qTIR using model series and real-world heartbeats.<ref name="ref_84258f80">[https://aip.scitation.org/doi/10.1063/1.5133419 Shannon entropy and quantitative time irreversibility for different and even contradictory aspects of complex systems]</ref>
# We find that the permutation-based Shannon entropy (PEn) and time irreversibility (PYs) detect nonlinearities in the model series differently according to the surrogate theory.<ref name="ref_84258f80" />
# In classical physics, the entropy of a physical system is proportional to the quantity of energy no longer available to do physical work.<ref name="ref_0dddc0c9">[http://www.scholarpedia.org/article/Entropy Scholarpedia]</ref>
# In quantum mechanics, von Neumann entropy extends the notion of entropy to quantum systems by means of the density matrix.<ref name="ref_0dddc0c9" />
# In the theory of dynamical systems, entropy quantifies the exponential complexity of a dynamical system or the average flow of information per unit of time.<ref name="ref_0dddc0c9" />
# The term entropy is now used in many other sciences (such as sociology), sometimes distant from physics or mathematics, where it no longer maintains its rigorous quantitative character.<ref name="ref_0dddc0c9" />
# Shannon (the man, not the entropy) was one of those annoying people that excels at everything he touches.<ref name="ref_05cc708c">[https://robotwealth.com/shannon-entropy/ Shannon Entropy: A Genius Gambler’s Guide to Market Randomness]</ref>
# Like the number of questions we need to arrive at the correct suit, Shannon Entropy decreases when order is imposed on a system and increases when the system is more random.<ref name="ref_05cc708c" />
# Once the (p)’s are known, Zorro simply implements the Shannon Entropy equation and returns the calculated value for (H), in bits.<ref name="ref_05cc708c" />
# A fair coin has an entropy of one bit.<ref name="ref_86e2070a">[https://www.chemeurope.com/en/encyclopedia/Information_entropy.html Information]</ref>
# However, if the coin is not fair, then the uncertainty is lower (if asked to bet on the next outcome, we would bet preferentially on the most frequent result), and thus the Shannon entropy is lower.<ref name="ref_86e2070a" />
# A long string of repeating characters has an entropy of 0, since every character is predictable.<ref name="ref_86e2070a" />
# Additivity The amount of entropy should be the same independently of how the process is regarded as being divided into parts.<ref name="ref_86e2070a" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q204570 Q204570]
===Spacy 패턴 목록===
* [{'LOWER': 'shanon'}, {'LOWER': 'entropy'}]
* [{'LOWER': 'information'}, {'LOWER': 'entropy'}]
* [{'LOWER': 'entropy'}]
* [{'LOWER': 'shannon'}, {'LOWER': 'entropy'}]
* [{'LOWER': 'average'}, {'LOWER': 'information'}, {'LEMMA': 'content'}]
* [{'LOWER': 'negentropy'}]

페르마 소수

2022-09-16T10:47:18Z

Pythagoras0: /* 메타데이터 */

==개요==

* 페르마소수란 <math>F_n= 2^{2^n}+1</math> 형태의 소수
** 3,5,17,257, 65537 다섯 가지만 알려져 있음.
* 페르마는 <math>F_n= 2^{2^n}+1</math> 가 모두 소수일 것이라 추측하였으나, 후에 [[오일러(1707-1783)|오일러]]는 반례를 발견:<math>F_5=641 \times 6700417</math>

==정다각형의 작도==

* 정n각형이 자와 컴파스로 작도가능 <math>\iff</math> <math>n=2^k p_1 p_2 \cdots p_r</math> (k ,r은 0이상의 정수, <math>p_1, p_2, \cdots, p_r</math> 은 서로 다른 페르마소수)
* [[정다각형의 작도]]와 [[가우스와 정17각형의 작도]] 항목을 참조

==역사==

* [[수학사 연표]]

==관련된 항목들==

* [[정다각형의 작도]]
* [[메르센 소수]]

[[분류:소수]]

== 노트 ==

===말뭉치===
# In fact, it is known that numbers of this form are not prime for values of n from 5 through 30, placing doubt on the existence of any Fermat primes for values of n > 4.<ref name="ref_edbed968">[https://www.britannica.com/science/Fermat-prime Fermat prime | mathematics]</ref>
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_21863434">[https://mathworld.wolfram.com/FermatPrime.html Fermat Prime -- from Wolfram MathWorld]</ref>
# Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime.<ref name="ref_77005ae2">[https://artofproblemsolving.com/wiki/index.php/Fermat_prime Art of Problem Solving]</ref>
# To find the Fermat number F n for an integer n , you first find m = 2 n , and then calculate 2 m + 1.<ref name="ref_7c2c701e">[https://www.techtarget.com/whatis/definition/Fermat-prime Definition from WhatIs.com]</ref>
# Surprisingly, Fermat primes arise in deciding whether a regular n-gon (a convex polygon with n equal sides) can be constructed with a compass and a straightedge.<ref name="ref_a529a8a6">[https://sites.millersville.edu/bikenaga/number-theory/fermat-numbers/fermat-numbers.pdf Fermat numbers]</ref>
# No fermat primes beyond (cid:8)4 have been found.<ref name="ref_a8ac433d">[http://www.math.ualberta.ca/~isaac/math324/s12/fermat_numbers.pdf Math 324 summer 2012]</ref>
# There are in(cid:12)nitely many distinct Fermat numbers, each of which is divisible by an odd prime, and since any two Fermat numbers are relatively prime, these odd primes must all be distinct.<ref name="ref_a8ac433d" />
# + 1 is a Fermat number; such primes are called Fermat primes.<ref name="ref_44f407e6">[https://en.wikipedia.org/wiki/Fermat_number Fermat number]</ref>
# From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_44f407e6" />
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 OEIS) is irrational.<ref name="ref_44f407e6" />
# Indeed, the first five Fermat numbers F 0 , ..., F 4 are easily shown to be prime.<ref name="ref_44f407e6" />
# The only known Fermat primes are the first five Fermat numbers: F 0 =3, F 1 =5, F 2 =17, F 3 =257, and F 4 =65537.<ref name="ref_fbf82f86">[https://primes.utm.edu/glossary/xpage/FermatNumber.html The Prime Glossary: Fermat number]</ref>
# It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number F n with n greater than 2 has the form k.2n+1+1 (exponent improved to n+2 by Lucas).<ref name="ref_fbf82f86" />
# Now we know that all of the Fermat numbers are composite for the other n less than 31.<ref name="ref_fbf82f86" />
# In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1 ) is a Fermat number, and such primes are called Fermat primes.<ref name="ref_e81de8fb">[http://www.scientificlib.com/en/Mathematics/LX/FermatNumber.html Fermat number]</ref>
# From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_e81de8fb" />
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational.<ref name="ref_e81de8fb" />
# Indeed, the first five Fermat numbers F 0 ,...,F 4 are easily shown to be prime.<ref name="ref_e81de8fb" />
# Indeed, even today, no other Fermat numbers are known to be prime!<ref name="ref_c596120d">[https://johncarlosbaez.wordpress.com/2019/02/05/fermat-primes-and-pascals-triangle/ Fermat Primes and Pascal’s Triangle]</ref>
# Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2.<ref name="ref_c596120d" />
# Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number.<ref name="ref_c596120d" />
# Wantzel went further and showed that if n is odd, we can construct a regular n-gon using straight-edge and compass if and only if n is a product of distinct Fermat primes.<ref name="ref_c596120d" />
# There are two definitions of the Fermat number.<ref name="ref_25205313">[https://mathworld.wolfram.com/FermatNumber.html Fermat Number -- from Wolfram MathWorld]</ref>
# The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form .<ref name="ref_25205313" />
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein proposed as a problem in 1844 the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_25205313" />
# At present, however, only composite Fermat numbers are known for .<ref name="ref_25205313" />
# In other words, every prime of the form 2 n +1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_17d186c1">[https://www.rieselprime.de/ziki/Fermat_number Fermat number]</ref>
# From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor.<ref name="ref_17d186c1" />
# Indeed, the first five Fermat numbers F 0 ,..., F 4 are easily shown to be prime.<ref name="ref_17d186c1" />
# Although it is widely believed that there are only finitely many Fermat primes, it should be noted that there are some experts who disagree (John Cosgrave: "Fermat 6").<ref name="ref_17d186c1" />
# A019434 List of Fermat primes: primes of form, for someIt is conjectured that there are only 5 terms.<ref name="ref_b3f9ea5b">[https://oeis.org/wiki/Fermat_primes Fermat primes]</ref>
# Numbers of the form F n =22n+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes.<ref name="ref_bfc52e99">[https://blogs.scientificamerican.com/roots-of-unity/extrapolation-gone-wrong-the-case-of-the-fermat-primes/ Extrapolation Gone Wrong: the Case of the Fermat Primes]</ref>
# Fermat conjectured that all Fermat numbers are prime.<ref name="ref_bfc52e99" />
# In 1732, about 70 years after Fermat's death, Leonhard Euler factored the 5th Fermat number into 641×6,700,417, disproving Fermat’s conjecture.<ref name="ref_bfc52e99" />
# So far, the only known Fermat primes are the ones that were known to Fermat.<ref name="ref_bfc52e99" />
# Is there a formula or a way to know which of the Fermat numbers are prime?.<ref name="ref_dc7c427d">[https://math.stackexchange.com/questions/563390/fermat-numbers-are-they-all-prime Fermat numbers. Are they all prime?]</ref>
# Prologue What are the known Fermat primes?<ref name="ref_b2c9d49a">[https://arxiv.org/pdf/1605.01371 This is the extended version of a paper that has appeared in the Mathematical Intelligencer. The ﬁnal publication is available]</ref>
# Taking Fermat prime to mean prime of the form 2n + 1, there are six known Fermat primes, namely those for n = 0, 1, 2, 4, 8, 16.<ref name="ref_b2c9d49a" />
# We shall pronounce the last letter of Fermats name, as he did, when we include 2 among the Fermat primes, as he did.<ref name="ref_b2c9d49a" />
# The Fermat number Fn is either prime or not prime: the question of how to approximate the probability of primality for a general n is delicate.<ref name="ref_b2c9d49a" />
# Fb;n = b2n + 1 and are particularly interesting since they have many characteristics of the heavily studied standard Fermat numbers Fn = F2;n.<ref name="ref_a10cb56f">[https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf Mathematics of computation]</ref>
# The \Proth" program was created in 1997 to extend the search for large factors of Fermat numbers.<ref name="ref_a10cb56f" />
# They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 1665) who first studied numbers in this form.<ref name="ref_8f117ada">[https://wstein.org/edu/2010/414/projects/tsang.pdf Fermat]</ref>
# We will not be able to answer this question in this paper, but we will prove some basic properties of Fermat numbers and discuss their primality and divisibility.<ref name="ref_8f117ada" />
# Primes in this form are called Fermat primes.<ref name="ref_8f117ada" />
# Up-to-date there are only five known Fermat primes.<ref name="ref_8f117ada" />
# Moreover, no other Fermat number is known to be prime for n > 4 , so now it is conjectured that those are all prime Fermat numbers.<ref name="ref_6428f90c">[https://planetmath.org/fermatnumbers Fermat numbers]</ref>
# In honour of the inspired pioneers, the numbers of the form 2n-1 are now called the Mersenne numbers and the numbers of the form 2n+1 the Fermat numbers.<ref name="ref_e35d89c0">[http://yves.gallot.pagesperso-orange.fr/primes/ Generalized Fermat Primes Search]</ref>
# The search for Mersenne and Fermat primes has been greatly extended since the 17th century.<ref name="ref_e35d89c0" />
# Today, all the Mersenne primes having less than 2,000,000 digits are known and all the Fermat primes up to 2,000,000,000 digits!<ref name="ref_e35d89c0" />
# In 1994, R. Crandall and B. Fagin discovered that the Discrete Weighted Transforms could be used to double the speed of the search for Mersenne and Fermat numbers.<ref name="ref_e35d89c0" />
# Those are the only primes below 100,000 that I could show must be primitive for all but finitely many Fermat primes.<ref name="ref_9d437e74">[https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1204796915 Fermat Primes and the number 7]</ref>
# Fermat numbers are named after Pierre de Fermat.<ref name="ref_a462b554">[https://kids.kiddle.co/Fermat_number Fermat number facts for kids]</ref>
# Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_a462b554" />
# Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.<ref name="ref_a462b554" />
# Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers.<ref name="ref_e5ed5592">[https://exploringnumbertheory.wordpress.com/2016/10/25/fermat-numbers/ Exploring Number Theory]</ref>
# Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set.<ref name="ref_e5ed5592" />
# The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2.<ref name="ref_e5ed5592" />
# He demonstrated that the first 5 Fermat numbers , , , , are prime and conjectured that all Fermat numbers are prime.<ref name="ref_e5ed5592" />
# Chapter 4 Fermat and Mersenne Primes 4.1 Fermat primes Theorem 4.1.<ref name="ref_7befcda2">[https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/FermatMersenne.pdf Chapter 4]</ref>
# Fermat conjectured that the Fermat numbers are all prime.<ref name="ref_7befcda2" />
# We will come back to their solution shortly as we must first introduce the notion of a Fermat prime!<ref name="ref_62f9f1ec">[https://vrs.amsi.org.au/fermat-primes-gauss-wantzel/ Fermat Primes and the Gauss-Wantzel Theorem]</ref>
# In the 1600s, a mathematician and lawyer named Pierre de Fermat studied numbers of the form 2^n+1 (where n=2^k) which are now called Fermat numbers.<ref name="ref_62f9f1ec" />
# So, he conjectured that all Fermat numbers are prime.<ref name="ref_62f9f1ec" />
# Funnily enough, it was shown by Leonhard Euler in 1732 (another famous mathematician), that only the next Fermat number is not prime.<ref name="ref_62f9f1ec" />
# It is still open whether there exist innitely many Fermat primes or innitely many composite Fermat numbers.<ref name="ref_0b05fdb4">[https://arxiv.org/pdf/2204.08302 MINIMALITY CONDITIONS EQUIVALENT TO THE FINITUDE OF FERMAT AND MERSENNE PRIMES]</ref>
# Especially, the formula for modulo Fermat primes is given. MSC2010: 11A07.<ref name="ref_9c963bc5">[https://arxiv.org/pdf/1601.06509 The largest cycles consist by the quadratic residues and Fermat primes]</ref>
# For the Fermat primes, i.e., the prime numbers of the form p 22k ` 1, we have Lppq 1. Proof.<ref name="ref_9c963bc5" />
# We have veried it for k 0, 1, 2, 3, 4, the known Fermat primes.<ref name="ref_9c963bc5" />
# As of 2021, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537.<ref name="ref_49cdcbc7">[https://arxiv.org/pdf/1912.12088 MINIMALITY OF TOPOLOGICAL MATRIX GROUPS AND FERMAT PRIMES]</ref>
# For an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p 1, (Q, p)) is minimal; (3) SL(p 1, Q(i)) is minimal.<ref name="ref_49cdcbc7" />
# We determine all the Carmichael numbers m with a Fermat prime factor such that L = 2P 2, where k N and P is an odd prime number.<ref name="ref_f1f90697">[https://arxiv.org/pdf/1710.01321 On the ﬁniteness of Carmichael numbers with Fermat factors and L = 2αP 2.]</ref>
# We assume that m is divisible by at least one of the known Fermat prime numbers.<ref name="ref_f1f90697" />
# If m is divisible by one of the known Fermat primes, then m must be one of the following 11 Carmichael numbers.<ref name="ref_f1f90697" />
# Let 2 R R 22, there exists a generalized Fermat prime p = r2 2 be an integer.<ref name="ref_25a71c73">[https://arxiv.org/pdf/1502.02800 FAST INTEGER MULTIPLICATION USING GENERALIZED FERMAT PRIMES]</ref>
# The key concept of our algorithm is the use of a chain of generalized Fermat primes (of the form r2 +1) to handle recursive calls.<ref name="ref_25a71c73" />
# First, we encode integers to be multiplied as integers modulo generalized Fermat primes, and not as polynomials.<ref name="ref_25a71c73" />
# Section 4 studies generalized Fermat primes, and their relation to the Bateman-Horn conjecture.<ref name="ref_25a71c73" />
# 1. Introduction + 1 for n The Fermat numbers are given by Fn = 22n 0.<ref name="ref_4a2a6fd4">[https://arxiv.org/pdf/2102.00906 ON UPPER BOUNDS FOR THE COUNT OF ELITE PRIMES Matthew Just]</ref>
# Notice that the rst ve Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such num- bers are prime.<ref name="ref_4a2a6fd4" />
# The sixth Fermat number is not prime, and no other Fermat primes are known.<ref name="ref_4a2a6fd4" />
# An ecient test exists to determine whether or not a Fermat number is prime, called Pepins test.<ref name="ref_4a2a6fd4" />
# This characterization uses uniquely values at most equal to tested Fermat number.<ref name="ref_bca905b7">[https://arxiv.org/pdf/2104.04875 A NEW CHARACTERIZATION OF PRIME FERMAT’S NUMBERS]</ref>
# We actually are able to establish the direct implication which is a real important result in the primality tests for Fermat numbers.<ref name="ref_bca905b7" />
# Condition 2m + = pn requires p to be either a Mersenne or Fermat prime.<ref name="ref_4393ff84">[https://arxiv.org/pdf/1809.03328 On Upper Bounds with ABC = 2mpn and ABC = 2mpnqr with p and q as Mersenne or Fermat]</ref>

===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q207264 Q207264]
===Spacy 패턴 목록===
* [{'LOWER': 'fermat'}, {'LEMMA': 'prime'}]
* [{'LOWER': 'fermat'}, {'LEMMA': 'number'}]

페르마 소수

2022-09-16T10:43:32Z

Pythagoras0: /* 메타데이터 */ 새 문단

==개요==

* 페르마소수란 <math>F_n= 2^{2^n}+1</math> 형태의 소수
** 3,5,17,257, 65537 다섯 가지만 알려져 있음.
* 페르마는 <math>F_n= 2^{2^n}+1</math> 가 모두 소수일 것이라 추측하였으나, 후에 [[오일러(1707-1783)|오일러]]는 반례를 발견:<math>F_5=641 \times 6700417</math>

==정다각형의 작도==

* 정n각형이 자와 컴파스로 작도가능 <math>\iff</math> <math>n=2^k p_1 p_2 \cdots p_r</math> (k ,r은 0이상의 정수, <math>p_1, p_2, \cdots, p_r</math> 은 서로 다른 페르마소수)
* [[정다각형의 작도]]와 [[가우스와 정17각형의 작도]] 항목을 참조

==역사==

* [[수학사 연표]]

==관련된 항목들==

* [[정다각형의 작도]]
* [[메르센 소수]]

[[분류:소수]]

== 노트 ==

===말뭉치===
# In fact, it is known that numbers of this form are not prime for values of n from 5 through 30, placing doubt on the existence of any Fermat primes for values of n > 4.<ref name="ref_edbed968">[https://www.britannica.com/science/Fermat-prime Fermat prime | mathematics]</ref>
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_21863434">[https://mathworld.wolfram.com/FermatPrime.html Fermat Prime -- from Wolfram MathWorld]</ref>
# Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime.<ref name="ref_77005ae2">[https://artofproblemsolving.com/wiki/index.php/Fermat_prime Art of Problem Solving]</ref>
# To find the Fermat number F n for an integer n , you first find m = 2 n , and then calculate 2 m + 1.<ref name="ref_7c2c701e">[https://www.techtarget.com/whatis/definition/Fermat-prime Definition from WhatIs.com]</ref>
# Surprisingly, Fermat primes arise in deciding whether a regular n-gon (a convex polygon with n equal sides) can be constructed with a compass and a straightedge.<ref name="ref_a529a8a6">[https://sites.millersville.edu/bikenaga/number-theory/fermat-numbers/fermat-numbers.pdf Fermat numbers]</ref>
# No fermat primes beyond (cid:8)4 have been found.<ref name="ref_a8ac433d">[http://www.math.ualberta.ca/~isaac/math324/s12/fermat_numbers.pdf Math 324 summer 2012]</ref>
# There are in(cid:12)nitely many distinct Fermat numbers, each of which is divisible by an odd prime, and since any two Fermat numbers are relatively prime, these odd primes must all be distinct.<ref name="ref_a8ac433d" />
# + 1 is a Fermat number; such primes are called Fermat primes.<ref name="ref_44f407e6">[https://en.wikipedia.org/wiki/Fermat_number Fermat number]</ref>
# From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_44f407e6" />
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 OEIS) is irrational.<ref name="ref_44f407e6" />
# Indeed, the first five Fermat numbers F 0 , ..., F 4 are easily shown to be prime.<ref name="ref_44f407e6" />
# The only known Fermat primes are the first five Fermat numbers: F 0 =3, F 1 =5, F 2 =17, F 3 =257, and F 4 =65537.<ref name="ref_fbf82f86">[https://primes.utm.edu/glossary/xpage/FermatNumber.html The Prime Glossary: Fermat number]</ref>
# It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number F n with n greater than 2 has the form k.2n+1+1 (exponent improved to n+2 by Lucas).<ref name="ref_fbf82f86" />
# Now we know that all of the Fermat numbers are composite for the other n less than 31.<ref name="ref_fbf82f86" />
# In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1 ) is a Fermat number, and such primes are called Fermat primes.<ref name="ref_e81de8fb">[http://www.scientificlib.com/en/Mathematics/LX/FermatNumber.html Fermat number]</ref>
# From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.<ref name="ref_e81de8fb" />
# The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational.<ref name="ref_e81de8fb" />
# Indeed, the first five Fermat numbers F 0 ,...,F 4 are easily shown to be prime.<ref name="ref_e81de8fb" />
# Indeed, even today, no other Fermat numbers are known to be prime!<ref name="ref_c596120d">[https://johncarlosbaez.wordpress.com/2019/02/05/fermat-primes-and-pascals-triangle/ Fermat Primes and Pascal’s Triangle]</ref>
# Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2.<ref name="ref_c596120d" />
# Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number.<ref name="ref_c596120d" />
# Wantzel went further and showed that if n is odd, we can construct a regular n-gon using straight-edge and compass if and only if n is a product of distinct Fermat primes.<ref name="ref_c596120d" />
# There are two definitions of the Fermat number.<ref name="ref_25205313">[https://mathworld.wolfram.com/FermatNumber.html Fermat Number -- from Wolfram MathWorld]</ref>
# The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form .<ref name="ref_25205313" />
# Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein proposed as a problem in 1844 the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).<ref name="ref_25205313" />
# At present, however, only composite Fermat numbers are known for .<ref name="ref_25205313" />
# In other words, every prime of the form 2 n +1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_17d186c1">[https://www.rieselprime.de/ziki/Fermat_number Fermat number]</ref>
# From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor.<ref name="ref_17d186c1" />
# Indeed, the first five Fermat numbers F 0 ,..., F 4 are easily shown to be prime.<ref name="ref_17d186c1" />
# Although it is widely believed that there are only finitely many Fermat primes, it should be noted that there are some experts who disagree (John Cosgrave: "Fermat 6").<ref name="ref_17d186c1" />
# A019434 List of Fermat primes: primes of form, for someIt is conjectured that there are only 5 terms.<ref name="ref_b3f9ea5b">[https://oeis.org/wiki/Fermat_primes Fermat primes]</ref>
# Numbers of the form F n =22n+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes.<ref name="ref_bfc52e99">[https://blogs.scientificamerican.com/roots-of-unity/extrapolation-gone-wrong-the-case-of-the-fermat-primes/ Extrapolation Gone Wrong: the Case of the Fermat Primes]</ref>
# Fermat conjectured that all Fermat numbers are prime.<ref name="ref_bfc52e99" />
# In 1732, about 70 years after Fermat's death, Leonhard Euler factored the 5th Fermat number into 641×6,700,417, disproving Fermat’s conjecture.<ref name="ref_bfc52e99" />
# So far, the only known Fermat primes are the ones that were known to Fermat.<ref name="ref_bfc52e99" />
# Is there a formula or a way to know which of the Fermat numbers are prime?.<ref name="ref_dc7c427d">[https://math.stackexchange.com/questions/563390/fermat-numbers-are-they-all-prime Fermat numbers. Are they all prime?]</ref>
# Prologue What are the known Fermat primes?<ref name="ref_b2c9d49a">[https://arxiv.org/pdf/1605.01371 This is the extended version of a paper that has appeared in the Mathematical Intelligencer. The ﬁnal publication is available]</ref>
# Taking Fermat prime to mean prime of the form 2n + 1, there are six known Fermat primes, namely those for n = 0, 1, 2, 4, 8, 16.<ref name="ref_b2c9d49a" />
# We shall pronounce the last letter of Fermats name, as he did, when we include 2 among the Fermat primes, as he did.<ref name="ref_b2c9d49a" />
# The Fermat number Fn is either prime or not prime: the question of how to approximate the probability of primality for a general n is delicate.<ref name="ref_b2c9d49a" />
# Fb;n = b2n + 1 and are particularly interesting since they have many characteristics of the heavily studied standard Fermat numbers Fn = F2;n.<ref name="ref_a10cb56f">[https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf Mathematics of computation]</ref>
# The \Proth" program was created in 1997 to extend the search for large factors of Fermat numbers.<ref name="ref_a10cb56f" />
# They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 1665) who first studied numbers in this form.<ref name="ref_8f117ada">[https://wstein.org/edu/2010/414/projects/tsang.pdf Fermat]</ref>
# We will not be able to answer this question in this paper, but we will prove some basic properties of Fermat numbers and discuss their primality and divisibility.<ref name="ref_8f117ada" />
# Primes in this form are called Fermat primes.<ref name="ref_8f117ada" />
# Up-to-date there are only five known Fermat primes.<ref name="ref_8f117ada" />
# Moreover, no other Fermat number is known to be prime for n > 4 , so now it is conjectured that those are all prime Fermat numbers.<ref name="ref_6428f90c">[https://planetmath.org/fermatnumbers Fermat numbers]</ref>
# In honour of the inspired pioneers, the numbers of the form 2n-1 are now called the Mersenne numbers and the numbers of the form 2n+1 the Fermat numbers.<ref name="ref_e35d89c0">[http://yves.gallot.pagesperso-orange.fr/primes/ Generalized Fermat Primes Search]</ref>
# The search for Mersenne and Fermat primes has been greatly extended since the 17th century.<ref name="ref_e35d89c0" />
# Today, all the Mersenne primes having less than 2,000,000 digits are known and all the Fermat primes up to 2,000,000,000 digits!<ref name="ref_e35d89c0" />
# In 1994, R. Crandall and B. Fagin discovered that the Discrete Weighted Transforms could be used to double the speed of the search for Mersenne and Fermat numbers.<ref name="ref_e35d89c0" />
# Those are the only primes below 100,000 that I could show must be primitive for all but finitely many Fermat primes.<ref name="ref_9d437e74">[https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=display;num=1204796915 Fermat Primes and the number 7]</ref>
# Fermat numbers are named after Pierre de Fermat.<ref name="ref_a462b554">[https://kids.kiddle.co/Fermat_number Fermat number facts for kids]</ref>
# Every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes.<ref name="ref_a462b554" />
# Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.<ref name="ref_a462b554" />
# Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers.<ref name="ref_e5ed5592">[https://exploringnumbertheory.wordpress.com/2016/10/25/fermat-numbers/ Exploring Number Theory]</ref>
# Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set.<ref name="ref_e5ed5592" />
# The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2.<ref name="ref_e5ed5592" />
# He demonstrated that the first 5 Fermat numbers , , , , are prime and conjectured that all Fermat numbers are prime.<ref name="ref_e5ed5592" />
# Chapter 4 Fermat and Mersenne Primes 4.1 Fermat primes Theorem 4.1.<ref name="ref_7befcda2">[https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/FermatMersenne.pdf Chapter 4]</ref>
# Fermat conjectured that the Fermat numbers are all prime.<ref name="ref_7befcda2" />
# We will come back to their solution shortly as we must first introduce the notion of a Fermat prime!<ref name="ref_62f9f1ec">[https://vrs.amsi.org.au/fermat-primes-gauss-wantzel/ Fermat Primes and the Gauss-Wantzel Theorem]</ref>
# In the 1600s, a mathematician and lawyer named Pierre de Fermat studied numbers of the form 2^n+1 (where n=2^k) which are now called Fermat numbers.<ref name="ref_62f9f1ec" />
# So, he conjectured that all Fermat numbers are prime.<ref name="ref_62f9f1ec" />
# Funnily enough, it was shown by Leonhard Euler in 1732 (another famous mathematician), that only the next Fermat number is not prime.<ref name="ref_62f9f1ec" />
# It is still open whether there exist innitely many Fermat primes or innitely many composite Fermat numbers.<ref name="ref_0b05fdb4">[https://arxiv.org/pdf/2204.08302 MINIMALITY CONDITIONS EQUIVALENT TO THE FINITUDE OF FERMAT AND MERSENNE PRIMES]</ref>
# Especially, the formula for modulo Fermat primes is given. MSC2010: 11A07.<ref name="ref_9c963bc5">[https://arxiv.org/pdf/1601.06509 The largest cycles consist by the quadratic residues and Fermat primes]</ref>
# For the Fermat primes, i.e., the prime numbers of the form p 22k ` 1, we have Lppq 1. Proof.<ref name="ref_9c963bc5" />
# We have veried it for k 0, 1, 2, 3, 4, the known Fermat primes.<ref name="ref_9c963bc5" />
# As of 2021, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537.<ref name="ref_49cdcbc7">[https://arxiv.org/pdf/1912.12088 MINIMALITY OF TOPOLOGICAL MATRIX GROUPS AND FERMAT PRIMES]</ref>
# For an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p 1, (Q, p)) is minimal; (3) SL(p 1, Q(i)) is minimal.<ref name="ref_49cdcbc7" />
# We determine all the Carmichael numbers m with a Fermat prime factor such that L = 2P 2, where k N and P is an odd prime number.<ref name="ref_f1f90697">[https://arxiv.org/pdf/1710.01321 On the ﬁniteness of Carmichael numbers with Fermat factors and L = 2αP 2.]</ref>
# We assume that m is divisible by at least one of the known Fermat prime numbers.<ref name="ref_f1f90697" />
# If m is divisible by one of the known Fermat primes, then m must be one of the following 11 Carmichael numbers.<ref name="ref_f1f90697" />
# Let 2 R R 22, there exists a generalized Fermat prime p = r2 2 be an integer.<ref name="ref_25a71c73">[https://arxiv.org/pdf/1502.02800 FAST INTEGER MULTIPLICATION USING GENERALIZED FERMAT PRIMES]</ref>
# The key concept of our algorithm is the use of a chain of generalized Fermat primes (of the form r2 +1) to handle recursive calls.<ref name="ref_25a71c73" />
# First, we encode integers to be multiplied as integers modulo generalized Fermat primes, and not as polynomials.<ref name="ref_25a71c73" />
# Section 4 studies generalized Fermat primes, and their relation to the Bateman-Horn conjecture.<ref name="ref_25a71c73" />
# 1. Introduction + 1 for n The Fermat numbers are given by Fn = 22n 0.<ref name="ref_4a2a6fd4">[https://arxiv.org/pdf/2102.00906 ON UPPER BOUNDS FOR THE COUNT OF ELITE PRIMES Matthew Just]</ref>
# Notice that the rst ve Fermat numbers are prime, and it was initially conjectured (by Fermat) that all such num- bers are prime.<ref name="ref_4a2a6fd4" />
# The sixth Fermat number is not prime, and no other Fermat primes are known.<ref name="ref_4a2a6fd4" />
# An ecient test exists to determine whether or not a Fermat number is prime, called Pepins test.<ref name="ref_4a2a6fd4" />
# This characterization uses uniquely values at most equal to tested Fermat number.<ref name="ref_bca905b7">[https://arxiv.org/pdf/2104.04875 A NEW CHARACTERIZATION OF PRIME FERMAT’S NUMBERS]</ref>
# We actually are able to establish the direct implication which is a real important result in the primality tests for Fermat numbers.<ref name="ref_bca905b7" />
# Condition 2m + = pn requires p to be either a Mersenne or Fermat prime.<ref name="ref_4393ff84">[https://arxiv.org/pdf/1809.03328 On Upper Bounds with ABC = 2mpn and ABC = 2mpnqr with p and q as Mersenne or Fermat]</ref>

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q207264 Q207264]
===Spacy 패턴 목록===
* [{'LOWER': 'fermat'}, {'LEMMA': 'prime'}]
* [{'LOWER': 'fermat'}, {'LEMMA': 'number'}]

페르마 소수

2022-09-16T10:43:30Z

Pythagoras0: /* 노트 */ 새 문단

가우스와 정17각형의 작도

2022-09-16T10:36:09Z

Pythagoras0:

==개요==

* 가우스는 정17각형이 자와 컴파스로 작도가능함을 증명함
* 대수적으로, <math>z^{16}+z^{15}+\cdots+z+1=0</math>의 풀이를 반복적인 2차방정식의 풀이로 환원할 수 있는가의 문제
* 16차 방정식을 2차방정식 네번 푸는 문제로 바꾸는 것
** 이 아이디어를 좀더 간단한 예를 통해 이해하기 위해서는 [[정오각형]] 항목 중 꼭지점의 평면좌표를 참조

==증명==

* <math>\zeta=e^{2\pi i \over 17}</math> 로 두자. 이 값을 대수적으로 구하는 것이 목표.
* <math>(3^1, 3^2,3^3, 3^4, 3^5, 3^7, 3^8, 3^9, 3^{10}, 3^{11}, 3^{12}, 3^{13}, 3^{14}, 3^{15}, 3^{16}) \equiv (3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4,12, 2, 6, 1) \pmod {17}</math>
* 이 순서대로 2로 나눈 나머지에 따라서 분류
** <math>A_0 = \zeta^{9} + \zeta^{13} + \zeta^{15} + \zeta^{16}+\zeta^{8} + \zeta^{4} + \zeta^{2} +\zeta^{1}</math>
** <math>A_1 = \zeta^3 + \zeta^{10} + \zeta^{5} + \zeta^{11}+\zeta^{14} + \zeta^{7} + \zeta^{12} +\zeta^{6}</math>
** <math>A_0+A_1= -1</math>, <math>A_{0}A_{1} = -4</math>, <math>A_0>A_1</math>
** <math>A_0 = \frac{-1 + \sqrt{17}}{2}</math> , <math>A_1= \frac{-1 - \sqrt{17}}{2}</math>
* 이번에는 4로 나눈 나머지에 따라서 분류
** <math>B_0 = \zeta^{13}+ \zeta^{16}+ \zeta^4 + \zeta^1 </math>
** <math>B_1= \zeta^3 + \zeta^5 + \zeta^{14} + \zeta^{12}</math>
** <math>B_2= \zeta^9 + \zeta^{15} + \zeta^8 +\zeta^2</math>
** <math>B_3 =\zeta^{10} + \zeta^{11} + \zeta^{7} +\zeta^{6}</math>
** <math>B_0+B_2=A_0</math>, <math>B_0B_2= -1</math>, <math>B_0>0</math>
** <math>B_0 = \frac{-1 + \sqrt{17} + \sqrt{34 - 2\sqrt{17}}}{4}</math>, <math>B_2 = \frac{-1 + \sqrt{17} - \sqrt{34 - 2\sqrt{17}}}{4}</math>
** <math>B_1+B_3=A_1</math>, <math>B_1B_3= -1</math>, <math>B_{1}> 0</math>
** <math>B_1 = \frac{-1 - \sqrt{17} + \sqrt{34 + 2\sqrt{17}}}{4}</math>, <math>B_3 = \frac{-1 - \sqrt{17} - \sqrt{34 + 2\sqrt{17}}}{4}</math>
* 이번에는 8로 나눈 나머지에 따라서 분류
** <math>C_0= \zeta^{16}+ \zeta^1</math>, <math>C_4= \zeta^{13} +\zeta^4</math>, <math>C_0 > C_1</math>
** <math>C_0+C_4=B_0</math>, <math>C_0C_4=B_1</math>
** <math>C_0= \frac{B_0+\sqrt{B_0^2-4B_1}}{2}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{8}</math>
** <math>C_4= \frac{B_0 - \sqrt{B_0^2-4B_1}}{2}</math>
* 이제 마무리
** <math>\zeta =\frac{{C_0} + \sqrt{{C_0}^2 - 4}}{2}</math>
** <math>\cos \frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{16}</math>

==가우스합과의 관계==

* 참고로 위에서 <math>A_0-A_1</math> 은 [[가우스 합|가우스합]] 임을 알 수 있음.
** <math>\{3, 10, 5, 11, 14, 7, 12, 6\}</math> 는 <math>\pmod {17}</math> 에 대하여 이차비잉여
** <math>\{9, 13, 15, 16, 8, 4, 2, 1\}</math>는 <math>\pmod {17}</math> 에 대하여 이차잉여
** 따라서 <math>A_{0}A_{1}</math>를 계산하는 대신에 <math>A_0-A_1=\sqrt{17}</math> 를 활용할 수도 있음.

==재미있는 사실==

* 17은 [[페르마 소수|페르마소수]]이다

==역사==
* 1796 가우스
* [[수학사 연표]]

==메모==

* http://pballew.blogspot.com/2011/04/constructructable-polygons-and-x17-1.html

==관련된 학부 과목과 미리 알고 있으면 좋은 것들==

* [[추상대수학]]
** [[순환군]]
** 가해군 (solvable groups)
* [[초등정수론]]
** [[합동식과 군론]]
** [[원시근(primitive root)]]
** [[오일러의 totient 함수]]

==관련된 항목들==

* [[그리스 3대 작도 불가능문제]]
* [[정다각형의 작도]]
* [[원시근(primitive root)]]
* [[가우스(1777 - 1855)]]

==매스매티카 파일 및 계산 리소스==
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxZGVmNzY1NmEtMThiNi00OTdkLTgxYTQtYTZhMzdlZTM4Mzkw&sort=name&layout=list&num=50

==사전형태의 자료==

* http://en.wikipedia.org/wiki/Heptadecagon

==관련논문==
* Garcia, Stephan Ramon, Trevor Hyde, and Bob Lutz. ‘Gauss’ Hidden Menagerie: From Cyclotomy to Supercharacters’. arXiv:1501.07507 [math], 29 January 2015. http://arxiv.org/abs/1501.07507.

==관련도서==

* [http://www.amazon.com/Problems-Elementary-Geometry-Phoenix-Editions/dp/0486495515 Famous Problems of Elementary Geometry] (Dover Phoenix Editions)
** 펠릭스 클라인 Felix Klein
** 얇은 책으로, 대수방정식과 함께 고대 그리스 3대 작도 불가능문제를 소개함.
* [http://www.amazon.com/Functions-Integrals-Translations-Mathematical-Monographs/dp/0821805878 Elliptic functions and elliptic integrals]
** Viktor Prasolov, Yuri Solovyev
* Lectures on Elementary Number Theory
** Hans Rademacher

==동영상==

* [http://www.youtube.com/watch?v=E2QMzRcjKkM 정17각형의 작도 과정을 보여주는 동영상], Youtube

[[분류:작도]]
[[분류:추상대수학]]

== 노트 ==

===말뭉치===
# Then, on 30 March 1796, the 19 year old Gauss discovered that it was possible to construct the regular heptadecagon (17-gon).<ref name="ref_47ccf09f">[https://mathpages.com/home/kmath487.htm Constructing the Heptadecagon]</ref>
# One of the nicest actual constructions of the 17-gon is Richmond's (1893), as reproduced in Stewart's "Galois Theory".<ref name="ref_47ccf09f" />
# Gauss was clearly fond of this discovery, and there's a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes' tombstone.<ref name="ref_47ccf09f" />
# On the other hand, if proximity to the actual remains is not important, then the heptadecagon on the monument to Gauss in his native town of Brunswick, or even the figure above, may suffice.<ref name="ref_47ccf09f" />
# Gauss's "heptadecagon', a 17-sided polygon that showed the relationship between geometry and algebra.<ref name="ref_275a9462">[https://www.inverse.com/article/44309-johann-carl-friedrich-gauss-math-statistics-accomplishments Johann Carl Friedrich Gauß Changed History With His 17-Sided Shape]</ref>
# But what’s widely considered his first important discovery is his construction of a 17-sided polygon called a heptadecagon, using only a ruler and a compass.<ref name="ref_275a9462" />
# A regular heptadecagon is represented by the Schläfli symbol {17}.<ref name="ref_3d3204f1">[https://en.wikipedia.org/wiki/Heptadecagon Heptadecagon]</ref>
# Constructing a regular heptadecagon thus involves finding the cosine of 2 π / 17 {\displaystyle 2\pi /17} in terms of square roots, which involves an equation of degree 17—a Fermat prime.<ref name="ref_3d3204f1" />
# The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893.<ref name="ref_3d3204f1" />
# These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon.<ref name="ref_3d3204f1" />
# Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is constructible with a compass and straightedge.<ref name="ref_05209e8b">[https://mathworld.wolfram.com/Heptadecagon.html Heptadecagon -- from Wolfram MathWorld]</ref>
# The first explicit construction of a heptadecagon was given by Erchinger in about 1800.<ref name="ref_05209e8b" />
# You now have points and of a heptadecagon.<ref name="ref_05209e8b" />
# Connect the adjacent points for to 17, forming the heptadecagon.<ref name="ref_05209e8b" />
# A regular heptadecagon, or 17 sided polygon, was known to have existed by mathematicians for many years, but creating one proved to be a greater challenge.<ref name="ref_c4d14414">[https://interestingengineering.com/create-regular-heptadecagon-using-math How to Create a Regular Heptadecagon using Math!]</ref>
# A regular heptadecagon was first created by 19-year-old Carl Friedrich Gauss in 1796 in a groundbreaking proof.<ref name="ref_c4d14414" />
# The proof in regards to this 17-gon's construction marked the first major breakthrough in polygon construction in over 2,000 years.<ref name="ref_c4d14414" />
# Apart from a heptadecagon, there are also heptadecagrams, which are 17 sided star polygons.<ref name="ref_c4d14414" />
# A regular heptadecagon has internal angles each measuring 158.823529411765 degrees.<ref name="ref_fc48496f">[http://academickids.com/encyclopedia/index.php/Heptadecagon Academic Kids]</ref>
# The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796.<ref name="ref_fc48496f" />
# Gauß proved in 1796 (when he was 19 years old) that the heptadecagon is Constructible with a Compass and Straightedge.<ref name="ref_b1207714">[https://archive.lib.msu.edu/crcmath/math/math/h/h188.htm Heptadecagon]</ref>
# The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1992) was first given by Richmond (1893).<ref name="ref_b1207714" />
# The following animation of a heptadecagon editing.<ref name="ref_8efd6d53">[https://kids.kiddle.co/Heptadecagon Heptadecagon facts for kids]</ref>
# Go to google.com and search about Gauss' 17-gon construction.<ref name="ref_5c6bd9ed">[http://mathgardenblog.blogspot.com/2014/06/construct-15gon.html Math Garden: How to construct a regular polygon with 15 sides]</ref>
# He described in his Disquitiones Arithmeticae, a major work on number theory, how to construct a regular 17-gon with Euclidean tools.<ref name="ref_853b9c26">[https://mathcs.clarku.edu/~djoyce/elements/bookIV/propIV16.html Euclid's Elements, Book IV, Proposition 16]</ref>
# The eighteen year old Gauss began his scientific diary with his construction of the regular 17-gon.<ref name="ref_b543aaf5">[https://www.maa.org/news/on-this-day/1796-3-30 Mathematical Association of America]</ref>
# The regular 17-sided polygon (heptadecagon) can be constructed with the help of a compass and a ruler.<ref name="ref_6575c93b">[https://en.wiktionary.org/wiki/heptadecagon heptadecagon]</ref>
# This document presents Gausss insight that it is possible to construct a heptadecagon a regular polygon with 17 sideswith straightedge and compass.<ref name="ref_3c71bab1">[https://www.weizmann.ac.il/sci-tea/benari/sites/sci-tea.benari/files/uploads/softwareAndLearningMaterials/heptadecagon-en.pdf Construction of a regular heptadecagon]</ref>
# 3 Gausss proof that a heptadecagon is constructable What Gauss saw is the one need not work with the roots in the natural order r, r2, . . .<ref name="ref_3c71bab1" />
# The Regular Polygon of 17 sides is called the Heptadecagon, or sometimes the Heptakaidecagon.<ref name="ref_b1207714" />
# In 1796, a 19 years old Gauss showed how to construct a heptadecagon using only a compass and an unmarked straightedge.<ref name="ref_29c607c6">[https://medium.com/@youssef.housni21/heptadecagon-3efb657a43b0 Heptadecagon]</ref>
# Key Words: constructing a regular heptadecagon, theory of cyclotonic equa- tions, modulo, prime number and primitive root, Constructing roots and frac- tions 1.<ref name="ref_d94ae8b4">[http://ijpam.eu/contents/2013-82-5/3/3.pdf International journal of pure and applied mathematics]</ref>
# til recently I did not know the proof supporting Gauss method for constructing a regular heptadecagon - a polygon with 17 sides.<ref name="ref_d94ae8b4" />
# Gauss Theory of Cyclotomic Equations We have seen how the value of cos needed for the construction of a regular heptadecagon can be obtained, but this calculation was just a conrmation.<ref name="ref_d94ae8b4" />
# That all changed in 1796 when a teenage Carl Friedrich Gauss proved the constructibility of the regular seventeen-sided polygon, or heptadecagon.<ref name="ref_535f5554">[https://alephoneplex.com/2021/08/22/gauss-and-the-regular-heptadecagon/ Gauss and the Regular Heptadecagon]</ref>
# This allows for the construction of the rest of the heptadecagon as shown bellow.<ref name="ref_535f5554" />
# However, on March 30th, 1796, a 19 year old Carl Gauss rose from bed and was struck by an idea regarding how to prove that the regular 17-gon was constructable.<ref name="ref_e227163f">[https://sites.math.washington.edu/~morrow/336_20/papers20/danielh.pdf The constructability of the regular]</ref>
# We can now use these results to prove the main theorem, that the regular heptadecagon is constructable.<ref name="ref_e227163f" />
# Upon seeing the number 17, I immediately thought of the Gauss construction of the heptadecagon.<ref name="ref_0e255cbb">[https://mathcircle.berkeley.edu/sites/default/files/archivedocs/2009_2010/lectures/0910lecturespdf/HeptadecagonBMC10.pdf Gauss and the heptadecagon]</ref>
# In any case, by just following the method of Gauss through the rst stage of the heptadecagon construction, I was able to solve the problem.<ref name="ref_0e255cbb" />
# I have just completed my first construction of the regular heptadecagon — a construction that even the ancient Greeks were never able to figure out.<ref name="ref_788dc367">[https://robertlovespi.net/2014/06/11/constructing-the-heptadecagon/ Constructing the Heptadecagon]</ref>
# The regular heptadecagon construction, however, I did not figure out independently.<ref name="ref_788dc367" />
# As a result, the regular heptadecagon is one of the few prime sided figures constructable using an unmarked ruler and pair of compasses - that is using a classical construction.<ref name="ref_07a9fbaa">[https://socratic.org/questions/58655ba911ef6b25b68845fd What is the internal angle of a regular #17#-sided polygon?]</ref>
# 1 2 HUGO TAVARES AND PEDRO J. FREITAS Gauss proved, in his early years, that the 17-gon is constructible.<ref name="ref_3798ab74">[https://arxiv.org/pdf/1507.07970 DIVIDING THE CIRCLE HUGO TAVARES AND PEDRO J. FREITAS]</ref>

===소스===
<references />

==메타데이터==
===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q542476 Q542476]
===Spacy 패턴 목록===
* [{'LEMMA': 'heptadecagon'}]
* [{'LOWER': '17'}, {'OP': '*'}, {'LEMMA': 'gon'}]
* [{'LOWER': 'heptakaidecagon'}]

가우스와 정17각형의 작도

2022-09-16T10:36:07Z

Pythagoras0:

==개요==

* 가우스는 정17각형이 자와 컴파스로 작도가능함을 증명함
* 대수적으로, <math>z^{16}+z^{15}+\cdots+z+1=0</math>의 풀이를 반복적인 2차방정식의 풀이로 환원할 수 있는가의 문제
* 16차 방정식을 2차방정식 네번 푸는 문제로 바꾸는 것
** 이 아이디어를 좀더 간단한 예를 통해 이해하기 위해서는 [[정오각형]] 항목 중 꼭지점의 평면좌표를 참조

==증명==

* <math>\zeta=e^{2\pi i \over 17}</math> 로 두자. 이 값을 대수적으로 구하는 것이 목표.
* <math>(3^1, 3^2,3^3, 3^4, 3^5, 3^7, 3^8, 3^9, 3^{10}, 3^{11}, 3^{12}, 3^{13}, 3^{14}, 3^{15}, 3^{16}) \equiv (3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4,12, 2, 6, 1) \pmod {17}</math>
* 이 순서대로 2로 나눈 나머지에 따라서 분류
** <math>A_0 = \zeta^{9} + \zeta^{13} + \zeta^{15} + \zeta^{16}+\zeta^{8} + \zeta^{4} + \zeta^{2} +\zeta^{1}</math>
** <math>A_1 = \zeta^3 + \zeta^{10} + \zeta^{5} + \zeta^{11}+\zeta^{14} + \zeta^{7} + \zeta^{12} +\zeta^{6}</math>
** <math>A_0+A_1= -1</math>, <math>A_{0}A_{1} = -4</math>, <math>A_0>A_1</math>
** <math>A_0 = \frac{-1 + \sqrt{17}}{2}</math> , <math>A_1= \frac{-1 - \sqrt{17}}{2}</math>
* 이번에는 4로 나눈 나머지에 따라서 분류
** <math>B_0 = \zeta^{13}+ \zeta^{16}+ \zeta^4 + \zeta^1 </math>
** <math>B_1= \zeta^3 + \zeta^5 + \zeta^{14} + \zeta^{12}</math>
** <math>B_2= \zeta^9 + \zeta^{15} + \zeta^8 +\zeta^2</math>
** <math>B_3 =\zeta^{10} + \zeta^{11} + \zeta^{7} +\zeta^{6}</math>
** <math>B_0+B_2=A_0</math>, <math>B_0B_2= -1</math>, <math>B_0>0</math>
** <math>B_0 = \frac{-1 + \sqrt{17} + \sqrt{34 - 2\sqrt{17}}}{4}</math>, <math>B_2 = \frac{-1 + \sqrt{17} - \sqrt{34 - 2\sqrt{17}}}{4}</math>
** <math>B_1+B_3=A_1</math>, <math>B_1B_3= -1</math>, <math>B_{1}> 0</math>
** <math>B_1 = \frac{-1 - \sqrt{17} + \sqrt{34 + 2\sqrt{17}}}{4}</math>, <math>B_3 = \frac{-1 - \sqrt{17} - \sqrt{34 + 2\sqrt{17}}}{4}</math>
* 이번에는 8로 나눈 나머지에 따라서 분류
** <math>C_0= \zeta^{16}+ \zeta^1</math>, <math>C_4= \zeta^{13} +\zeta^4</math>, <math>C_0 > C_1</math>
** <math>C_0+C_4=B_0</math>, <math>C_0C_4=B_1</math>
** <math>C_0= \frac{B_0+\sqrt{B_0^2-4B_1}}{2}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{8}</math>
** <math>C_4= \frac{B_0 - \sqrt{B_0^2-4B_1}}{2}</math>
* 이제 마무리
** <math>\zeta =\frac{{C_0} + \sqrt{{C_0}^2 - 4}}{2}</math>
** <math>\cos \frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{16}</math>

==가우스합과의 관계==

* 참고로 위에서 <math>A_0-A_1</math> 은 [[가우스 합|가우스합]] 임을 알 수 있음.
** <math>\{3, 10, 5, 11, 14, 7, 12, 6\}</math> 는 <math>\pmod {17}</math> 에 대하여 이차비잉여
** <math>\{9, 13, 15, 16, 8, 4, 2, 1\}</math>는 <math>\pmod {17}</math> 에 대하여 이차잉여
** 따라서 <math>A_{0}A_{1}</math>를 계산하는 대신에 <math>A_0-A_1=\sqrt{17}</math> 를 활용할 수도 있음.

==재미있는 사실==

* 17은 [[페르마 소수|페르마소수]]이다

==역사==
* 1796 가우스
* [[수학사 연표]]

==메모==

* http://pballew.blogspot.com/2011/04/constructructable-polygons-and-x17-1.html

==관련된 학부 과목과 미리 알고 있으면 좋은 것들==

* [[추상대수학]]
** [[순환군]]
** 가해군 (solvable groups)
* [[초등정수론]]
** [[합동식과 군론]]
** [[원시근(primitive root)]]
** [[오일러의 totient 함수]]

==관련된 항목들==

* [[그리스 3대 작도 불가능문제]]
* [[정다각형의 작도]]
* [[원시근(primitive root)]]
* [[가우스(1777 - 1855)]]

==매스매티카 파일 및 계산 리소스==
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxZGVmNzY1NmEtMThiNi00OTdkLTgxYTQtYTZhMzdlZTM4Mzkw&sort=name&layout=list&num=50

==사전형태의 자료==

* http://en.wikipedia.org/wiki/Heptadecagon

==관련논문==
* Garcia, Stephan Ramon, Trevor Hyde, and Bob Lutz. ‘Gauss’ Hidden Menagerie: From Cyclotomy to Supercharacters’. arXiv:1501.07507 [math], 29 January 2015. http://arxiv.org/abs/1501.07507.

==관련도서==

* [http://www.amazon.com/Problems-Elementary-Geometry-Phoenix-Editions/dp/0486495515 Famous Problems of Elementary Geometry] (Dover Phoenix Editions)
** 펠릭스 클라인 Felix Klein
** 얇은 책으로, 대수방정식과 함께 고대 그리스 3대 작도 불가능문제를 소개함.
* [http://www.amazon.com/Functions-Integrals-Translations-Mathematical-Monographs/dp/0821805878 Elliptic functions and elliptic integrals]
** Viktor Prasolov, Yuri Solovyev
* Lectures on Elementary Number Theory
** Hans Rademacher

==동영상==

* [http://www.youtube.com/watch?v=E2QMzRcjKkM 정17각형의 작도 과정을 보여주는 동영상], Youtube

[[분류:작도]]
[[분류:추상대수학]]

== 노트 ==

===말뭉치===
# Then, on 30 March 1796, the 19 year old Gauss discovered that it was possible to construct the regular heptadecagon (17-gon).<ref name="ref_47ccf09f">[https://mathpages.com/home/kmath487.htm Constructing the Heptadecagon]</ref>
# One of the nicest actual constructions of the 17-gon is Richmond's (1893), as reproduced in Stewart's "Galois Theory".<ref name="ref_47ccf09f" />
# Gauss was clearly fond of this discovery, and there's a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes' tombstone.<ref name="ref_47ccf09f" />
# On the other hand, if proximity to the actual remains is not important, then the heptadecagon on the monument to Gauss in his native town of Brunswick, or even the figure above, may suffice.<ref name="ref_47ccf09f" />
# Gauss's "heptadecagon', a 17-sided polygon that showed the relationship between geometry and algebra.<ref name="ref_275a9462">[https://www.inverse.com/article/44309-johann-carl-friedrich-gauss-math-statistics-accomplishments Johann Carl Friedrich Gauß Changed History With His 17-Sided Shape]</ref>
# But what’s widely considered his first important discovery is his construction of a 17-sided polygon called a heptadecagon, using only a ruler and a compass.<ref name="ref_275a9462" />
# A regular heptadecagon is represented by the Schläfli symbol {17}.<ref name="ref_3d3204f1">[https://en.wikipedia.org/wiki/Heptadecagon Heptadecagon]</ref>
# Constructing a regular heptadecagon thus involves finding the cosine of 2 π / 17 {\displaystyle 2\pi /17} in terms of square roots, which involves an equation of degree 17—a Fermat prime.<ref name="ref_3d3204f1" />
# The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893.<ref name="ref_3d3204f1" />
# These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon.<ref name="ref_3d3204f1" />
# Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is constructible with a compass and straightedge.<ref name="ref_05209e8b">[https://mathworld.wolfram.com/Heptadecagon.html Heptadecagon -- from Wolfram MathWorld]</ref>
# The first explicit construction of a heptadecagon was given by Erchinger in about 1800.<ref name="ref_05209e8b" />
# You now have points and of a heptadecagon.<ref name="ref_05209e8b" />
# Connect the adjacent points for to 17, forming the heptadecagon.<ref name="ref_05209e8b" />
# A regular heptadecagon, or 17 sided polygon, was known to have existed by mathematicians for many years, but creating one proved to be a greater challenge.<ref name="ref_c4d14414">[https://interestingengineering.com/create-regular-heptadecagon-using-math How to Create a Regular Heptadecagon using Math!]</ref>
# A regular heptadecagon was first created by 19-year-old Carl Friedrich Gauss in 1796 in a groundbreaking proof.<ref name="ref_c4d14414" />
# The proof in regards to this 17-gon's construction marked the first major breakthrough in polygon construction in over 2,000 years.<ref name="ref_c4d14414" />
# Apart from a heptadecagon, there are also heptadecagrams, which are 17 sided star polygons.<ref name="ref_c4d14414" />
# A regular heptadecagon has internal angles each measuring 158.823529411765 degrees.<ref name="ref_fc48496f">[http://academickids.com/encyclopedia/index.php/Heptadecagon Academic Kids]</ref>
# The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796.<ref name="ref_fc48496f" />
# Gauß proved in 1796 (when he was 19 years old) that the heptadecagon is Constructible with a Compass and Straightedge.<ref name="ref_b1207714">[https://archive.lib.msu.edu/crcmath/math/math/h/h188.htm Heptadecagon]</ref>
# The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1992) was first given by Richmond (1893).<ref name="ref_b1207714" />
# The following animation of a heptadecagon editing.<ref name="ref_8efd6d53">[https://kids.kiddle.co/Heptadecagon Heptadecagon facts for kids]</ref>
# Go to google.com and search about Gauss' 17-gon construction.<ref name="ref_5c6bd9ed">[http://mathgardenblog.blogspot.com/2014/06/construct-15gon.html Math Garden: How to construct a regular polygon with 15 sides]</ref>
# He described in his Disquitiones Arithmeticae, a major work on number theory, how to construct a regular 17-gon with Euclidean tools.<ref name="ref_853b9c26">[https://mathcs.clarku.edu/~djoyce/elements/bookIV/propIV16.html Euclid's Elements, Book IV, Proposition 16]</ref>
# The eighteen year old Gauss began his scientific diary with his construction of the regular 17-gon.<ref name="ref_b543aaf5">[https://www.maa.org/news/on-this-day/1796-3-30 Mathematical Association of America]</ref>
# The regular 17-sided polygon (heptadecagon) can be constructed with the help of a compass and a ruler.<ref name="ref_6575c93b">[https://en.wiktionary.org/wiki/heptadecagon heptadecagon]</ref>
# This document presents Gausss insight that it is possible to construct a heptadecagon a regular polygon with 17 sideswith straightedge and compass.<ref name="ref_3c71bab1">[https://www.weizmann.ac.il/sci-tea/benari/sites/sci-tea.benari/files/uploads/softwareAndLearningMaterials/heptadecagon-en.pdf Construction of a regular heptadecagon]</ref>
# 3 Gausss proof that a heptadecagon is constructable What Gauss saw is the one need not work with the roots in the natural order r, r2, . . .<ref name="ref_3c71bab1" />
# The Regular Polygon of 17 sides is called the Heptadecagon, or sometimes the Heptakaidecagon.<ref name="ref_b1207714" />
# In 1796, a 19 years old Gauss showed how to construct a heptadecagon using only a compass and an unmarked straightedge.<ref name="ref_29c607c6">[https://medium.com/@youssef.housni21/heptadecagon-3efb657a43b0 Heptadecagon]</ref>
# Key Words: constructing a regular heptadecagon, theory of cyclotonic equa- tions, modulo, prime number and primitive root, Constructing roots and frac- tions 1.<ref name="ref_d94ae8b4">[http://ijpam.eu/contents/2013-82-5/3/3.pdf International journal of pure and applied mathematics]</ref>
# til recently I did not know the proof supporting Gauss method for constructing a regular heptadecagon - a polygon with 17 sides.<ref name="ref_d94ae8b4" />
# Gauss Theory of Cyclotomic Equations We have seen how the value of cos needed for the construction of a regular heptadecagon can be obtained, but this calculation was just a conrmation.<ref name="ref_d94ae8b4" />
# That all changed in 1796 when a teenage Carl Friedrich Gauss proved the constructibility of the regular seventeen-sided polygon, or heptadecagon.<ref name="ref_535f5554">[https://alephoneplex.com/2021/08/22/gauss-and-the-regular-heptadecagon/ Gauss and the Regular Heptadecagon]</ref>
# This allows for the construction of the rest of the heptadecagon as shown bellow.<ref name="ref_535f5554" />
# However, on March 30th, 1796, a 19 year old Carl Gauss rose from bed and was struck by an idea regarding how to prove that the regular 17-gon was constructable.<ref name="ref_e227163f">[https://sites.math.washington.edu/~morrow/336_20/papers20/danielh.pdf The constructability of the regular]</ref>
# We can now use these results to prove the main theorem, that the regular heptadecagon is constructable.<ref name="ref_e227163f" />
# Upon seeing the number 17, I immediately thought of the Gauss construction of the heptadecagon.<ref name="ref_0e255cbb">[https://mathcircle.berkeley.edu/sites/default/files/archivedocs/2009_2010/lectures/0910lecturespdf/HeptadecagonBMC10.pdf Gauss and the heptadecagon]</ref>
# In any case, by just following the method of Gauss through the rst stage of the heptadecagon construction, I was able to solve the problem.<ref name="ref_0e255cbb" />
# I have just completed my first construction of the regular heptadecagon — a construction that even the ancient Greeks were never able to figure out.<ref name="ref_788dc367">[https://robertlovespi.net/2014/06/11/constructing-the-heptadecagon/ Constructing the Heptadecagon]</ref>
# The regular heptadecagon construction, however, I did not figure out independently.<ref name="ref_788dc367" />
# As a result, the regular heptadecagon is one of the few prime sided figures constructable using an unmarked ruler and pair of compasses - that is using a classical construction.<ref name="ref_07a9fbaa">[https://socratic.org/questions/58655ba911ef6b25b68845fd What is the internal angle of a regular #17#-sided polygon?]</ref>
# 1 2 HUGO TAVARES AND PEDRO J. FREITAS Gauss proved, in his early years, that the 17-gon is constructible.<ref name="ref_3798ab74">[https://arxiv.org/pdf/1507.07970 DIVIDING THE CIRCLE HUGO TAVARES AND PEDRO J. FREITAS]</ref>

===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q542476 Q542476]
===Spacy 패턴 목록===
* [{'LEMMA': 'heptadecagon'}]
* [{'LOWER': '17'}, {'OP': '*'}, {'LEMMA': 'gon'}]
* [{'LEMMA': 'heptakaidecagon'}]

준결정 (quasicrystal)

2022-09-16T10:27:11Z

Pythagoras0: /* 말뭉치 */

==사전 형태의 자료==
* http://ko.wikipedia.org/wiki/준결정

==리뷰, 에세이, 강의노트==
* Faustin Adiceam, Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280 [math-ph], April 16 2016, http://arxiv.org/abs/1604.06280
* Baake, Michael, David Damanik, and Uwe Grimm. “What Is Aperiodic Order?” arXiv:1512.05104 [math-Ph], December 16, 2015. http://arxiv.org/abs/1512.05104.

==관련논문==
* Fang Fang, Klee Irwin, An Icosahedral Quasicrystal as a Golden Modification of the Icosagrid and its Connection to the E8 Lattice, arXiv:1511.07786 [math.MG], November 20 2015, http://arxiv.org/abs/1511.07786
* Michael Baake, David Ecija, Uwe Grimm, A guide to lifting aperiodic structures, arXiv:1606.07647 [cond-mat.mtrl-sci], June 24 2016, http://arxiv.org/abs/1606.07647
* Emilio Zappa, Eric C. Dykeman, James A. Geraets, Reidun Twarock, A group theoretical approach to structural transitions of icosahedral quasicrystals and point arrays, http://arxiv.org/abs/1512.02101v2
* Palamodov, Victor P. “Uniformly Discrete Quasicrystals Are Crystals.” arXiv:1601.07049 [math], January 26, 2016. http://arxiv.org/abs/1601.07049.
* Lev, Nir, and Alexander Olevskii. “Fourier Quasicrystals and Discreteness of the Diffraction Spectrum.” arXiv:1512.08735 [math-Ph], December 29, 2015. http://arxiv.org/abs/1512.08735.
* Bédaride, Nicolas, and Thomas Fernique. “Weak Local Rules for Planar Octagonal Tilings.” arXiv:1512.04679 [math-Ph], December 15, 2015. http://arxiv.org/abs/1512.04679.
* Puelz, Charles, Mark Embree, and Jake Fillman. “Spectral Approximation for Quasiperiodic Jacobi Operators.” arXiv:1408.0370 [math], August 2, 2014. http://arxiv.org/abs/1408.0370.

== 노트 ==

===말뭉치===
# Forming a quasicrystal is a little like tiling a floor.<ref name="ref_dfb84d70">[https://www.brown.edu/news/2018-12-20/quasicrystal Chemists create new quasicrystal material from nanoparticle building blocks]</ref>
# The same goes for this new quasicrystal structure — they require secondary “tiles” that can fill the gaps between decagons.<ref name="ref_dfb84d70" />
# The arrangement of atoms in a quasicrystal displays a property called long-range order, which is lacking in amorphous metals.<ref name="ref_63f3a1e5">[https://www.britannica.com/science/quasicrystal quasicrystal]</ref>
# The 1D quasicrystal is obtained as a section of the decorated periodic lattice by E par : each time the E par line intercepts a segment line, an atomic position is generated.<ref name="ref_95a81cbf">[https://www.sciencedirect.com/topics/materials-science/quasicrystal Quasicrystal - an overview]</ref>
# More complex structures can be generated: for instance, the segment line can be given a longer length; this will generate additional positions in the 1D quasicrystal.<ref name="ref_95a81cbf" />
# Illustration of the 2D description of a 1D quasicrystal, here the Fibonacci chain (see text).<ref name="ref_95a81cbf" />
# This procedure generalizes to the case of a 3D quasicrystal such as icosahedral phases.<ref name="ref_95a81cbf" />
# In a quasicrystal, imagine atoms are at the points of the objects you’re using.<ref name="ref_c6929e49">[https://www.pbs.org/newshour/science/quasicrystals-win-chemistry-nobel What are Quasicrystals, and What Makes Them Nobel-Worthy?]</ref>
# The first synthetic quasicrystal was grown in the lab in 1982, and there are now more than 100 types of lab-grown ones.<ref name="ref_e2fa2383">[https://www.newscientist.com/article/2115570-third-ever-natural-quasicrystal-found-in-siberian-meteorite/ Third-ever natural quasicrystal found in Siberian meteorite]</ref>
# The new quasicrystal has a similar molecular structure to the first one, but slightly different chemistry: both are made of aluminium, copper and iron, but in different proportions.<ref name="ref_e2fa2383" />
# With the composition of this new quasicrystal in hand, it should be easy to synthesise it.<ref name="ref_e2fa2383" />
# A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.<ref name="ref_0e9191f1">[https://en.wikipedia.org/wiki/Quasicrystal#:~:text=A%20quasiperiodic%20crystal%2C%20or%20quasicrystal,but%20it%20lacks%20translational%20symmetry. Quasicrystal]</ref>
# In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures.<ref name="ref_0e9191f1" />
# This quasicrystal, with a composition of Al 63 Cu 24 Fe 13 , was named icosahedrite and it was approved by the International Mineralogical Association in 2010.<ref name="ref_0e9191f1" />
# A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al 71 Ni 24 Fe 5 .<ref name="ref_0e9191f1" />
# He didn’t know it yet, but he had just discovered the first quasicrystal.<ref name="ref_ad945b07">[https://www.nature.com/articles/d41586-019-00026-y Quasicrystals: the thrill of the chase]</ref>
# Steel hardened by small quasicrystal particles is used in needles for acupuncture and surgery, dental instruments and razor blades.<ref name="ref_ad945b07" />
# Perhaps, he surmised, one of these was a misidentified quasicrystal.<ref name="ref_ad945b07" />
# On 2 January 2009, the researchers became certain that they had discovered a natural quasicrystal (later named icosahedrite).<ref name="ref_ad945b07" />
# A Ho-Mg-Zn icosahedral quasicrystal formed as a dodecahedron, the dual of the icosahedron.<ref name="ref_122643af">[https://en.wiktionary.org/wiki/quasicrystal quasicrystal]</ref>
# The force of the Trinity test had forged a new quasicrystal.<ref name="ref_5419c0fc">[https://www.merriam-webster.com/dictionary/quasicrystal Quasicrystal Definition & Meaning]</ref>
# The data show that electron momenta and energies are correlated with the structure of the quasicrystal.<ref name="ref_61bba954">[https://www2.lbl.gov/Science-Articles/Archive/quasicrystal-states.html Quasicrystal Electronic State Studies]</ref>
# IN THE PLANE, THE AlNiCo QUASICRYSTAL, WHICH CONSISTS OF OVERLAPPING DECAGONS, IS APERIODIC.<ref name="ref_61bba954" />
# Peter Gille of the Ludwig-Maximilians-University, Munich, grew the quasicrystal, and the samples were prepared and characterized by Horn and by Wolfgang Theis of the Free University of Berlin.<ref name="ref_61bba954" />
# The electrons aren't localized to clusters, instead they feel the long-range quasicrystal potential," Rotenberg says.<ref name="ref_61bba954" />
# A quasicrystal, however, is a permanent physical fingerprint of the conditions inside the nuclear fireball it formed within.<ref name="ref_b6fd6550">[https://www.discovermagazine.com/the-sciences/the-first-atomic-bomb-created-this-forbidden-quasicrystal The First Atomic Bomb Created This ‘Forbidden’ Quasicrystal]</ref>
# As scientists whip up more elemental combinations in a lab or uncover them in remote atomic blast sites, they may stumble across a quasicrystal useful for all sorts of applications, he says.<ref name="ref_b6fd6550" />
# So, by examining all of the polygons, you can locate the verticies of your quasicrystal.<ref name="ref_440e2f00">[http://www.physics.emory.edu/~weeks/software/exquasi.html How to make a quasicrystal]</ref>
# Each polygon corresponds to a vertex of the quasicrystal you are trying to make.<ref name="ref_440e2f00" />
# In fact, the indices of these four polygons are very similar, and these four polygons correspond to the four vertices of a single tile in your quasicrystal.<ref name="ref_440e2f00" />
# Since in principle these imaginary sets of lines can be infinitely big, the program stops at some arbitrary point (basically some point past when the page is filled with a quasicrystal).<ref name="ref_440e2f00" />
# For one, they realized a 2D quasicrystal optical lattice tuned far from any internal atomic resonance, reducing problematic atom-light scattering effects.<ref name="ref_63c054fc">[https://physics.aps.org/articles/v12/31 A Quasicrystal for Quantum Simulations]</ref>
# Schneider’s team also showed that the time evolution of the BEC on the quasicrystal is quite distinct from that on a periodic lattice.<ref name="ref_63c054fc" />
# The successive population of these smaller momentum states constitutes a quantum walk in the quasicrystal’s momentum space.<ref name="ref_63c054fc" />
# In the present work, Schneider and colleagues interpret their 2D quasicrystal as an incommensurate projection of a 4D cubic lattice onto a 2D plane.<ref name="ref_63c054fc" />
# This was the discovery (later recognized by Nobel Prize) of a "quasicrystal" (QC), a curious solid that shows long-range ordering similar to crystals but lacks their periodicity.<ref name="ref_3452f0db">[https://www.tus.ac.jp/en/mediarelations/archive/20211119_0222.html Clear as (Quasi) Crystal: Scientists Discover the First Ferromagnetic Quasicrystals]</ref>
# This enables us to describe the whole quasicrystal structure with a finite set of parameters.<ref name="ref_3015322b">[http://www.jcrystal.com/steffenweber/qc.html Introduction to Quasicrystals]</ref>
# After the discovery of quasicrystals in 1984 a close resemblance was noted between the icosahedral quasicrystal and the 3D-Penrose pattern.<ref name="ref_3015322b" />
# By putting atoms at the vertices of a 3D-Penrose pattern one can obtain a Fourier Transform which explains very well the diffraction patterns of the found Al-Mn quasicrystal.<ref name="ref_3015322b" />
# However, the structural solution of the quasicrystal is still under debate and it will provide a broad aspect for future development and guide the investigations of different aspects of quasicrystals.<ref name="ref_10b51f7b">[https://medcraveonline.com/MSEIJ/quasicrystal-a-beautiful-morphology-and-diffraction-pattern.html Quasicrystal: a beautiful morphology and diffraction pattern]</ref>
# The diffraction patterns simulated from these structures are startlingly close to those observed for the icosahederal phase and this phase was termed as a quasicrystal.<ref name="ref_10b51f7b" />
# The transmission electron microscopy selected area electron diffraction pattern form Al-Cu-Fe icosahedral quasicrystal.<ref name="ref_10b51f7b" />
# Moreover, the quantum quasicrystal patterns are found to emerge as the ground state with no need of moderate thermal uctuations.<ref name="ref_98408bf0">[https://arxiv.org/pdf/2110.12299?context=cond-mat.soft Exploring quantum quasicrystal patterns: a variational study A. Mendoza-Coto,1, ∗ R. Turcati,1 V. Zampronio,2 R. D´ıaz-M´endez,3, 4 T. Macr`ı,5 and F. Cinti6, 7, 8, †]</ref>
# Our calculations show that, in an intermediate region between the homogeneous superuid and the normal quasicrystal phases, these exotic states indeed exist at zero temperature.<ref name="ref_98408bf0" />
# Yet, by increasing uctuations, a structural transition from quasicrystal to cluster triangular crystal takes place.<ref name="ref_98408bf0" />
# The blue curve separates the homogeneous from the dodecagonal cluster quasicrystal phase.<ref name="ref_98408bf0" />
# It's been nearly four decades since he set out to convince the chemist community of a discovery most considered impossible – a material called a quasicrystal.<ref name="ref_23666aed">[https://www.sciencealert.com/quasicrystals-were-once-impossible-we-re-still-finding-new-ways-to-make-them Physicists Just Created a Strange New Type of 'Quasicrystal' in The Lab]</ref>
# "It's a fundamentally new type of quasicrystal, and we've been able to figure out the rules for making it, which will be useful in the continued study of quasicrystal structures.<ref name="ref_23666aed" />
# A quasicrystal is an aperiodic crystal that is not an incommensurate modulated structure, nor an aperiodic composite crystal.<ref name="ref_fdab6711">[https://dictionary.iucr.org/Quasicrystal Online Dictionary of Crystallography]</ref>
# However, presence of such a forbidden symmetry is not required for a quasicrystal.<ref name="ref_fdab6711" />
# The term quasicrystal stems from the property of quasiperiodicity observed for the first alloys found with forbidden symmetries.<ref name="ref_fdab6711" />
# The material was a quasicrystal, a solid in which atoms could exist in stable patterns of peculiar irregular symmetry.<ref name="ref_3ea0bc4e">[https://www.cbc.ca/radio/quirks/may-22-solving-our-sand-crisis-nuclear-quasicrystals-voyager-hears-an-interstellar-hum-and-more-1.6035006/world-s-first-nuclear-detonation-forged-the-first-human-made-quasicrystal-1.6035016 World's first nuclear detonation forged the first human-made quasicrystal]</ref>
# His theorizing was soon proved right, as another researcher, Dan Schectman, created a quasicrystal in his lab, a feat that won Shechtman the Nobel Prize in Chemistry in 2011.<ref name="ref_3ea0bc4e" />
# Now in new work they've found a quasicrystal unlike anything ever seen before from ground zero of the Trinity atomic test.<ref name="ref_3ea0bc4e" />
# After analyzing the sample that came from the nuclear blast site, Steinhardt and his colleagues found the quasicrystal had fivefold, threefold and twofold symmetries.<ref name="ref_3ea0bc4e" />
# Other puzzling cases have been reported, but until the concept of quasicrystal came to be established they were explained away or simply denied.<ref name="ref_42005e72">[https://www.chemeurope.com/en/encyclopedia/Quasicrystal.html Quasicrystal]</ref>
# Since 2004 different research groups have reported evidence for quasicrystal ordering in liquids and polymers.<ref name="ref_42005e72" />
# The new quasicrystal, formed of silicon, copper, calcium and iron, is “brand new to science,” says mineralogist Chi Ma of Caltech, who was not involved with the study.<ref name="ref_752d6908">[https://www.sciencenews.org/article/new-quasi-crystal-formed-first-atomic-bomb-test A newfound quasicrystal formed in the first atomic bomb test]</ref>
# Asimow and colleagues hypothesized that the energy released by the shock could have caused the quasicrystal's formation by triggering a rapid cycle of compression, heating, decompression, and cooling.<ref name="ref_12563628">[https://www.caltech.edu/about/news/natural-quasicrystals-may-be-result-collisions-between-objects-asteroid-belt-50984 Natural Quasicrystals May Be the Result of Collisions Between Objects in the Asteroid Belt]</ref>
# And now we know that when you shock the starting materials that were available in that meteorite, you get a quasicrystal.<ref name="ref_12563628" />
# For example, it is unclear at what point the quasicrystal formed during the shock's pressure and temperature cycle.<ref name="ref_12563628" />
# This was something thought to be impossible until their Nobel prize-winning discovery by Dan Schechtman in 1982, with Steinhardt suggesting the name ‘quasicrystal’.<ref name="ref_33420604">[https://www.chemistryworld.com/news/oldest-human-made-quasicrystal-discovered-in-remains-of-first-nuclear-blast/4013708.article Oldest human-made quasicrystal discovered in remains of first nuclear blast]</ref>
# The trinitite sample is likely to be the first quasicrystal humans ever synthesised, albeit unknowingly.<ref name="ref_33420604" />
# At low temperatures motion of atoms within the solid is difficult, and phason strain may be easily frozen into the quasicrystal, limiting its perfection.<ref name="ref_6193386d">[https://www.britannica.com/science/quasicrystal/Properties quasicrystal - Properties]</ref>
# The term quasicrystal should simply be regarded as an abbreviation for quasiperiodic crystal, possibly with two provisos, as discussed below.<ref name="ref_6c743825">[https://arxiv.org/pdf/cond-mat/0008152 The Definition of Quasicrystals RON LIFSHITZ]</ref>
# Here we construct the one-dimensional Anbry-Andre-Harper (AAH) model based on the coupled ring chain structure to reveal diusive quasicrystal.<ref name="ref_43826de8">[http://arxiv.org/pdf/2208.06765 Non-Hermitian Diﬀusive Quasicrystal Zhoufei Liu1 and Jiping Huang1,]</ref>
# The quasicrystal is an ordered but not periodic phase that has received much attention theoretically and experimentally over the last several decades.<ref name="ref_43826de8" />
# Here, we reveal through real-time and 3D imaging the formation of a single decagonal quasicrystal arising from a hard collision between multiple growing quasicrystals in an Al-Co-Ni liquid.<ref name="ref_a9d75b9b">[https://arxiv.org/pdf/2106.14074 Formation of a single quasicrystal upon collision of multiple grains Insung Han1, †, Kelly L. Wang2, †, Andrew T. Cadotte3, Zhucong Xi1, Hadi Parsamehr1,]</ref>
# 6. 2 Structures of quasicrystals Quasicrystal was a special form of solid matter that is ordered but not periodic.<ref name="ref_fc185657">[https://arxiv.org/pdf/2108.01560 TOPOLOGICAL STATES IN QUASICRYSTALS Jiahao Fan]</ref>
# One can ask what the eects of disorder are in a quasicrystal, and how critical states are aected by randomness.<ref name="ref_ed2a6e8c">[https://arxiv.org/pdf/2012.14744 The Fibonacci quasicrystal: case study of hidden dimensions and multifractality]</ref>
# When is irrational, the potential Vn = V cos(2n + ) is quasiperiodic in n, and HAAH describes a 1D quasicrystal.<ref name="ref_249e08bf">[https://arxiv.org/pdf/2105.03302 Non-Hermitian quasicrystal in dimerized lattices Longwen Zhou1, ∗ and Wenqian Han1]</ref>
# In this work, we consider another extension of the AAH quasicrystal by introducing hopping dimerizations.<ref name="ref_249e08bf" />
# DEPLOYABLE QUASICRYSTAL DESIGN Kirigami is a traditional Japanese paper crafting art that has recently become popular among scientists and engineers.<ref name="ref_62f928d3">[https://arxiv.org/pdf/2104.13399 Quasicrystal kirigami Lucy Liu,1, ∗ Gary P. T. Choi,2, ∗ and L. Mahadevan3, 4, †]</ref>
# Here we show that we can achieve deployable symmetry-preserving patterns, with the special quasicrystal rotation orders preserved upon deployment in all three approaches.<ref name="ref_62f928d3" />
# Deployable quasicrystal patterns created using the expansion tile method.<ref name="ref_62f928d3" />
# B. The tile removal method Our second approach for achieving deployability is re- moving tiles from a given quasicrystal pattern, changing the lattice connectivity and introducing negative space.<ref name="ref_62f928d3" />

===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q263214 Q263214]
===Spacy 패턴 목록===
* [{'LOWER': 'quasicrystal'}]

준결정 (quasicrystal)

2022-09-16T10:25:56Z

Pythagoras0: /* 메타데이터 */ 새 문단

==사전 형태의 자료==
* http://ko.wikipedia.org/wiki/준결정

==리뷰, 에세이, 강의노트==
* Faustin Adiceam, Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280 [math-ph], April 16 2016, http://arxiv.org/abs/1604.06280
* Baake, Michael, David Damanik, and Uwe Grimm. “What Is Aperiodic Order?” arXiv:1512.05104 [math-Ph], December 16, 2015. http://arxiv.org/abs/1512.05104.

==관련논문==
* Fang Fang, Klee Irwin, An Icosahedral Quasicrystal as a Golden Modification of the Icosagrid and its Connection to the E8 Lattice, arXiv:1511.07786 [math.MG], November 20 2015, http://arxiv.org/abs/1511.07786
* Michael Baake, David Ecija, Uwe Grimm, A guide to lifting aperiodic structures, arXiv:1606.07647 [cond-mat.mtrl-sci], June 24 2016, http://arxiv.org/abs/1606.07647
* Emilio Zappa, Eric C. Dykeman, James A. Geraets, Reidun Twarock, A group theoretical approach to structural transitions of icosahedral quasicrystals and point arrays, http://arxiv.org/abs/1512.02101v2
* Palamodov, Victor P. “Uniformly Discrete Quasicrystals Are Crystals.” arXiv:1601.07049 [math], January 26, 2016. http://arxiv.org/abs/1601.07049.
* Lev, Nir, and Alexander Olevskii. “Fourier Quasicrystals and Discreteness of the Diffraction Spectrum.” arXiv:1512.08735 [math-Ph], December 29, 2015. http://arxiv.org/abs/1512.08735.
* Bédaride, Nicolas, and Thomas Fernique. “Weak Local Rules for Planar Octagonal Tilings.” arXiv:1512.04679 [math-Ph], December 15, 2015. http://arxiv.org/abs/1512.04679.
* Puelz, Charles, Mark Embree, and Jake Fillman. “Spectral Approximation for Quasiperiodic Jacobi Operators.” arXiv:1408.0370 [math], August 2, 2014. http://arxiv.org/abs/1408.0370.

== 노트 ==

===말뭉치===
# Forming a quasicrystal is a little like tiling a floor.<ref name="ref_dfb84d70">[https://www.brown.edu/news/2018-12-20/quasicrystal Chemists create new quasicrystal material from nanoparticle building blocks]</ref>
# The same goes for this new quasicrystal structure — they require secondary “tiles” that can fill the gaps between decagons.<ref name="ref_dfb84d70" />
# The arrangement of atoms in a quasicrystal displays a property called long-range order, which is lacking in amorphous metals.<ref name="ref_63f3a1e5">[https://www.britannica.com/science/quasicrystal quasicrystal]</ref>
# The 1D quasicrystal is obtained as a section of the decorated periodic lattice by E par : each time the E par line intercepts a segment line, an atomic position is generated.<ref name="ref_95a81cbf">[https://www.sciencedirect.com/topics/materials-science/quasicrystal Quasicrystal - an overview]</ref>
# More complex structures can be generated: for instance, the segment line can be given a longer length; this will generate additional positions in the 1D quasicrystal.<ref name="ref_95a81cbf" />
# Illustration of the 2D description of a 1D quasicrystal, here the Fibonacci chain (see text).<ref name="ref_95a81cbf" />
# This procedure generalizes to the case of a 3D quasicrystal such as icosahedral phases.<ref name="ref_95a81cbf" />
# In a quasicrystal, imagine atoms are at the points of the objects you’re using.<ref name="ref_c6929e49">[https://www.pbs.org/newshour/science/quasicrystals-win-chemistry-nobel What are Quasicrystals, and What Makes Them Nobel-Worthy?]</ref>
# The first synthetic quasicrystal was grown in the lab in 1982, and there are now more than 100 types of lab-grown ones.<ref name="ref_e2fa2383">[https://www.newscientist.com/article/2115570-third-ever-natural-quasicrystal-found-in-siberian-meteorite/ Third-ever natural quasicrystal found in Siberian meteorite]</ref>
# The new quasicrystal has a similar molecular structure to the first one, but slightly different chemistry: both are made of aluminium, copper and iron, but in different proportions.<ref name="ref_e2fa2383" />
# With the composition of this new quasicrystal in hand, it should be easy to synthesise it.<ref name="ref_e2fa2383" />
# A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.<ref name="ref_0e9191f1">[https://en.wikipedia.org/wiki/Quasicrystal#:~:text=A%20quasiperiodic%20crystal%2C%20or%20quasicrystal,but%20it%20lacks%20translational%20symmetry. Quasicrystal]</ref>
# In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures.<ref name="ref_0e9191f1" />
# This quasicrystal, with a composition of Al 63 Cu 24 Fe 13 , was named icosahedrite and it was approved by the International Mineralogical Association in 2010.<ref name="ref_0e9191f1" />
# A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al 71 Ni 24 Fe 5 .<ref name="ref_0e9191f1" />
# He didn’t know it yet, but he had just discovered the first quasicrystal.<ref name="ref_ad945b07">[https://www.nature.com/articles/d41586-019-00026-y Quasicrystals: the thrill of the chase]</ref>
# Steel hardened by small quasicrystal particles is used in needles for acupuncture and surgery, dental instruments and razor blades.<ref name="ref_ad945b07" />
# Perhaps, he surmised, one of these was a misidentified quasicrystal.<ref name="ref_ad945b07" />
# On 2 January 2009, the researchers became certain that they had discovered a natural quasicrystal (later named icosahedrite).<ref name="ref_ad945b07" />
# A Ho-Mg-Zn icosahedral quasicrystal formed as a dodecahedron, the dual of the icosahedron.<ref name="ref_122643af">[https://en.wiktionary.org/wiki/quasicrystal quasicrystal]</ref>
# The force of the Trinity test had forged a new quasicrystal.<ref name="ref_5419c0fc">[https://www.merriam-webster.com/dictionary/quasicrystal Quasicrystal Definition & Meaning]</ref>
# The data show that electron momenta and energies are correlated with the structure of the quasicrystal.<ref name="ref_61bba954">[https://www2.lbl.gov/Science-Articles/Archive/quasicrystal-states.html Quasicrystal Electronic State Studies]</ref>
# IN THE PLANE, THE AlNiCo QUASICRYSTAL, WHICH CONSISTS OF OVERLAPPING DECAGONS, IS APERIODIC.<ref name="ref_61bba954" />
# Peter Gille of the Ludwig-Maximilians-University, Munich, grew the quasicrystal, and the samples were prepared and characterized by Horn and by Wolfgang Theis of the Free University of Berlin.<ref name="ref_61bba954" />
# The electrons aren't localized to clusters, instead they feel the long-range quasicrystal potential," Rotenberg says.<ref name="ref_61bba954" />
# A quasicrystal, however, is a permanent physical fingerprint of the conditions inside the nuclear fireball it formed within.<ref name="ref_b6fd6550">[https://www.discovermagazine.com/the-sciences/the-first-atomic-bomb-created-this-forbidden-quasicrystal The First Atomic Bomb Created This ‘Forbidden’ Quasicrystal]</ref>
# As scientists whip up more elemental combinations in a lab or uncover them in remote atomic blast sites, they may stumble across a quasicrystal useful for all sorts of applications, he says.<ref name="ref_b6fd6550" />
# So, by examining all of the polygons, you can locate the verticies of your quasicrystal.<ref name="ref_440e2f00">[http://www.physics.emory.edu/~weeks/software/exquasi.html How to make a quasicrystal]</ref>
# Each polygon corresponds to a vertex of the quasicrystal you are trying to make.<ref name="ref_440e2f00" />
# In fact, the indices of these four polygons are very similar, and these four polygons correspond to the four vertices of a single tile in your quasicrystal.<ref name="ref_440e2f00" />
# Since in principle these imaginary sets of lines can be infinitely big, the program stops at some arbitrary point (basically some point past when the page is filled with a quasicrystal).<ref name="ref_440e2f00" />
# For one, they realized a 2D quasicrystal optical lattice tuned far from any internal atomic resonance, reducing problematic atom-light scattering effects.<ref name="ref_63c054fc">[https://physics.aps.org/articles/v12/31 A Quasicrystal for Quantum Simulations]</ref>
# Schneider’s team also showed that the time evolution of the BEC on the quasicrystal is quite distinct from that on a periodic lattice.<ref name="ref_63c054fc" />
# The successive population of these smaller momentum states constitutes a quantum walk in the quasicrystal’s momentum space.<ref name="ref_63c054fc" />
# In the present work, Schneider and colleagues interpret their 2D quasicrystal as an incommensurate projection of a 4D cubic lattice onto a 2D plane.<ref name="ref_63c054fc" />
# This was the discovery (later recognized by Nobel Prize) of a "quasicrystal" (QC), a curious solid that shows long-range ordering similar to crystals but lacks their periodicity.<ref name="ref_3452f0db">[https://www.tus.ac.jp/en/mediarelations/archive/20211119_0222.html Clear as (Quasi) Crystal: Scientists Discover the First Ferromagnetic Quasicrystals]</ref>
# This enables us to describe the whole quasicrystal structure with a finite set of parameters.<ref name="ref_3015322b">[http://www.jcrystal.com/steffenweber/qc.html Introduction to Quasicrystals]</ref>
# After the discovery of quasicrystals in 1984 a close resemblance was noted between the icosahedral quasicrystal and the 3D-Penrose pattern.<ref name="ref_3015322b" />
# By putting atoms at the vertices of a 3D-Penrose pattern one can obtain a Fourier Transform which explains very well the diffraction patterns of the found Al-Mn quasicrystal.<ref name="ref_3015322b" />
# However, the structural solution of the quasicrystal is still under debate and it will provide a broad aspect for future development and guide the investigations of different aspects of quasicrystals.<ref name="ref_10b51f7b">[https://medcraveonline.com/MSEIJ/quasicrystal-a-beautiful-morphology-and-diffraction-pattern.html Quasicrystal: a beautiful morphology and diffraction pattern]</ref>
# The diffraction patterns simulated from these structures are startlingly close to those observed for the icosahederal phase and this phase was termed as a quasicrystal.<ref name="ref_10b51f7b" />
# The transmission electron microscopy selected area electron diffraction pattern form Al-Cu-Fe icosahedral quasicrystal.<ref name="ref_10b51f7b" />
# Moreover, the quantum quasicrystal patterns are found to emerge as the ground state with no need of moderate thermal uctuations.<ref name="ref_98408bf0">[https://arxiv.org/pdf/2110.12299?context=cond-mat.soft Exploring quantum quasicrystal patterns: a variational study A. Mendoza-Coto,1, ∗ R. Turcati,1 V. Zampronio,2 R. D´ıaz-M´endez,3, 4 T. Macr`ı,5 and F. Cinti6, 7, 8, †]</ref>
# Our calculations show that, in an intermediate region between the homogeneous superuid and the normal quasicrystal phases, these exotic states indeed exist at zero temperature.<ref name="ref_98408bf0" />
# Yet, by increasing uctuations, a structural transition from quasicrystal to cluster triangular crystal takes place.<ref name="ref_98408bf0" />
# The blue curve separates the homogeneous from the dodecagonal cluster quasicrystal phase.<ref name="ref_98408bf0" />
# It's been nearly four decades since he set out to convince the chemist community of a discovery most considered impossible – a material called a quasicrystal.<ref name="ref_23666aed">[https://www.sciencealert.com/quasicrystals-were-once-impossible-we-re-still-finding-new-ways-to-make-them Physicists Just Created a Strange New Type of 'Quasicrystal' in The Lab]</ref>
# "It's a fundamentally new type of quasicrystal, and we've been able to figure out the rules for making it, which will be useful in the continued study of quasicrystal structures.<ref name="ref_23666aed" />
# A quasicrystal is an aperiodic crystal that is not an incommensurate modulated structure, nor an aperiodic composite crystal.<ref name="ref_fdab6711">[https://dictionary.iucr.org/Quasicrystal Online Dictionary of Crystallography]</ref>
# However, presence of such a forbidden symmetry is not required for a quasicrystal.<ref name="ref_fdab6711" />
# The term quasicrystal stems from the property of quasiperiodicity observed for the first alloys found with forbidden symmetries.<ref name="ref_fdab6711" />
# The material was a quasicrystal, a solid in which atoms could exist in stable patterns of peculiar irregular symmetry.<ref name="ref_3ea0bc4e">[https://www.cbc.ca/radio/quirks/may-22-solving-our-sand-crisis-nuclear-quasicrystals-voyager-hears-an-interstellar-hum-and-more-1.6035006/world-s-first-nuclear-detonation-forged-the-first-human-made-quasicrystal-1.6035016 World's first nuclear detonation forged the first human-made quasicrystal]</ref>
# His theorizing was soon proved right, as another researcher, Dan Schectman, created a quasicrystal in his lab, a feat that won Shechtman the Nobel Prize in Chemistry in 2011.<ref name="ref_3ea0bc4e" />
# Now in new work they've found a quasicrystal unlike anything ever seen before from ground zero of the Trinity atomic test.<ref name="ref_3ea0bc4e" />
# After analyzing the sample that came from the nuclear blast site, Steinhardt and his colleagues found the quasicrystal had fivefold, threefold and twofold symmetries.<ref name="ref_3ea0bc4e" />
# Other puzzling cases have been reported, but until the concept of quasicrystal came to be established they were explained away or simply denied.<ref name="ref_42005e72">[https://www.chemeurope.com/en/encyclopedia/Quasicrystal.html Quasicrystal]</ref>
# Since 2004 different research groups have reported evidence for quasicrystal ordering in liquids and polymers.<ref name="ref_42005e72" />
# The new quasicrystal, formed of silicon, copper, calcium and iron, is “brand new to science,” says mineralogist Chi Ma of Caltech, who was not involved with the study.<ref name="ref_752d6908">[https://www.sciencenews.org/article/new-quasi-crystal-formed-first-atomic-bomb-test A newfound quasicrystal formed in the first atomic bomb test]</ref>
# Asimow and colleagues hypothesized that the energy released by the shock could have caused the quasicrystal's formation by triggering a rapid cycle of compression, heating, decompression, and cooling.<ref name="ref_12563628">[https://www.caltech.edu/about/news/natural-quasicrystals-may-be-result-collisions-between-objects-asteroid-belt-50984 Natural Quasicrystals May Be the Result of Collisions Between Objects in the Asteroid Belt]</ref>
# And now we know that when you shock the starting materials that were available in that meteorite, you get a quasicrystal.<ref name="ref_12563628" />
# For example, it is unclear at what point the quasicrystal formed during the shock's pressure and temperature cycle.<ref name="ref_12563628" />
# This was something thought to be impossible until their Nobel prize-winning discovery by Dan Schechtman in 1982, with Steinhardt suggesting the name ‘quasicrystal’.<ref name="ref_33420604">[https://www.chemistryworld.com/news/oldest-human-made-quasicrystal-discovered-in-remains-of-first-nuclear-blast/4013708.article Oldest human-made quasicrystal discovered in remains of first nuclear blast]</ref>
# The trinitite sample is likely to be the first quasicrystal humans ever synthesised, albeit unknowingly.<ref name="ref_33420604" />
# At low temperatures motion of atoms within the solid is difficult, and phason strain may be easily frozen into the quasicrystal, limiting its perfection.<ref name="ref_6193386d">[https://www.britannica.com/science/quasicrystal/Properties quasicrystal - Properties]</ref>
# The term quasicrystal should simply be regarded as an abbreviation for quasiperiodic crystal, possibly with two provisos, as discussed below.<ref name="ref_6c743825">[https://arxiv.org/pdf/cond-mat/0008152 The Definition of Quasicrystals RON LIFSHITZ]</ref>
# Here we construct the one-dimensional Anbry-Andre-Harper (AAH) model based on the coupled ring chain structure to reveal diusive quasicrystal.<ref name="ref_43826de8">[http://arxiv.org/pdf/2208.06765 Non-Hermitian Diﬀusive Quasicrystal Zhoufei Liu1 and Jiping Huang1,]</ref>
# The quasicrystal is an ordered but not periodic phase that has received much attention theoretically and experimentally over the last several decades.<ref name="ref_43826de8" />
# Here, we reveal through real-time and 3D imaging the formation of a single decagonal quasicrystal arising from a hard collision between multiple growing quasicrystals in an Al-Co-Ni liquid.<ref name="ref_a9d75b9b">[https://arxiv.org/pdf/2106.14074 Formation of a single quasicrystal upon collision of multiple grains Insung Han1, †, Kelly L. Wang2, †, Andrew T. Cadotte3, Zhucong Xi1, Hadi Parsamehr1,]</ref>
# 6. 2 Structures of quasicrystals Quasicrystal was a special form of solid matter that is ordered but not periodic.<ref name="ref_fc185657">[https://arxiv.org/pdf/2108.01560 TOPOLOGICAL STATES IN QUASICRYSTALS Jiahao Fan]</ref>
# One can ask what the eects of disorder are in a quasicrystal, and how critical states are aected by randomness.<ref name="ref_ed2a6e8c">[https://arxiv.org/pdf/2012.14744 The Fibonacci quasicrystal: case study of hidden dimensions and multifractality]</ref>
# When is irrational, the potential Vn = V cos(2n + ) is quasiperiodic in n, and HAAH describes a 1D quasicrystal.<ref name="ref_249e08bf">[https://arxiv.org/pdf/2105.03302 Non-Hermitian quasicrystal in dimerized lattices Longwen Zhou1, ∗ and Wenqian Han1]</ref>
# In this work, we consider another extension of the AAH quasicrystal by introducing hopping dimerizations.<ref name="ref_249e08bf" />
# DEPLOYABLE QUASICRYSTAL DESIGN Kirigami is a traditional Japanese paper crafting art that has recently become popular among scientists and engineers.<ref name="ref_62f928d3">[https://arxiv.org/pdf/2104.13399 Quasicrystal kirigami Lucy Liu,1, ∗ Gary P. T. Choi,2, ∗ and L. Mahadevan3, 4, †]</ref>
# Here we show that we can achieve deployable symmetry-preserving patterns, with the special quasicrystal rotation orders preserved upon deployment in all three approaches.<ref name="ref_62f928d3" />
# Deployable quasicrystal patterns created using the expansion tile method.<ref name="ref_62f928d3" />
# B. The tile removal method Our second approach for achieving deployability is re- moving tiles from a given quasicrystal pattern, changing the lattice connectivity and introducing negative space.<ref name="ref_62f928d3" />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q263214 Q263214]
===Spacy 패턴 목록===
* [{'LOWER': 'quasicrystal'}]

준결정 (quasicrystal)

2022-09-16T10:25:55Z

Pythagoras0: /* 노트 */ 새 문단

타원적분

2022-09-16T10:19:44Z

Pythagoras0:

==개요==

* 먼저 [[타원적분론 입문]] 참조
* <math>R(x,y)</math>는 <math>x,y</math>의 유리함수이고, <math>y^2</math>은 <math>x</math>의 3차 또는 4차식:<math>\int R(x,\sqrt{ax^3+bx^2+cx+d}) \,dx</math> 또는:<math>\int R(x,\sqrt{ax^4+bx^3+cx^2+dx+e}) \,dx</math>

==타원 둘레의 길이==

* 역사적으로 [[타원 둘레의 길이]]를 구하는 적분에서 그 이름이 기원함.
* 타원 <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>의 둘레의 길이는 <math>4aE(k)</math> 로 주어짐.:<math>k=\sqrt{1-\frac{b^2}{a^2}}</math>:<math>E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>

==정의==

* 일반적으로 다음과 같은 형태로 주어지는 적분을 타원적분이라 부름
:<math>\int R(x,y)\,dx</math>
여기서 <math>R(x,y)</math>는 <math>x,y</math>의 유리함수, <math>y^2</math>= 중근을 갖지 않는 <math>x</math>의 3차식 또는 4차식.

* 예를 들자면,
:<math>\int \frac{dx}{\sqrt{1-x^4}}</math>
:<math>\int \frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>

==일종타원적분과 이종타원적분==

* [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}=\int_0^1\frac{1}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math>
* [[제2종타원적분 E (complete elliptic integral of the second kind)]]:<math>E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>:<math>E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)</math>
* <math>\,_2F_1(a,b;c;z)</math>는 [[초기하급수(Hypergeometric series)]]

==르장드르의 항등식==

* 일종타원적분과 이종타원적분 사이에는 다음과 같은 관계가 성립
:<math>E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}</math>

또는 <math>\theta+\phi=\frac{\pi}{2}</math> 에 대하여
:<math>E(\sin\theta)K(\sin\phi)+E(\sin\phi)K(\sin\theta)-K(\sin\theta)K(\sin\phi)=\frac{\pi}{2}</math>

* 특별히 다음과 같은 관계가 성립함
:<math>2K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})-K(\frac{1}{\sqrt{2}})^2=\frac{\pi}{2}</math>

[[산술기하평균함수(AGM)와 파이값의 계산]]에 응용

==덧셈공식==

* 파그나노의 공식 ([[렘니스케이트 곡선과 Lemniscatomy]] 항목 참조)
:<math>\int_0^x{\frac{1}{\sqrt{1-x^4}}}dx+\int_0^y{\frac{1}{\sqrt{1-x^4}}}dx = \int_0^{A(x,y)}{\frac{1}{\sqrt{1-x^4}}}dx</math>
여기서 <math>A(x,y)=\frac{x\sqrt{1-y^4}+y\sqrt{1-x^4}}{1+x^2y^2}</math>
* 오일러의 일반화
<math>p(x)=1+mx^2+nx^4</math>일 때,
:<math>\int_0^x{\frac{1}{\sqrt{p(x)}}}dx+\int_0^y{\frac{1}{\sqrt{p(x)}}}dx = \int_0^{B(x,y)}{\frac{1}{\sqrt{p(x)}}}dx</math>
여기서
:<math>B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}</math>

==메모==

* 타원적분의 응용으로 [[단진자의 주기와 타원적분]] 항목 참조

==관련된 항목들==

* [[타원곡선]]
* [[타원함수]]
** [[바이어슈트라스 타원함수 ℘]]
* [[자코비 세타함수]]
* [[초기하급수(Hypergeometric series)]]
* [[대수적 함수와 아벨적분]]
* [[오일러 치환|오일러치환]]

==사전 형태의 자료==

* http://ko.wikipedia.org/wiki/타원적분
* http://en.wikipedia.org/wiki/Elliptic_integral
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
** [http://dlmf.nist.gov/19 Chapter 19 Elliptic Integrals]

==관련논문==

* [http://www.springerlink.com/content/b365w3511067g184/ In Search of the "Birthday" of Elliptic Functions - Bit by bit, the discoverers decided what it was they had discovered.]
** Rice, Adrian, 48-57, The Mathematical Intelligencer, Volume 30, Number 2, 2008-3
* Totaro, Burt. 2007. “Euler and Algebraic Geometry.” Bulletin of the American Mathematical Society 44 (4): 541–559. doi:[http://dx.doi.org/10.1090/S0273-0979-07-01178-0 10.1090/S0273-0979-07-01178-0].
* [http://www.springerlink.com/content/t32h69374h887w33/ The Lemniscate and Fagnano's Contributions to Elliptic Integrals]
** AYOUB R
* [http://www.math.tulane.edu/%7Evhm/papers_html/EU.pdf A property of Euler's elastic curve]
* [http://www.springerlink.com/content/911pnwauaeggxk13/ The story of Landen, the hyperbola and the ellipse]
** Elemente der Mathematik, Volume 57, Number 1 / 2002년 2월
* [http://www.jstor.org/stable/2687483 Three Fermat Trails to Elliptic Curves]
** Ezra Brown, <cite style="line-height: 2em;">The College Mathematics Journal</cite>, Vol. 31, No. 3 (May, 2000), pp. 162-172
* [http://www.jstor.org/stable/2974515 Elliptic Curves]
** John Stillwell, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
* [http://www.jstor.org/stable/2321821 Abel's Theorem on the Lemniscate]
** Michael Rosen, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 88, No. 6 (Jun. - Jul., 1981), pp. 387-395

==관련도서==

* [http://www.amazon.com/Functions-Integrals-Translations-Mathematical-Monographs/dp/0821805878 Elliptic functions and elliptic integrals]
** Viktor Prasolov, Yuri Solovyev
* [http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM]
** Jonathan M. Borwein, Peter B. Borwein

==블로그==

* [http://bomber0.byus.net/index.php/2009/08/19/1428 삼각치환에서 타원적분으로] 피타고라스의 창, 2009-8-19
[[분류:리만곡면론]]
[[분류:특수함수]]

==메타데이터==
===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1126603 Q1126603]
===Spacy 패턴 목록===
* [{'LOWER': 'elliptic'}, {'LOWER': 'integral'}]

== 노트 ==

===말뭉치===
# For instance, while the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral.<ref name="ref_156bfa3e">[https://mathworld.wolfram.com/EllipticIntegral.html Elliptic Integral -- from Wolfram MathWorld]</ref>
# An elliptic integral is written when the parameter is used, when the elliptic modulus is used, and when the modular angle is used.<ref name="ref_156bfa3e" />
# Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping.<ref name="ref_f2288109">[https://en.wikipedia.org/wiki/Elliptic_integral Elliptic integral]</ref>
# These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral).<ref name="ref_f2288109" />
# This is referred to as the incomplete Legendre elliptic integral.<ref name="ref_8a03a90c">[http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap3.pdf Elliptic integrals, elliptic functions and]</ref>
# The integral is also called Legendres form for the elliptic integral of the first kind.<ref name="ref_728c7c82">[https://www.pearson.com/content/dam/one-dot-com/one-dot-com/us/en/files/Jay-Villanuevaictcm3013.pdf Elliptic integrals and some applications]</ref>
# = 0 (1+2)122 , 0 < < 1, 0, also called Legendres form for the elliptic integral of the third kind.<ref name="ref_728c7c82" />
# The upper limit x in the Jacobi form of the elliptic integral of the first kind is related to the upper limit in the Legendre form by = sin .<ref name="ref_728c7c82" />
# returns values of the complete elliptic integral E(K).<ref name="ref_13b5d81f">[https://people.sc.fsu.edu/~jburkardt/f77_src/elliptic_integral/elliptic_integral.html Elliptic Integrals]</ref>
# returns values of the complete elliptic integral E(M).<ref name="ref_13b5d81f" />
# returns values of the complete elliptic integral F(K).<ref name="ref_13b5d81f" />
# returns values of the complete elliptic integral F(M).<ref name="ref_13b5d81f" />
# Any elliptic integral can be expressed as a sum of elementary functions and linear combinations of canonical elliptic integrals of the first, second and third kinds.<ref name="ref_053ce1aa">[https://encyclopediaofmath.org/wiki/Elliptic_integral Encyclopedia of Mathematics]</ref>
# TOMS577, a C++ library which evaluates Carlson's elliptic integral functions RC, RD, RF and RJ.<ref name="ref_adbb6bc9">[https://people.math.sc.edu/Burkardt/cpp_src/elliptic_integral/elliptic_integral.html Elliptic Integrals]</ref>
# The point here is simply show one of the uses of the elliptic integral of the first kind.<ref name="ref_957e1726">[https://www.codeproject.com/Articles/566614/Elliptic-integrals Elliptic integrals]</ref>
# However, the complete elliptic integral is taken from 0 to 90 degrees, but since that is exactly 1/4 over the total arc length due to symmetry.<ref name="ref_957e1726" />
# Complete elliptic integral of the second kind Math.<ref name="ref_a0c3c644">[https://github.com/duetosymmetry/elliptic-integrals-js duetosymmetry/elliptic-integrals-js: Complete elliptic integrals in javascript]</ref>
# Complete elliptic integral of the first kind.<ref name="ref_2f73b86e">[https://solitaryroad.com/c684.html Elliptic integrals of the first, second and third kinds. Jacobian elliptic functions. Identities, formulas, series expansions, derivatives, integrals.]</ref>
# Complete elliptic integral of the third kind.<ref name="ref_2f73b86e" />
# Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively.<ref name="ref_4885c2a7">[https://arxiv.org/pdf/0801.4813 Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function]</ref>
# dy 1 + a cos x + b cos y (1) Keywords: Double elliptic integral, hypergeometric function 1 .<ref name="ref_ce2c00f8">[https://arxiv.org/pdf/0709.1289 A two-parameter generalization of the complete elliptic integral of the second kind M. L. Glasser]</ref>
# v i X r a SHARP DOUBLE INEQUALITY FOR COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND QI BAO r2 sin2 t)1/2dt is known as the Abstract.<ref name="ref_740b93c5">[https://arxiv.org/pdf/2104.11630 SHARP DOUBLE INEQUALITY FOR COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND]</ref>
# Abstract We prove a simple relation for a special case of Carlsons elliptic integral RD.<ref name="ref_a6ffe9ff">[https://arxiv.org/pdf/2001.02203 Short note on a relation between the inverse of the cosine and Carlson’s elliptic integral RD]</ref>
# Let K be the complete elliptic integral of the rst kind.<ref name="ref_11a7f590">[https://arxiv.org/pdf/2103.04072 On a Conjecture Concerning the Approximates of Complete Elliptic Integral of the First Kind by Inverse Hyperbolic Tangent]</ref>

타원적분

2022-09-16T10:19:43Z

Pythagoras0: /* 노트 */ 새 문단

==개요==

* 먼저 [[타원적분론 입문]] 참조
* <math>R(x,y)</math>는 <math>x,y</math>의 유리함수이고, <math>y^2</math>은 <math>x</math>의 3차 또는 4차식:<math>\int R(x,\sqrt{ax^3+bx^2+cx+d}) \,dx</math> 또는:<math>\int R(x,\sqrt{ax^4+bx^3+cx^2+dx+e}) \,dx</math>

==타원 둘레의 길이==

* 역사적으로 [[타원 둘레의 길이]]를 구하는 적분에서 그 이름이 기원함.
* 타원 <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>의 둘레의 길이는 <math>4aE(k)</math> 로 주어짐.:<math>k=\sqrt{1-\frac{b^2}{a^2}}</math>:<math>E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>

==정의==

* 일반적으로 다음과 같은 형태로 주어지는 적분을 타원적분이라 부름
:<math>\int R(x,y)\,dx</math>
여기서 <math>R(x,y)</math>는 <math>x,y</math>의 유리함수, <math>y^2</math>= 중근을 갖지 않는 <math>x</math>의 3차식 또는 4차식.

* 예를 들자면,
:<math>\int \frac{dx}{\sqrt{1-x^4}}</math>
:<math>\int \frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>

==일종타원적분과 이종타원적분==

* [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}=\int_0^1\frac{1}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math>
* [[제2종타원적분 E (complete elliptic integral of the second kind)]]:<math>E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx</math>:<math>E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)</math>
* <math>\,_2F_1(a,b;c;z)</math>는 [[초기하급수(Hypergeometric series)]]

==르장드르의 항등식==

* 일종타원적분과 이종타원적분 사이에는 다음과 같은 관계가 성립
:<math>E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}</math>

또는 <math>\theta+\phi=\frac{\pi}{2}</math> 에 대하여
:<math>E(\sin\theta)K(\sin\phi)+E(\sin\phi)K(\sin\theta)-K(\sin\theta)K(\sin\phi)=\frac{\pi}{2}</math>

* 특별히 다음과 같은 관계가 성립함
:<math>2K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})-K(\frac{1}{\sqrt{2}})^2=\frac{\pi}{2}</math>

[[산술기하평균함수(AGM)와 파이값의 계산]]에 응용

==덧셈공식==

* 파그나노의 공식 ([[렘니스케이트 곡선과 Lemniscatomy]] 항목 참조)
:<math>\int_0^x{\frac{1}{\sqrt{1-x^4}}}dx+\int_0^y{\frac{1}{\sqrt{1-x^4}}}dx = \int_0^{A(x,y)}{\frac{1}{\sqrt{1-x^4}}}dx</math>
여기서 <math>A(x,y)=\frac{x\sqrt{1-y^4}+y\sqrt{1-x^4}}{1+x^2y^2}</math>
* 오일러의 일반화
<math>p(x)=1+mx^2+nx^4</math>일 때,
:<math>\int_0^x{\frac{1}{\sqrt{p(x)}}}dx+\int_0^y{\frac{1}{\sqrt{p(x)}}}dx = \int_0^{B(x,y)}{\frac{1}{\sqrt{p(x)}}}dx</math>
여기서
:<math>B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}</math>

==메모==

* 타원적분의 응용으로 [[단진자의 주기와 타원적분]] 항목 참조

==관련된 항목들==

* [[타원곡선]]
* [[타원함수]]
** [[바이어슈트라스 타원함수 ℘]]
* [[자코비 세타함수]]
* [[초기하급수(Hypergeometric series)]]
* [[대수적 함수와 아벨적분]]
* [[오일러 치환|오일러치환]]

==사전 형태의 자료==

* http://ko.wikipedia.org/wiki/타원적분
* http://en.wikipedia.org/wiki/Elliptic_integral
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
** [http://dlmf.nist.gov/19 Chapter 19 Elliptic Integrals]

==관련논문==

* [http://www.springerlink.com/content/b365w3511067g184/ In Search of the "Birthday" of Elliptic Functions - Bit by bit, the discoverers decided what it was they had discovered.]
** Rice, Adrian, 48-57, The Mathematical Intelligencer, Volume 30, Number 2, 2008-3
* Totaro, Burt. 2007. “Euler and Algebraic Geometry.” Bulletin of the American Mathematical Society 44 (4): 541–559. doi:[http://dx.doi.org/10.1090/S0273-0979-07-01178-0 10.1090/S0273-0979-07-01178-0].
* [http://www.springerlink.com/content/t32h69374h887w33/ The Lemniscate and Fagnano's Contributions to Elliptic Integrals]
** AYOUB R
* [http://www.math.tulane.edu/%7Evhm/papers_html/EU.pdf A property of Euler's elastic curve]
* [http://www.springerlink.com/content/911pnwauaeggxk13/ The story of Landen, the hyperbola and the ellipse]
** Elemente der Mathematik, Volume 57, Number 1 / 2002년 2월
* [http://www.jstor.org/stable/2687483 Three Fermat Trails to Elliptic Curves]
** Ezra Brown, <cite style="line-height: 2em;">The College Mathematics Journal</cite>, Vol. 31, No. 3 (May, 2000), pp. 162-172
* [http://www.jstor.org/stable/2974515 Elliptic Curves]
** John Stillwell, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
* [http://www.jstor.org/stable/2321821 Abel's Theorem on the Lemniscate]
** Michael Rosen, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 88, No. 6 (Jun. - Jul., 1981), pp. 387-395

==관련도서==

* [http://www.amazon.com/Functions-Integrals-Translations-Mathematical-Monographs/dp/0821805878 Elliptic functions and elliptic integrals]
** Viktor Prasolov, Yuri Solovyev
* [http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM]
** Jonathan M. Borwein, Peter B. Borwein

==블로그==

* [http://bomber0.byus.net/index.php/2009/08/19/1428 삼각치환에서 타원적분으로] 피타고라스의 창, 2009-8-19
[[분류:리만곡면론]]
[[분류:특수함수]]

==메타데이터==
===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1126603 Q1126603]
===Spacy 패턴 목록===
* [{'LOWER': 'elliptic'}, {'LEMMA': 'integral'}]

== 노트 ==

===말뭉치===
# For instance, while the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral.<ref name="ref_156bfa3e">[https://mathworld.wolfram.com/EllipticIntegral.html Elliptic Integral -- from Wolfram MathWorld]</ref>
# An elliptic integral is written when the parameter is used, when the elliptic modulus is used, and when the modular angle is used.<ref name="ref_156bfa3e" />
# Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping.<ref name="ref_f2288109">[https://en.wikipedia.org/wiki/Elliptic_integral Elliptic integral]</ref>
# These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral).<ref name="ref_f2288109" />
# This is referred to as the incomplete Legendre elliptic integral.<ref name="ref_8a03a90c">[http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap3.pdf Elliptic integrals, elliptic functions and]</ref>
# The integral is also called Legendres form for the elliptic integral of the first kind.<ref name="ref_728c7c82">[https://www.pearson.com/content/dam/one-dot-com/one-dot-com/us/en/files/Jay-Villanuevaictcm3013.pdf Elliptic integrals and some applications]</ref>
# = 0 (1+2)122 , 0 < < 1, 0, also called Legendres form for the elliptic integral of the third kind.<ref name="ref_728c7c82" />
# The upper limit x in the Jacobi form of the elliptic integral of the first kind is related to the upper limit in the Legendre form by = sin .<ref name="ref_728c7c82" />
# returns values of the complete elliptic integral E(K).<ref name="ref_13b5d81f">[https://people.sc.fsu.edu/~jburkardt/f77_src/elliptic_integral/elliptic_integral.html Elliptic Integrals]</ref>
# returns values of the complete elliptic integral E(M).<ref name="ref_13b5d81f" />
# returns values of the complete elliptic integral F(K).<ref name="ref_13b5d81f" />
# returns values of the complete elliptic integral F(M).<ref name="ref_13b5d81f" />
# Any elliptic integral can be expressed as a sum of elementary functions and linear combinations of canonical elliptic integrals of the first, second and third kinds.<ref name="ref_053ce1aa">[https://encyclopediaofmath.org/wiki/Elliptic_integral Encyclopedia of Mathematics]</ref>
# TOMS577, a C++ library which evaluates Carlson's elliptic integral functions RC, RD, RF and RJ.<ref name="ref_adbb6bc9">[https://people.math.sc.edu/Burkardt/cpp_src/elliptic_integral/elliptic_integral.html Elliptic Integrals]</ref>
# The point here is simply show one of the uses of the elliptic integral of the first kind.<ref name="ref_957e1726">[https://www.codeproject.com/Articles/566614/Elliptic-integrals Elliptic integrals]</ref>
# However, the complete elliptic integral is taken from 0 to 90 degrees, but since that is exactly 1/4 over the total arc length due to symmetry.<ref name="ref_957e1726" />
# Complete elliptic integral of the second kind Math.<ref name="ref_a0c3c644">[https://github.com/duetosymmetry/elliptic-integrals-js duetosymmetry/elliptic-integrals-js: Complete elliptic integrals in javascript]</ref>
# Complete elliptic integral of the first kind.<ref name="ref_2f73b86e">[https://solitaryroad.com/c684.html Elliptic integrals of the first, second and third kinds. Jacobian elliptic functions. Identities, formulas, series expansions, derivatives, integrals.]</ref>
# Complete elliptic integral of the third kind.<ref name="ref_2f73b86e" />
# Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively.<ref name="ref_4885c2a7">[https://arxiv.org/pdf/0801.4813 Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function]</ref>
# dy 1 + a cos x + b cos y (1) Keywords: Double elliptic integral, hypergeometric function 1 .<ref name="ref_ce2c00f8">[https://arxiv.org/pdf/0709.1289 A two-parameter generalization of the complete elliptic integral of the second kind M. L. Glasser]</ref>
# v i X r a SHARP DOUBLE INEQUALITY FOR COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND QI BAO r2 sin2 t)1/2dt is known as the Abstract.<ref name="ref_740b93c5">[https://arxiv.org/pdf/2104.11630 SHARP DOUBLE INEQUALITY FOR COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND]</ref>
# Abstract We prove a simple relation for a special case of Carlsons elliptic integral RD.<ref name="ref_a6ffe9ff">[https://arxiv.org/pdf/2001.02203 Short note on a relation between the inverse of the cosine and Carlson’s elliptic integral RD]</ref>
# Let K be the complete elliptic integral of the rst kind.<ref name="ref_11a7f590">[https://arxiv.org/pdf/2103.04072 On a Conjecture Concerning the Approximates of Complete Elliptic Integral of the First Kind by Inverse Hyperbolic Tangent]</ref>

Test234

2022-09-16T10:09:58Z

Test234

2022-09-16T09:44:16Z

Pythagoras0:

Test234

2022-09-16T07:20:34Z

Pythagoras0: 새 문서: ==개요== * j-불변량 ** 클라인의 absolute j-invariant 라는 이름으로 불리기도 함 * 타원 모듈라 함수(elliptic modular function) 로 불리기도 함 * 복소...

==개요==

* j-불변량
** 클라인의 absolute j-invariant 라는 이름으로 불리기도 함
* 타원 모듈라 함수(elliptic modular function) 로 불리기도 함
* 복소 이차 수체의 class field 이론에서 중요한 역할
* [[몬스터 군]]의 monstrous moonshine에 등장

==정의==
* <math>q=e^{2\pi i\tau},\tau\in \mathbb{H}</math>라 두자
* 타원 모듈라 j-함수는 다음과 같이 정의된다
:<math>
j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+21493760q^2+\cdots
</math> 여기서
:<math> E_ 4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots,\quad \sigma_3(n)=\sum_{d|n}d^3</math>는 [[아이젠슈타인 급수(Eisenstein series)]],
:<math>\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots</math> 는 [[판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)|판별식 함수]]
* 다음과 같이 쓰기도 한다
:<math>j(\tau)=1728\frac{g_ 2^3}{g_ 2^3-27g_ 3^2}</math>
여기서 <math>g_2,g_3</math>는 [[아이젠슈타인 급수(Eisenstein series)]], [[바이어슈트라스 타원함수 ℘]] 항목 참조

==singular moduli==
* quadratic imaginary number 에서의 값
* 예 :
:<math>j(\frac {-1+\sqrt{-163}} {2})=-262537412640768000=-640320^3</math>
* [[타원 모듈라 j-함수의 singular moduli]] 참조
* 판별식이 -23인 세 이차형식 ([[숫자 23과 다항식 x³-x+1]] 참조)
:<math>
x^2+x+6,2 x^2-x+3,2 x^2+x+3
</math>
의 상반평면에서의 해를 구하여, 다음의 값을 생각하자
:<math>
j\left(\frac{1}{2} \left(-1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(-1+i \sqrt{23}\right)\right)</math>
* 이들은 대수적 정수이며, 다음 다항식의 해가 된다
:<math>
x^3+3491750 x^2-5151296875 x+12771880859375
</math>

에드워즈 곡선 디지털 서명 알고리듬

2022-09-16T04:14:34Z

Pythagoras0: /* 메타데이터 */ 새 문단

== 노트 ==

===말뭉치===
# Abstract This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA).<ref name="ref_2e69d3c6">[https://datatracker.ietf.org/doc/rfc8032/ Edwards-Curve Digital Signature Algorithm (EdDSA)]</ref>
# EdDSA needs to be instantiated with certain parameters, and this document describes some recommended variants.<ref name="ref_2e69d3c6" />
# To facilitate adoption of EdDSA in the Internet community, this document describes the signature scheme in an implementation-oriented way and provides sample code and test vectors.<ref name="ref_2e69d3c6" />
# The advantages with EdDSA are as follows: 1. EdDSA provides high performance on a variety of platforms; 2. The use of a unique random number for each signature is not required; 3.<ref name="ref_2e69d3c6" />
# Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using twisted Edwards curves.<ref name="ref_f6224868">[https://infocenter.nordicsemi.com/topic/sdk_nrf5_v17.1.0/lib_crypto_eddsa.html Edwards-curve Digital Signature Algorithm]</ref>
# This module provides support for EdDSA (Edwards-curve Digital Signature Algorithm) using SHA-512 and Ed25519.<ref name="ref_f6224868" />
# 1. An odd prime power p. EdDSA uses an elliptic curve over the finite field GF(p).<ref name="ref_c615f04d">[https://www.rfc-editor.org/rfc/rfc8032 RFC 8032: Edwards-Curve Digital Signature Algorithm (EdDSA)]</ref>
# EdDSA public keys have exactly b bits, and EdDSA signatures have exactly 2*b bits.<ref name="ref_c615f04d" />
# Conservative hash functions (i.e., hash functions where it is infeasible to create collisions) are recommended and do not have much impact on the total cost of EdDSA. 5.<ref name="ref_c615f04d" />
# Secret EdDSA scalars have exactly n + 1 bits, with the top bit (the 2^n position) always set and the bottom c bits always cleared.<ref name="ref_c615f04d" />
# Herein, Edwards-curve digital signature algorithm or shortly EdDSA offers slightly faster signatures than ECDSA.<ref name="ref_2cef576e">[https://sefiks.com/2018/12/24/a-gentle-introduction-to-edwards-curve-digital-signature-algorithm-eddsa/ A Gentle Introduction to Edwards-curve Digital Signature Algorithm (EdDSA)]</ref>
# In EdDSA, this is handled by generating random key based on the hash of the message.<ref name="ref_2cef576e" />
# This issue is handled in EdDSA.<ref name="ref_2cef576e" />
# However, ECDSA/EdDSA and DSA differ in that DSA uses a mathematical operation known as modular exponentiation while ECDSA/EdDSA uses elliptic curves.<ref name="ref_fe1a3782">[https://goteleport.com/blog/comparing-ssh-keys/ Comparing SSH Keys - RSA, DSA, ECDSA, or EdDSA?]</ref>
# EdDSA solves the same discrete log problem as DSA/ECDSA, but uses a different family of elliptic curves known as the Edwards Curve (EdDSA uses a Twisted Edwards Curve).<ref name="ref_fe1a3782" />
# The EdDSA signatures use the Edwards form of the elliptic curves (for performance reasons), respectively edwards25519 and edwards448 .<ref name="ref_5dcd89c0">[https://cryptobook.nakov.com/digital-signatures/eddsa-and-ed25519 EdDSA and Ed25519]</ref>
# The hash function H {\displaystyle H} is normally modelled as a random oracle in formal analyses of EdDSA's security.<ref name="ref_c73f9309">[https://en.wikipedia.org/wiki/EdDSA#:~:text=In%20public%2Dkey%20cryptography%2C%20Edwards,signature%20schemes%20without%20sacrificing%20security. Wikipedia]</ref>
# Like other discrete-log-based signature schemes, EdDSA uses a secret value called a nonce unique to each signature.<ref name="ref_c73f9309" />
# In contrast, EdDSA chooses the nonce deterministically as the hash of a part of the private key and the message.<ref name="ref_c73f9309" />
# The Edwards-curve Digital Signature Algorithm (EdDSA) scheme uses a variant of the Schnorr signature based on twisted Edwards curves.<ref name="ref_b44dad56">[https://doc.primekey.com/ejbca/ejbca-operations/ejbca-ca-concept-guide/certificate-authority-overview/eddsa-keys-and-signatures EdDSA Keys and Signatures]</ref>
# EdDSA is designed to be faster than existing digital signature schemes without sacrificing security.<ref name="ref_b44dad56" />
# EJBCA supports EdDSA signature keys and you can create a Certificate Authority (CA) using EdDSA keys both using the EJBCA Admin UI and the CLI ( bin/ejbca.sh ca init ).<ref name="ref_b44dad56" />
# PKCS#11 did not standardize support for EdDSA until PKCS#11v3, while most HSMs still (October 2020) are still on PKCS#11v2.40.<ref name="ref_b44dad56" />
# Although EdDSA is employed in many widely used protocols, such as TLS and SSH, there appear to be extremely few hardware implementations that focus only on EdDSA.<ref name="ref_e6c15893">[https://cse.usf.edu/~mehran2/Papers/J46.pdf Ieee transactions on very large scale integration (vlsi) systems, vol. 29, no. 7, july 2021]</ref>
# I. INTRODUCTION E DWARDS curve digital signature algorithm (EdDSA) developed by Bernstein et al.<ref name="ref_e6c15893" />
# The Ed25519, as the most popular instance of EdDSA, is widely used as a digital signature method to guarantee the validity of the communications.<ref name="ref_e6c15893" />
# However, EdDSA has not got sufcient study, especially in the eld of hard- ware implementation based on eld-programmable gate arrays (FPGAs).<ref name="ref_e6c15893" />
# The API for EdDSA and the implementation in SunEC will not support arbitrary domain parameters.<ref name="ref_a0a94417">[https://openjdk.org/jeps/339 JEP 339: Edwards-Curve Digital Signature Algorithm (EdDSA)]</ref>
# Typical uses of EdDSA only use standardized parameter sets such as Ed25519 and Ed448 which can be specified using identifiers, and support for arbitrary curve parameters is not typically needed.<ref name="ref_a0a94417" />
# The EdDSA API should permit, through extension, the specification of arbitrary domain parameters.<ref name="ref_a0a94417" />
# Some users may have EdDSA certificates, and may have a strong preference to use EdDSA.<ref name="ref_a0a94417" />
# If you’re working on embedded systems, the determinism inherent to EdDSA might be undesirable due to the possibility of fault attacks.<ref name="ref_e17b9766">[https://soatok.blog/2022/05/19/guidance-for-choosing-an-elliptic-curve-signature-algorithm-in-2022/ Guidance for Choosing an Elliptic Curve Signature Algorithm in 2022]</ref>
# Maybe you've seen the landslide of acronyms that go along with it: ECC, ECDSA, ECDH, EdDSA, Ed25519, etc.<ref name="ref_ba58ee33">[https://fission.codes/blog/everything-you-wanted-to-know-about-elliptic-curve-cryptography/ Everything you wanted to know about Elliptic Curve Cryptography – Fission]</ref>
# Hence implementing EdDSA over Galois field provides more security compared to the conventional EdDSA signature.<ref name="ref_4f5df1eb">[http://www.sdiarticle3.com/wp-content/uploads/2019/05/Revised-ms_JERR_48655_v2.pdf Eddsa over galois field gf((cid:2198)(cid:2195)) for multimedia data]</ref>
# EdDSA needs to be instantiated with certain parameters.<ref name="ref_4f5df1eb" />
# Creation of signature is deterministic in EdDSA and it has higher security due to intractability of some discrete logarithm problems.<ref name="ref_4f5df1eb" />
# For the EdDSA authenticator to function, it needs to know its own private key.<ref name="ref_4f5df1eb" />
# It means that EdDSA is similar to other elliptic curve signature algorithms, but has some different algorithmic details.<ref name="ref_3c6dfb07">[https://medium.com/@qinwen228/eddsa-a-good-signature-algorithm-717499a305 EdDSA, a good signature algorithm]</ref>
# On some other occasions, the EdDSA is also called ed25519.<ref name="ref_3c6dfb07" />
# But the security of EdDSA does not depend on a random number generator, which is very different from ECDSA.<ref name="ref_3c6dfb07" />
# Last but not least, EdDSA is very fast during the key generation process to sign a signature, make a verification.<ref name="ref_3c6dfb07" />
# In this paper, we make a comparative study of these methods for the Edwards curve digital signature algorithm (EdDSA).<ref name="ref_0f7f5980">[https://link.springer.com/chapter/10.1007/978-3-319-12060-7_17 Batch Verification of EdDSA Signatures]</ref>
# We describe the adaptation of Algorithms N, N′, S2′ and SP for EdDSA signatures.<ref name="ref_0f7f5980" />
# More precisely, we study seminumeric scalar multiplication and Montgomery ladders during randomization of EdDSA signatures.<ref name="ref_0f7f5980" />
# Each EdDSA signature verification involves a square-root computation.<ref name="ref_0f7f5980" />
# Signing a message with EdDSA proves to the recipient that the sender of the message is in possession of the private key corresponding to the transmitted public key used during verification.<ref name="ref_f5a772cf">[https://software-dl.ti.com/simplelink/esd/simplelink_cc13x2_26x2_sdk/5.10.00.48/exports/docs/drivers/doxygen/html/_e_d_d_s_a_8h.html EDDSA.h File Reference]</ref>
# The sender generates an EdDSA private-public keypair with private key k and public key A. For Ed25519, these are 32 bytes in little endian.<ref name="ref_f5a772cf" />
# This result is used as a scalar to generate EdDSA signature component R which is a point on Ed25519.<ref name="ref_f5a772cf" />
# The signature component R, public key A, and message M are hashed to find a value that is used to generate the EdDSA signature component S which is a scalar.<ref name="ref_f5a772cf" />
# Using EdDSA has a few advantages over ECDSA, mostly due to it being easier to implement and, therefore, more secure.<ref name="ref_6ead77c0">[https://www.scottbrady91.com/c-sharp/eddsa-for-jwt-signing-in-dotnet-core EdDSA for JWT Signing in .NET Core]</ref>
# To learn more about EdDSA and these variants, I recommend checking out David Wong’s article “EdDSA, Ed25519, Ed25519-IETF, Ed25519ph, Ed25519ctx, HashEdDSA, PureEdDSA, WTF?”.<ref name="ref_6ead77c0" />
# Otherwise, check out ed25519.cr.yp.to, which lists the benefits of using EdDSA (some are debatable).<ref name="ref_6ead77c0" />
# With EdDSA, both Ed25519 and Ed448 use an alg value of EdDSA .<ref name="ref_6ead77c0" />
# This document specifies the conventions for using the Edwards-curve Digital Signature Algorithm (EdDSA) for curve25519 and curve448 in the Cryptographic Message Syntax (CMS).<ref name="ref_cfc89118">[https://www.hjp.at/(en)/doc/rfc/rfc8419.html hjp: doc: RFC 8419: Use of Edwards-Curve Digital Signature Algorithm (EdDSA) Signatures in the Cryptographic Message Syntax (CMS)]</ref>
# For each curve, EdDSA defines the PureEdDSA and HashEdDSA modes.<ref name="ref_cfc89118" />
# The id-Ed25519 and id-Ed448 object identifiers are used to identify EdDSA public keys in certificates.<ref name="ref_cfc89118" />
# The SignerInfo signature field contains the octet string resulting from the EdDSA private key signing operation.<ref name="ref_cfc89118" />
# You've heard of EdDSA right?<ref name="ref_24deef46">[https://www.cryptologie.net/article/497/eddsa-ed25519-ed25519-ietf-ed25519ph-ed25519ctx-hasheddsa-pureeddsa-wtf/ EdDSA, Ed25519, Ed25519-IETF, Ed25519ph, Ed25519ctx, HashEdDSA, PureEdDSA, WTF?]</ref>
# Since its inception, EdDSA has evolved quite a lot, and some amount of standardization process has happened to it.<ref name="ref_24deef46" />
# Using EdDSA Signatures with CMS August 2018 Table of Contents 1. Introduction ....................................................2 1.1.<ref name="ref_0a6951d3">[http://www.muonics.com/rfc/rfc8419.php RFC 8419 - Use of Edwards-Curve Digital Signature Algorithm (EdDSA) Signatures in the Cryptographic Message Syntax (CMS)]</ref>
# EdDSA with curve25519 is referred to as "Ed25519", and EdDSA with curve448 is referred to as "Ed448".<ref name="ref_0a6951d3" />
# Using EdDSA Signatures with CMS August 2018 2.3.<ref name="ref_0a6951d3" />
# EdDSA needs to be instantiated with certain parameters and this document describe some recommended variants.<ref name="ref_6e4f16c1">[http://www.watersprings.org/pub/id/draft-irtf-cfrg-eddsa-00.html Edwards-curve Digital Signature Algorithm (EdDSA)]</ref>
# This obviates the need for EdDSA to perform expensive point validation on untrusted public values.<ref name="ref_6e4f16c1" />
# The generic EdDSA digital signature system with its eleven input parameters is not intended to be implemented directly.<ref name="ref_6e4f16c1" />
# EdDSA public keys have exactly b bits, and EdDSA signatures have exactly 2b bits.<ref name="ref_6e4f16c1" />
# EDDSA Specifies to generate a digital signature using the EDDSA algorithm.<ref name="ref_b1905011">[https://www.ibm.com/docs/en/zos/2.5.0?topic=signatures-digital-signature-generate-csnddsg-csnfdsg Digital Signature Generate (CSNDDSG and CSNFDSG)]</ref>
# This keyword is required with EDDSA, EC-SDSA , and CRDL-DSA keywords.<ref name="ref_b1905011" />
# This keyword is required with the EDDSA keyword.<ref name="ref_b1905011" />
# EdDSA is used in TLS 1.3.<ref name="ref_9bf6be08">[https://www.googlecloudcommunity.com/gc/-/-/td-p/23063 Edwards-Curve Digital Signature Algorithm (EdDSA) ...]</ref>
# So in that regard, no, EdDSA is not supported.<ref name="ref_9bf6be08" />
# According to our knowledge, this is the rst two-part cryptography scheme designed for Edwards-curve digital signature algorithm without sacricing security.<ref name="ref_6baef6d0">[http://ijns.jalaxy.com.tw/contents/ijns-v23-n4/ijns-2021-v23-n4-p558-568.pdf International journal of network security, vol.23, no.4, pp.558-568, july 2021 (doi: 10.6633/ijns.202107 23(4).02)]</ref>
# For the sake of improving above deciency, we present a two-party Edwards-curve digital signature algorithm.<ref name="ref_6baef6d0" />
# However, EdDSA signatures are defined on twisted Edwards curves, where a public key is a compressed point consisting of a twisted Edwards y-coordinate and a sign bit s which is either 0 or 1.<ref name="ref_3a3cf059">[https://signal.org/docs/specifications/xeddsa/ Signal >> Specifications >> The XEdDSA and VXEdDSA Signature Schemes]</ref>
# Abstract We present an EdDSA-compatible multi-party digital signature scheme that supports an oine participant during the key-generation phase, without relying on a trusted third party.<ref name="ref_9cb6e7a0">[https://arxiv.org/pdf/2009.01631 Springer Nature 2021 LATEX template A Provably-Unforgeable Threshold EdDSA]</ref>
# A Provably-Unforgeable Threshold EdDSA with an Oine Recovery Party 3 Organization We present some preliminaries in Section 2.<ref name="ref_9cb6e7a0" />
# Our protocol works with both ECDSA and EdDSA signature schemes and prioritizes efcient computation and communication.<ref name="ref_960480d9">[https://arxiv.org/pdf/2106.10972 Improving security for users of decentralized exchanges through multiparty computation]</ref>
# z, the rst part of the signature r, and the nonce k as follows: s k1 (z + r d) mod n. C. EdDSA Signature generation in EdDSA works similar to ECDSA.<ref name="ref_960480d9" />
# An EdDSA signature also consists of a tuple of integers (r, s), but computation differs slightly: 1) First, the secret key is hashed.<ref name="ref_960480d9" />
# 2) A cryptographically secure nonce is also required in EdDSA, but generating it is not left to the implementer.<ref name="ref_960480d9" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q16966748 Q16966748]
===Spacy 패턴 목록===
* [{'LOWER': 'edwards'}, {'OP': '*'}, {'LOWER': 'curve'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LOWER': 'algorithm'}]
* [{'LOWER': 'eddsa'}]

에드워즈 곡선 디지털 서명 알고리듬

2022-09-16T04:14:33Z

Pythagoras0: /* 노트 */ 새 문단

슈노르 서명

2022-09-16T03:22:44Z

Pythagoras0:

== 노트 ==

===말뭉치===
# In the near future, Bitcoin will enable Schnorr signatures in addition to ECDSA signatures.<ref name="ref_db591c4c">[https://river.com/learn/what-are-schnorr-signatures/ What Do Schnorr Signatures Do for Bitcoin?]</ref>
# Schnorr signatures will be introduced to Bitcoin through Taproot upgrade, which will hopefully be activated around 2022.<ref name="ref_db591c4c" />
# Although developers have added all necessary code to Bitcoin Core, Bitcoin nodes must accept the upgrade in order to consider Schnorr signatures valid.<ref name="ref_db591c4c" />
# Schnorr signatures are quite simple compared to other schemes.<ref name="ref_dd62bb89">[https://academy.binance.com/en/articles/what-do-schnorr-signatures-mean-for-bitcoin What do Schnorr Signatures Mean for Bitcoin?]</ref>
# Schnorr signatures have been touted as a solution to these privacy and scalability issues.<ref name="ref_dd62bb89" />
# As with most upgrades to the Bitcoin protocol, it could take time for the broader community of Bitcoin users to agree on the Schnorr signature inclusion.<ref name="ref_dd62bb89" />
# Schnorr signatures could be merged into the code as a soft fork , meaning that a change would not split the network.<ref name="ref_dd62bb89" />
# In this post I will explain what Schnorr signatures are and how they intuitively work.<ref name="ref_3440f6d6">[https://suredbits.com/introduction-to-schnorr-signatures/ Introduction to Schnorr Signatures]</ref>
# And that is all there is to the actual computation surrounding “vanilla” Schnorr signatures!<ref name="ref_3440f6d6" />
# But Schnorr signatures are much more elegant and simple, and it has one more magical property; linearity.<ref name="ref_27782bb7">[https://medium.com/bitbees/what-the-heck-is-schnorr-52ef5dba289f What The Heck Is Schnorr]</ref>
# Schnorr signature was invented by Claus-Peter Schnorr back in the 1980s.<ref name="ref_27782bb7" />
# Because of his patent, the Schnorr signature algorithm did not see any widespread use for decades.<ref name="ref_27782bb7" />
# Six more years later, in 2014 the first talk of implementing Schnorr signature on Bitcoin protocol came up in the bitcoin-talk forum.<ref name="ref_27782bb7" />
# In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr.<ref name="ref_492c923b">[https://www.geeksforgeeks.org/schnorr-digital-signature/ Schnorr Digital Signature]</ref>
# Note: When you construct the signature like this, it’s known as a Schnorr signature, which is discussed in a following section.<ref name="ref_440c07d2">[https://tlu.tarilabs.com/cryptography/introduction-schnorr-signatures Introduction to Schnorr Signatures]</ref>
# The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model.<ref name="ref_440c07d2" />
# The main function of Schnorr signatures is to allow multiple users to create a single signature for all parties involved.<ref name="ref_3bdb1e89">[https://academy.bit2me.com/en/que-son-las-firmas-schnorr/ What are Schnorr signatures?]</ref>
# So the implementation of Schnorr signatures represents a real solution to this problem.<ref name="ref_3bdb1e89" />
# Historically, EdDSA is known as a variant of Schnorr signatures, which are well-studied and suitable for efficient thresholdization,...<ref name="ref_6a94b9af">[https://csrc.nist.gov/publications/detail/nistir/8214b/draft NISTIR 8214B (Draft), Notes on Threshold EdDSA/Schnorr Signatures]</ref>
# We aim at designing a leakage-resilient variant of the Schnorr signature scheme whose secret key’s storage space is constant, independently of the amount of leakage that it can tolerate.<ref name="ref_4e437487">[https://link.springer.com/chapter/10.1007/978-3-642-45239-0_11 A Leakage-Resilient Pairing-Based Variant of the Schnorr Signature Scheme]</ref>
# We proceed by first proposing a pairing analogue of the Schnorr signature scheme, that we next transform to include split signing key updates.<ref name="ref_4e437487" />
# As the increased uptake in connected devices revives the interest in resource-constrained signature algorithms, we introduce a variant of Schnorr signatures that mutualises exponentiation eorts.<ref name="ref_d70872a7">[https://eprint.iacr.org/2018/069.pdf Reusing nonces in schnorr signatures]</ref>
# Sharing a nonce is a deadly blow to Schnorr signatures, but is not a security concern for our variant.<ref name="ref_d70872a7" />
# We start by reminding how the original Schnorr signature scheme works and explain how we extend it assuming that k is randomly drawn from Zp1.<ref name="ref_d70872a7" />
# 3.1 Our Signature Scheme Similar to the Schnorr signature scheme, our scheme is a tuple of algorithms (Setup, KeyGen, Sign, and Verify), which we dene as follows: Setup(1): Generate primes q1, . .<ref name="ref_d70872a7" />
# This has lead to a long line of research investigating the existence of tighter security proofs for Schnorr signatures.<ref name="ref_1b944f9d">[https://eprint.iacr.org/2013/418.pdf On tight security proofs for schnorr signatures]</ref>
# We begin with the hypothesis that there exists a tight generic re- duction R from some hard non-interactive problem to the UUF-NMA-security of Schnorr signatures.<ref name="ref_1b944f9d" />
# So Schnorr signature solves these 2 problems, it is non-malleable, which means #Bitcoin network becomes more secure.<ref name="ref_3c9b8920">[https://www.linkedin.com/pulse/cryptography-digital-signatures-schnorr-taproot-upgrade-nitesh-balusu Cryptography: Digital Signatures and Schnorr Signatures Explained-#Bitcoin Taproot Upgrade]</ref>
# I'm trying to understand the security of the short schnorr signature a little bit better.<ref name="ref_d5ffc39f">[https://crypto.stackexchange.com/questions/95345/security-proof-of-short-schnorr-signature Security Proof of Short Schnorr Signature]</ref>
# A Schnorr signature is a digital signature produced by the Schnorr signature algorithm.<ref name="ref_02d5a128">[https://www.bitstamp.net/learn/blockchain/what-are-schnorr-signatures/ What are Schnorr Signatures?]</ref>
# Another advantage of Schnorr signatures is increased privacy in terms of securing your bitcoins.<ref name="ref_02d5a128" />
# By reducing the amount of signature data stored on the blockchain, Schnorr signatures free up block storage space.<ref name="ref_02d5a128" />
# But scaling is not the only way Schnorr signatures can improve the Bitcoin protocol.<ref name="ref_02d5a128" />
# The Schnorr signatures (Schnorr, n.d.) have been known before ECDSA signatures, yet they were not so widely used due to the patent which expired in the year 2008.<ref name="ref_a00e3450">[https://mareknarozniak.com/2021/05/25/schnorr-signature/ Schnorr Signature]</ref>
# One of the advantages is the existence of proof that breaking the Schnorr signature is equivalent to breaking the discrete logarithm problem.<ref name="ref_a00e3450" />
# If you like to know more, I based this tutorial on what the heck is Schnorr medium article and cryptography fandom Schnorr signature page.<ref name="ref_a00e3450" />
# We have implemented Schnorr signatures on Bitcoin.<ref name="ref_2b5e61bd">[https://coingeek.com/schnorr-signatures-on-bitcoin/ Schnorr signatures on Bitcoin]</ref>
# FROST is a threshold Schnorr signature protocol that contains two important components.<ref name="ref_adf8fce0">[https://blog.coinbase.com/frost-flexible-round-optimized-schnorr-threshold-signatures-b2e950164ee1 FROST: Flexible Round-Optimized Schnorr Threshold Signatures]</ref>
# Afterwards, any t-out-of-n participants can run a threshold signing protocol to collaboratively generate a valid Schnorr signature.<ref name="ref_adf8fce0" />
# In addition, FROST also requires each participant to demonstrate knowledge of their own secret by sending to other participants a zero-knowledge proof, which itself is a Schnorr signature.<ref name="ref_adf8fce0" />
# To create a valid Schnorr signature, any t participants work together to execute this round.<ref name="ref_adf8fce0" />
# In 2005, when elliptic curve cryptography was being standardized people built on top of DSA rather than Schnorr signatures that had advantages.<ref name="ref_3fd754e6">[https://diyhpl.us/wiki/transcripts/scalingbitcoin/milan/schnorr-signatures/ schnorr-signatures]</ref>
# What I want you to take away from this is Schnorr signatures are not an established standard.<ref name="ref_3fd754e6" />
# The security proof of Schnorr signatures says that they are existentially unforgeable under the assumptions I mentioned before.<ref name="ref_3fd754e6" />
# It turns out if you take Schnorr signatures naively and apply it to an elliptic curve group it has a really annoying interaction with BIP 32 when used with public derivation.<ref name="ref_3fd754e6" />
# But Schnorr signatures can add a new advantage to CoinJoin.<ref name="ref_626bfe1f">[https://bitcoinmagazine.com/culture/the-power-of-schnorr-the-signature-algorithm-to-increase-bitcoin-s-scale-and-privacy-1460642496 The Power of Schnorr: The Signature Algorithm to Increase Bitcoin's Scale and Privacy]</ref>
# Note: The process of implementing Schnorr signatures in Bitcoin is still in the concept phase.<ref name="ref_626bfe1f" />
# Schnorr signatures can be proved secure in the random oracle model (ROM) under the discrete logarithm assumption (DL) by rewinding the adversary; but this security proof is loose.<ref name="ref_d7c091b0">[https://www.semanticscholar.org/paper/Blind-Schnorr-Signatures-in-the-Algebraic-Group-Fuchsbauer-Plouviez/abfbac3d8b2de10803b9df6fe6625090feddb991 PDF Blind Schnorr Signatures in the Algebraic Group Model]</ref>
# The written specication for Schnorr signatures should fully describe the algorithm.<ref name="ref_b8368f56">[https://courses.csail.mit.edu/6.857/2020/projects/4-Elbahrawy-Lovejoy-Ouyang-Perez.pdf Analysis of bitcoin improvement proposal 340]</ref>
# In the Bitcoin specication of Schnorr signatures, the public key Q is 32 bytes, and it can be converted from existing generated public keys by dropping the rst byte (the prex).<ref name="ref_b8368f56" />
# The Schnorr signature scheme is constructed by applying the Fiat-Shamir heuristic to Schnorrs identication protocol.<ref name="ref_b8368f56" />
# Schnorr signature is an alternative algorithm to Bitcoin’s original ECDSA.<ref name="ref_ee937991">[https://www.telemediaonline.co.uk/schnorr-signatures-role-in-bitcoin-transactions/ Schnorr Signatures Role in Bitcoin Transactions]</ref>
# Schnorr signatures are the second type of signatures scheme introduced with the Taproot upgrade to address some of the flaws of the ECDSA protocol.<ref name="ref_ee937991" />
# Schnorr signatures offer that advantage, allowing the Bitcoin network to optimize payment processing and data storage.<ref name="ref_ee937991" />
# That makes it impossible for chain analysis to distinguish between multi-sig and single-sign Bitcoin transactions with Schnorr signatures, ensuring enhanced privacy.<ref name="ref_ee937991" />
# In this blog post we will explain one of the main advantages of Schnorr signatures’: its native support for Multi-Signatures (MultiSig).<ref name="ref_d65743ce">[https://hackernoon.com/a-brief-intro-to-bitcoin-schnorr-multi-signatures-b9ef052374c5 A brief intro to Bitcoin Schnorr Multi-signatures]</ref>
# Package schnorr implements the vanilla Schnorr signature scheme.<ref name="ref_a63ab04b">[https://pkg.go.dev/go.dedis.ch/kyber/sign/schnorr go.dedis.ch/kyber/sign/schnorr]</ref>
# We provide two necessary conditions on hash functions for the Schnorr signature scheme to be secure, assuming compact group rep- resentations such as those which occur in elliptic curve groups.<ref name="ref_6d25ea82">[http://www.neven.org/papers/schnorr.pdf Hash function requirements]</ref>
# First, since security does not rely on the hash function being collision resistant, Schnorr signatures can still be securely instantiated with SHA-1/SHA- 256, unlike DSA signatures.<ref name="ref_6d25ea82" />
# Apart from instantiation candidates for the hash function, our results have a number of other important implications for the eciency and security of Schnorr signatures.<ref name="ref_6d25ea82" />
# Our work uses Schnorr signatures and leverages Bitcoin recent Taproot upgrade, allowing us to create a checkpointing transaction of constant size.<ref name="ref_0d6bbf83">[https://arxiv.org/pdf/2208.05408 Pikachu: securing pos blockchains from long-range attacks by]</ref>
# To overcome these weaknesses in the Ma-Chen scheme, we propose a new scheme based on the Schnorr signature.<ref name="ref_cf714458">[https://arxiv.org/pdf/cs/0504019 5]</ref>
# == r = (H(m) + r.x) (H(m) + r.x)1 (cid:0)k1(cid:1)1 G = k G = R 2.1.2 Schnorr The Schnorr signature variant over ECC has multiple standards.<ref name="ref_398d6469">[https://arxiv.org/pdf/2110.00274 1]</ref>
# This property allows Schnorr signatures to be aggregated easily to construct a multi-party signature.<ref name="ref_398d6469" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1465057 Q1465057]
===Spacy 패턴 목록===
* [{'LOWER': 'schnorr'}, {'LEMMA': 'signature'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LEMMA': 'scheme'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'signature'}, {'LEMMA': 'algorithm'}]
* [{'LOWER': 'sdsa'}]

슈노르 서명

2022-09-16T03:19:57Z

Pythagoras0: /* 메타데이터 */ 새 문단

== 노트 ==

===말뭉치===
# But Schnorr signatures are much more elegant and simple, and it has one more magical property; linearity.<ref name="ref_27782bb7">[https://medium.com/bitbees/what-the-heck-is-schnorr-52ef5dba289f What The Heck Is Schnorr]</ref>
# Schnorr signature was invented by Claus-Peter Schnorr back in the 1980s.<ref name="ref_27782bb7" />
# Because of his patent, the Schnorr signature algorithm did not see any widespread use for decades.<ref name="ref_27782bb7" />
# Six more years later, in 2014 the first talk of implementing Schnorr signature on Bitcoin protocol came up in the bitcoin-talk forum.<ref name="ref_27782bb7" />
# In the near future, Bitcoin will enable Schnorr signatures in addition to ECDSA signatures.<ref name="ref_db591c4c">[https://river.com/learn/what-are-schnorr-signatures/ What Do Schnorr Signatures Do for Bitcoin?]</ref>
# Schnorr signatures will be introduced to Bitcoin through Taproot upgrade, which will hopefully be activated around 2022.<ref name="ref_db591c4c" />
# Although developers have added all necessary code to Bitcoin Core, Bitcoin nodes must accept the upgrade in order to consider Schnorr signatures valid.<ref name="ref_db591c4c" />
# Schnorr signatures are quite simple compared to other schemes.<ref name="ref_dd62bb89">[https://academy.binance.com/en/articles/what-do-schnorr-signatures-mean-for-bitcoin What do Schnorr Signatures Mean for Bitcoin?]</ref>
# Schnorr signatures have been touted as a solution to these privacy and scalability issues.<ref name="ref_dd62bb89" />
# As with most upgrades to the Bitcoin protocol, it could take time for the broader community of Bitcoin users to agree on the Schnorr signature inclusion.<ref name="ref_dd62bb89" />
# Schnorr signatures could be merged into the code as a soft fork , meaning that a change would not split the network.<ref name="ref_dd62bb89" />
# In this post I will explain what Schnorr signatures are and how they intuitively work.<ref name="ref_3440f6d6">[https://suredbits.com/introduction-to-schnorr-signatures/ Introduction to Schnorr Signatures]</ref>
# And that is all there is to the actual computation surrounding “vanilla” Schnorr signatures!<ref name="ref_3440f6d6" />
# In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr.<ref name="ref_492c923b">[https://www.geeksforgeeks.org/schnorr-digital-signature/ Schnorr Digital Signature]</ref>
# Note: When you construct the signature like this, it’s known as a Schnorr signature, which is discussed in a following section.<ref name="ref_440c07d2">[https://tlu.tarilabs.com/cryptography/introduction-schnorr-signatures Introduction to Schnorr Signatures]</ref>
# The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model.<ref name="ref_440c07d2" />
# Historically, EdDSA is known as a variant of Schnorr signatures, which are well-studied and suitable for efficient thresholdization,...<ref name="ref_6a94b9af">[https://csrc.nist.gov/publications/detail/nistir/8214b/draft NISTIR 8214B (Draft), Notes on Threshold EdDSA/Schnorr Signatures]</ref>
# We aim at designing a leakage-resilient variant of the Schnorr signature scheme whose secret key’s storage space is constant, independently of the amount of leakage that it can tolerate.<ref name="ref_4e437487">[https://link.springer.com/chapter/10.1007/978-3-642-45239-0_11 A Leakage-Resilient Pairing-Based Variant of the Schnorr Signature Scheme]</ref>
# We proceed by first proposing a pairing analogue of the Schnorr signature scheme, that we next transform to include split signing key updates.<ref name="ref_4e437487" />
# As the increased uptake in connected devices revives the interest in resource-constrained signature algorithms, we introduce a variant of Schnorr signatures that mutualises exponentiation eorts.<ref name="ref_d70872a7">[https://eprint.iacr.org/2018/069.pdf Reusing nonces in schnorr signatures]</ref>
# Sharing a nonce is a deadly blow to Schnorr signatures, but is not a security concern for our variant.<ref name="ref_d70872a7" />
# We start by reminding how the original Schnorr signature scheme works and explain how we extend it assuming that k is randomly drawn from Zp1.<ref name="ref_d70872a7" />
# 3.1 Our Signature Scheme Similar to the Schnorr signature scheme, our scheme is a tuple of algorithms (Setup, KeyGen, Sign, and Verify), which we dene as follows: Setup(1): Generate primes q1, . .<ref name="ref_d70872a7" />
# This has lead to a long line of research investigating the existence of tighter security proofs for Schnorr signatures.<ref name="ref_1b944f9d">[https://eprint.iacr.org/2013/418.pdf On tight security proofs for schnorr signatures]</ref>
# We begin with the hypothesis that there exists a tight generic re- duction R from some hard non-interactive problem to the UUF-NMA-security of Schnorr signatures.<ref name="ref_1b944f9d" />
# I'm trying to understand the security of the short schnorr signature a little bit better.<ref name="ref_d5ffc39f">[https://crypto.stackexchange.com/questions/95345/security-proof-of-short-schnorr-signature Security Proof of Short Schnorr Signature]</ref>
# The Schnorr signatures (Schnorr, n.d.) have been known before ECDSA signatures, yet they were not so widely used due to the patent which expired in the year 2008.<ref name="ref_a00e3450">[https://mareknarozniak.com/2021/05/25/schnorr-signature/ Schnorr Signature]</ref>
# One of the advantages is the existence of proof that breaking the Schnorr signature is equivalent to breaking the discrete logarithm problem.<ref name="ref_a00e3450" />
# If you like to know more, I based this tutorial on what the heck is Schnorr medium article and cryptography fandom Schnorr signature page.<ref name="ref_a00e3450" />
# So Schnorr signature solves these 2 problems, it is non-malleable, which means #Bitcoin network becomes more secure.<ref name="ref_3c9b8920">[https://www.linkedin.com/pulse/cryptography-digital-signatures-schnorr-taproot-upgrade-nitesh-balusu Cryptography: Digital Signatures and Schnorr Signatures Explained-#Bitcoin Taproot Upgrade]</ref>
# A Schnorr signature is a digital signature produced by the Schnorr signature algorithm.<ref name="ref_02d5a128">[https://www.bitstamp.net/learn/blockchain/what-are-schnorr-signatures/ What are Schnorr Signatures?]</ref>
# Another advantage of Schnorr signatures is increased privacy in terms of securing your bitcoins.<ref name="ref_02d5a128" />
# By reducing the amount of signature data stored on the blockchain, Schnorr signatures free up block storage space.<ref name="ref_02d5a128" />
# But scaling is not the only way Schnorr signatures can improve the Bitcoin protocol.<ref name="ref_02d5a128" />
# FROST is a threshold Schnorr signature protocol that contains two important components.<ref name="ref_adf8fce0">[https://blog.coinbase.com/frost-flexible-round-optimized-schnorr-threshold-signatures-b2e950164ee1 FROST: Flexible Round-Optimized Schnorr Threshold Signatures]</ref>
# Afterwards, any t-out-of-n participants can run a threshold signing protocol to collaboratively generate a valid Schnorr signature.<ref name="ref_adf8fce0" />
# In addition, FROST also requires each participant to demonstrate knowledge of their own secret by sending to other participants a zero-knowledge proof, which itself is a Schnorr signature.<ref name="ref_adf8fce0" />
# To create a valid Schnorr signature, any t participants work together to execute this round.<ref name="ref_adf8fce0" />
# We have implemented Schnorr signatures on Bitcoin.<ref name="ref_2b5e61bd">[https://coingeek.com/schnorr-signatures-on-bitcoin/ Schnorr signatures on Bitcoin]</ref>
# In this blog post we will explain one of the main advantages of Schnorr signatures’: its native support for Multi-Signatures (MultiSig).<ref name="ref_d65743ce">[https://hackernoon.com/a-brief-intro-to-bitcoin-schnorr-multi-signatures-b9ef052374c5 A brief intro to Bitcoin Schnorr Multi-signatures]</ref>
# But Schnorr signatures can add a new advantage to CoinJoin.<ref name="ref_626bfe1f">[https://bitcoinmagazine.com/culture/the-power-of-schnorr-the-signature-algorithm-to-increase-bitcoin-s-scale-and-privacy-1460642496 The Power of Schnorr: The Signature Algorithm to Increase Bitcoin's Scale and Privacy]</ref>
# Note: The process of implementing Schnorr signatures in Bitcoin is still in the concept phase.<ref name="ref_626bfe1f" />
# Schnorr signatures can be proved secure in the random oracle model (ROM) under the discrete logarithm assumption (DL) by rewinding the adversary; but this security proof is loose.<ref name="ref_d7c091b0">[https://www.semanticscholar.org/paper/Blind-Schnorr-Signatures-in-the-Algebraic-Group-Fuchsbauer-Plouviez/abfbac3d8b2de10803b9df6fe6625090feddb991 PDF Blind Schnorr Signatures in the Algebraic Group Model]</ref>
# The written specication for Schnorr signatures should fully describe the algorithm.<ref name="ref_b8368f56">[https://courses.csail.mit.edu/6.857/2020/projects/4-Elbahrawy-Lovejoy-Ouyang-Perez.pdf Analysis of bitcoin improvement proposal 340]</ref>
# In the Bitcoin specication of Schnorr signatures, the public key Q is 32 bytes, and it can be converted from existing generated public keys by dropping the rst byte (the prex).<ref name="ref_b8368f56" />
# The Schnorr signature scheme is constructed by applying the Fiat-Shamir heuristic to Schnorrs identication protocol.<ref name="ref_b8368f56" />
# Schnorr signature is an alternative algorithm to Bitcoin’s original ECDSA.<ref name="ref_ee937991">[https://www.telemediaonline.co.uk/schnorr-signatures-role-in-bitcoin-transactions/ Schnorr Signatures Role in Bitcoin Transactions]</ref>
# Schnorr signatures are the second type of signatures scheme introduced with the Taproot upgrade to address some of the flaws of the ECDSA protocol.<ref name="ref_ee937991" />
# Schnorr signatures offer that advantage, allowing the Bitcoin network to optimize payment processing and data storage.<ref name="ref_ee937991" />
# That makes it impossible for chain analysis to distinguish between multi-sig and single-sign Bitcoin transactions with Schnorr signatures, ensuring enhanced privacy.<ref name="ref_ee937991" />
# In 2005, when elliptic curve cryptography was being standardized people built on top of DSA rather than Schnorr signatures that had advantages.<ref name="ref_3fd754e6">[https://diyhpl.us/wiki/transcripts/scalingbitcoin/milan/schnorr-signatures/ schnorr-signatures]</ref>
# What I want you to take away from this is Schnorr signatures are not an established standard.<ref name="ref_3fd754e6" />
# The security proof of Schnorr signatures says that they are existentially unforgeable under the assumptions I mentioned before.<ref name="ref_3fd754e6" />
# It turns out if you take Schnorr signatures naively and apply it to an elliptic curve group it has a really annoying interaction with BIP 32 when used with public derivation.<ref name="ref_3fd754e6" />
# Package schnorr implements the vanilla Schnorr signature scheme.<ref name="ref_a63ab04b">[https://pkg.go.dev/go.dedis.ch/kyber/sign/schnorr go.dedis.ch/kyber/sign/schnorr]</ref>
# To analyze the security of Schnorr signatures, we model the hash function as a random oracle.<ref name="ref_565a2e8d">[https://web.stanford.edu/class/cs259c/lectures/schnorr.pdf Schnorr identiﬁcation and signatures]</ref>
# We provide two necessary conditions on hash functions for the Schnorr signature scheme to be secure, assuming compact group rep- resentations such as those which occur in elliptic curve groups.<ref name="ref_6d25ea82">[http://www.neven.org/papers/schnorr.pdf Hash function requirements]</ref>
# First, since security does not rely on the hash function being collision resistant, Schnorr signatures can still be securely instantiated with SHA-1/SHA- 256, unlike DSA signatures.<ref name="ref_6d25ea82" />
# Apart from instantiation candidates for the hash function, our results have a number of other important implications for the eciency and security of Schnorr signatures.<ref name="ref_6d25ea82" />
# ii Abstract This thesis investigates implicit multi-party protocols based on Schnorr signature scheme and their benefits to the Bitcoin ecosystem.<ref name="ref_6be9281a">[https://is.muni.cz/th/oaxta/thesis.pdf Masaryk university]</ref>
# To demonstrate the practicality of Schnorr signatures, a solution for Bitcoin transaction cosigning is designed and implemented.<ref name="ref_6be9281a" />
# signature, Schnorr signature scheme, Bitcoin, JavaCard iv Contents 1 Introduction 2 Schnorr Signature Scheme 2.1 Alternative Formulation . .<ref name="ref_6be9281a" />
# The most prominent alternative signature scheme with the desired properties is the Schnorr signature scheme.<ref name="ref_6be9281a" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1465057 Q1465057]
===Spacy 패턴 목록===
* [{'LOWER': 'schnorr'}, {'LEMMA': 'signature'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LEMMA': 'scheme'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'signature'}, {'LEMMA': 'algorithm'}]
* [{'LOWER': 'sdsa'}]

== 노트 ==

===말뭉치===
# In the near future, Bitcoin will enable Schnorr signatures in addition to ECDSA signatures.<ref name="ref_db591c4c">[https://river.com/learn/what-are-schnorr-signatures/ What Do Schnorr Signatures Do for Bitcoin?]</ref>
# Schnorr signatures will be introduced to Bitcoin through Taproot upgrade, which will hopefully be activated around 2022.<ref name="ref_db591c4c" />
# Although developers have added all necessary code to Bitcoin Core, Bitcoin nodes must accept the upgrade in order to consider Schnorr signatures valid.<ref name="ref_db591c4c" />
# In this post I will explain what Schnorr signatures are and how they intuitively work.<ref name="ref_3440f6d6">[https://suredbits.com/introduction-to-schnorr-signatures/ Introduction to Schnorr Signatures]</ref>
# And that is all there is to the actual computation surrounding “vanilla” Schnorr signatures!<ref name="ref_3440f6d6" />
# But Schnorr signatures are much more elegant and simple, and it has one more magical property; linearity.<ref name="ref_27782bb7">[https://medium.com/bitbees/what-the-heck-is-schnorr-52ef5dba289f What The Heck Is Schnorr]</ref>
# Schnorr signature was invented by Claus-Peter Schnorr back in the 1980s.<ref name="ref_27782bb7" />
# Because of his patent, the Schnorr signature algorithm did not see any widespread use for decades.<ref name="ref_27782bb7" />
# Six more years later, in 2014 the first talk of implementing Schnorr signature on Bitcoin protocol came up in the bitcoin-talk forum.<ref name="ref_27782bb7" />
# In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr.<ref name="ref_492c923b">[https://www.geeksforgeeks.org/schnorr-digital-signature/ Schnorr Digital Signature]</ref>
# Note: When you construct the signature like this, it’s known as a Schnorr signature, which is discussed in a following section.<ref name="ref_440c07d2">[https://tlu.tarilabs.com/cryptography/introduction-schnorr-signatures Introduction to Schnorr Signatures]</ref>
# The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model.<ref name="ref_440c07d2" />
# We have implemented Schnorr signatures on Bitcoin.<ref name="ref_2b5e61bd">[https://coingeek.com/schnorr-signatures-on-bitcoin/ Schnorr signatures on Bitcoin]</ref>
# Historically, EdDSA is known as a variant of Schnorr signatures, which are well-studied and suitable for efficient thresholdization,...<ref name="ref_6a94b9af">[https://csrc.nist.gov/publications/detail/nistir/8214b/draft NISTIR 8214B (Draft), Notes on Threshold EdDSA/Schnorr Signatures]</ref>
# Schnorr signatures are quite simple compared to other schemes.<ref name="ref_dd62bb89">[https://academy.binance.com/en/articles/what-do-schnorr-signatures-mean-for-bitcoin What do Schnorr Signatures Mean for Bitcoin?]</ref>
# Schnorr signatures have been touted as a solution to these privacy and scalability issues.<ref name="ref_dd62bb89" />
# As with most upgrades to the Bitcoin protocol, it could take time for the broader community of Bitcoin users to agree on the Schnorr signature inclusion.<ref name="ref_dd62bb89" />
# Schnorr signatures could be merged into the code as a soft fork , meaning that a change would not split the network.<ref name="ref_dd62bb89" />
# We aim at designing a leakage-resilient variant of the Schnorr signature scheme whose secret key’s storage space is constant, independently of the amount of leakage that it can tolerate.<ref name="ref_4e437487">[https://link.springer.com/chapter/10.1007/978-3-642-45239-0_11 A Leakage-Resilient Pairing-Based Variant of the Schnorr Signature Scheme]</ref>
# We proceed by first proposing a pairing analogue of the Schnorr signature scheme, that we next transform to include split signing key updates.<ref name="ref_4e437487" />
# As the increased uptake in connected devices revives the interest in resource-constrained signature algorithms, we introduce a variant of Schnorr signatures that mutualises exponentiation eorts.<ref name="ref_d70872a7">[https://eprint.iacr.org/2018/069.pdf Reusing nonces in schnorr signatures]</ref>
# Sharing a nonce is a deadly blow to Schnorr signatures, but is not a security concern for our variant.<ref name="ref_d70872a7" />
# We start by reminding how the original Schnorr signature scheme works and explain how we extend it assuming that k is randomly drawn from Zp1.<ref name="ref_d70872a7" />
# 3.1 Our Signature Scheme Similar to the Schnorr signature scheme, our scheme is a tuple of algorithms (Setup, KeyGen, Sign, and Verify), which we dene as follows: Setup(1): Generate primes q1, . .<ref name="ref_d70872a7" />
# This has lead to a long line of research investigating the existence of tighter security proofs for Schnorr signatures.<ref name="ref_1b944f9d">[https://eprint.iacr.org/2013/418.pdf On tight security proofs for schnorr signatures]</ref>
# We begin with the hypothesis that there exists a tight generic re- duction R from some hard non-interactive problem to the UUF-NMA-security of Schnorr signatures.<ref name="ref_1b944f9d" />
# I'm trying to understand the security of the short schnorr signature a little bit better.<ref name="ref_d5ffc39f">[https://crypto.stackexchange.com/questions/95345/security-proof-of-short-schnorr-signature Security Proof of Short Schnorr Signature]</ref>
# So Schnorr signature solves these 2 problems, it is non-malleable, which means #Bitcoin network becomes more secure.<ref name="ref_3c9b8920">[https://www.linkedin.com/pulse/cryptography-digital-signatures-schnorr-taproot-upgrade-nitesh-balusu Cryptography: Digital Signatures and Schnorr Signatures Explained-#Bitcoin Taproot Upgrade]</ref>
# A Schnorr signature is a digital signature produced by the Schnorr signature algorithm.<ref name="ref_02d5a128">[https://www.bitstamp.net/learn/blockchain/what-are-schnorr-signatures/ What are Schnorr Signatures?]</ref>
# Another advantage of Schnorr signatures is increased privacy in terms of securing your bitcoins.<ref name="ref_02d5a128" />
# By reducing the amount of signature data stored on the blockchain, Schnorr signatures free up block storage space.<ref name="ref_02d5a128" />
# But scaling is not the only way Schnorr signatures can improve the Bitcoin protocol.<ref name="ref_02d5a128" />
# The Schnorr signatures (Schnorr, n.d.) have been known before ECDSA signatures, yet they were not so widely used due to the patent which expired in the year 2008.<ref name="ref_a00e3450">[https://mareknarozniak.com/2021/05/25/schnorr-signature/ Schnorr Signature]</ref>
# One of the advantages is the existence of proof that breaking the Schnorr signature is equivalent to breaking the discrete logarithm problem.<ref name="ref_a00e3450" />
# If you like to know more, I based this tutorial on what the heck is Schnorr medium article and cryptography fandom Schnorr signature page.<ref name="ref_a00e3450" />
# FROST is a threshold Schnorr signature protocol that contains two important components.<ref name="ref_adf8fce0">[https://blog.coinbase.com/frost-flexible-round-optimized-schnorr-threshold-signatures-b2e950164ee1 FROST: Flexible Round-Optimized Schnorr Threshold Signatures]</ref>
# Afterwards, any t-out-of-n participants can run a threshold signing protocol to collaboratively generate a valid Schnorr signature.<ref name="ref_adf8fce0" />
# In addition, FROST also requires each participant to demonstrate knowledge of their own secret by sending to other participants a zero-knowledge proof, which itself is a Schnorr signature.<ref name="ref_adf8fce0" />
# To create a valid Schnorr signature, any t participants work together to execute this round.<ref name="ref_adf8fce0" />
# In 2005, when elliptic curve cryptography was being standardized people built on top of DSA rather than Schnorr signatures that had advantages.<ref name="ref_3fd754e6">[https://diyhpl.us/wiki/transcripts/scalingbitcoin/milan/schnorr-signatures/ schnorr-signatures]</ref>
# What I want you to take away from this is Schnorr signatures are not an established standard.<ref name="ref_3fd754e6" />
# The security proof of Schnorr signatures says that they are existentially unforgeable under the assumptions I mentioned before.<ref name="ref_3fd754e6" />
# It turns out if you take Schnorr signatures naively and apply it to an elliptic curve group it has a really annoying interaction with BIP 32 when used with public derivation.<ref name="ref_3fd754e6" />
# But Schnorr signatures can add a new advantage to CoinJoin.<ref name="ref_626bfe1f">[https://bitcoinmagazine.com/culture/the-power-of-schnorr-the-signature-algorithm-to-increase-bitcoin-s-scale-and-privacy-1460642496 The Power of Schnorr: The Signature Algorithm to Increase Bitcoin's Scale and Privacy]</ref>
# Note: The process of implementing Schnorr signatures in Bitcoin is still in the concept phase.<ref name="ref_626bfe1f" />
# Schnorr signatures can be proved secure in the random oracle model (ROM) under the discrete logarithm assumption (DL) by rewinding the adversary; but this security proof is loose.<ref name="ref_d7c091b0">[https://www.semanticscholar.org/paper/Blind-Schnorr-Signatures-in-the-Algebraic-Group-Fuchsbauer-Plouviez/abfbac3d8b2de10803b9df6fe6625090feddb991 PDF Blind Schnorr Signatures in the Algebraic Group Model]</ref>
# Schnorr signature is an alternative algorithm to Bitcoin’s original ECDSA.<ref name="ref_ee937991">[https://www.telemediaonline.co.uk/schnorr-signatures-role-in-bitcoin-transactions/ Schnorr Signatures Role in Bitcoin Transactions]</ref>
# Schnorr signatures are the second type of signatures scheme introduced with the Taproot upgrade to address some of the flaws of the ECDSA protocol.<ref name="ref_ee937991" />
# Schnorr signatures offer that advantage, allowing the Bitcoin network to optimize payment processing and data storage.<ref name="ref_ee937991" />
# That makes it impossible for chain analysis to distinguish between multi-sig and single-sign Bitcoin transactions with Schnorr signatures, ensuring enhanced privacy.<ref name="ref_ee937991" />
# The written specication for Schnorr signatures should fully describe the algorithm.<ref name="ref_b8368f56">[https://courses.csail.mit.edu/6.857/2020/projects/4-Elbahrawy-Lovejoy-Ouyang-Perez.pdf Analysis of bitcoin improvement proposal 340]</ref>
# In the Bitcoin specication of Schnorr signatures, the public key Q is 32 bytes, and it can be converted from existing generated public keys by dropping the rst byte (the prex).<ref name="ref_b8368f56" />
# The Schnorr signature scheme is constructed by applying the Fiat-Shamir heuristic to Schnorrs identication protocol.<ref name="ref_b8368f56" />
# Package schnorr implements the vanilla Schnorr signature scheme.<ref name="ref_a63ab04b">[https://pkg.go.dev/go.dedis.ch/kyber/sign/schnorr go.dedis.ch/kyber/sign/schnorr]</ref>
# In this blog post we will explain one of the main advantages of Schnorr signatures’: its native support for Multi-Signatures (MultiSig).<ref name="ref_d65743ce">[https://hackernoon.com/a-brief-intro-to-bitcoin-schnorr-multi-signatures-b9ef052374c5 A brief intro to Bitcoin Schnorr Multi-signatures]</ref>
# We provide two necessary conditions on hash functions for the Schnorr signature scheme to be secure, assuming compact group rep- resentations such as those which occur in elliptic curve groups.<ref name="ref_6d25ea82">[http://www.neven.org/papers/schnorr.pdf Hash function requirements]</ref>
# First, since security does not rely on the hash function being collision resistant, Schnorr signatures can still be securely instantiated with SHA-1/SHA- 256, unlike DSA signatures.<ref name="ref_6d25ea82" />
# Apart from instantiation candidates for the hash function, our results have a number of other important implications for the eciency and security of Schnorr signatures.<ref name="ref_6d25ea82" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1465057 Q1465057]
===Spacy 패턴 목록===
* [{'LOWER': 'schnorr'}, {'LEMMA': 'signature'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LEMMA': 'scheme'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'signature'}, {'LEMMA': 'algorithm'}]
* [{'LOWER': 'sdsa'}]

== 노트 ==

===말뭉치===
# In the near future, Bitcoin will enable Schnorr signatures in addition to ECDSA signatures.<ref name="ref_db591c4c">[https://river.com/learn/what-are-schnorr-signatures/ What Do Schnorr Signatures Do for Bitcoin?]</ref>
# Schnorr signatures will be introduced to Bitcoin through Taproot upgrade, which will hopefully be activated around 2022.<ref name="ref_db591c4c" />
# Although developers have added all necessary code to Bitcoin Core, Bitcoin nodes must accept the upgrade in order to consider Schnorr signatures valid.<ref name="ref_db591c4c" />
# Schnorr signatures are quite simple compared to other schemes.<ref name="ref_dd62bb89">[https://academy.binance.com/en/articles/what-do-schnorr-signatures-mean-for-bitcoin What do Schnorr Signatures Mean for Bitcoin?]</ref>
# Schnorr signatures have been touted as a solution to these privacy and scalability issues.<ref name="ref_dd62bb89" />
# As with most upgrades to the Bitcoin protocol, it could take time for the broader community of Bitcoin users to agree on the Schnorr signature inclusion.<ref name="ref_dd62bb89" />
# Schnorr signatures could be merged into the code as a soft fork , meaning that a change would not split the network.<ref name="ref_dd62bb89" />
# In this post I will explain what Schnorr signatures are and how they intuitively work.<ref name="ref_3440f6d6">[https://suredbits.com/introduction-to-schnorr-signatures/ Introduction to Schnorr Signatures]</ref>
# And that is all there is to the actual computation surrounding “vanilla” Schnorr signatures!<ref name="ref_3440f6d6" />
# But Schnorr signatures are much more elegant and simple, and it has one more magical property; linearity.<ref name="ref_27782bb7">[https://medium.com/bitbees/what-the-heck-is-schnorr-52ef5dba289f What The Heck Is Schnorr]</ref>
# Schnorr signature was invented by Claus-Peter Schnorr back in the 1980s.<ref name="ref_27782bb7" />
# Because of his patent, the Schnorr signature algorithm did not see any widespread use for decades.<ref name="ref_27782bb7" />
# Six more years later, in 2014 the first talk of implementing Schnorr signature on Bitcoin protocol came up in the bitcoin-talk forum.<ref name="ref_27782bb7" />
# In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr.<ref name="ref_492c923b">[https://www.geeksforgeeks.org/schnorr-digital-signature/ Schnorr Digital Signature]</ref>
# Note: When you construct the signature like this, it’s known as a Schnorr signature, which is discussed in a following section.<ref name="ref_440c07d2">[https://tlu.tarilabs.com/cryptography/introduction-schnorr-signatures Introduction to Schnorr Signatures]</ref>
# The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model.<ref name="ref_440c07d2" />
# The main function of Schnorr signatures is to allow multiple users to create a single signature for all parties involved.<ref name="ref_3bdb1e89">[https://academy.bit2me.com/en/que-son-las-firmas-schnorr/ What are Schnorr signatures?]</ref>
# So the implementation of Schnorr signatures represents a real solution to this problem.<ref name="ref_3bdb1e89" />
# Historically, EdDSA is known as a variant of Schnorr signatures, which are well-studied and suitable for efficient thresholdization,...<ref name="ref_6a94b9af">[https://csrc.nist.gov/publications/detail/nistir/8214b/draft NISTIR 8214B (Draft), Notes on Threshold EdDSA/Schnorr Signatures]</ref>
# We aim at designing a leakage-resilient variant of the Schnorr signature scheme whose secret key’s storage space is constant, independently of the amount of leakage that it can tolerate.<ref name="ref_4e437487">[https://link.springer.com/chapter/10.1007/978-3-642-45239-0_11 A Leakage-Resilient Pairing-Based Variant of the Schnorr Signature Scheme]</ref>
# We proceed by first proposing a pairing analogue of the Schnorr signature scheme, that we next transform to include split signing key updates.<ref name="ref_4e437487" />
# As the increased uptake in connected devices revives the interest in resource-constrained signature algorithms, we introduce a variant of Schnorr signatures that mutualises exponentiation eorts.<ref name="ref_d70872a7">[https://eprint.iacr.org/2018/069.pdf Reusing nonces in schnorr signatures]</ref>
# Sharing a nonce is a deadly blow to Schnorr signatures, but is not a security concern for our variant.<ref name="ref_d70872a7" />
# We start by reminding how the original Schnorr signature scheme works and explain how we extend it assuming that k is randomly drawn from Zp1.<ref name="ref_d70872a7" />
# 3.1 Our Signature Scheme Similar to the Schnorr signature scheme, our scheme is a tuple of algorithms (Setup, KeyGen, Sign, and Verify), which we dene as follows: Setup(1): Generate primes q1, . .<ref name="ref_d70872a7" />
# This has lead to a long line of research investigating the existence of tighter security proofs for Schnorr signatures.<ref name="ref_1b944f9d">[https://eprint.iacr.org/2013/418.pdf On tight security proofs for schnorr signatures]</ref>
# We begin with the hypothesis that there exists a tight generic re- duction R from some hard non-interactive problem to the UUF-NMA-security of Schnorr signatures.<ref name="ref_1b944f9d" />
# So Schnorr signature solves these 2 problems, it is non-malleable, which means #Bitcoin network becomes more secure.<ref name="ref_3c9b8920">[https://www.linkedin.com/pulse/cryptography-digital-signatures-schnorr-taproot-upgrade-nitesh-balusu Cryptography: Digital Signatures and Schnorr Signatures Explained-#Bitcoin Taproot Upgrade]</ref>
# I'm trying to understand the security of the short schnorr signature a little bit better.<ref name="ref_d5ffc39f">[https://crypto.stackexchange.com/questions/95345/security-proof-of-short-schnorr-signature Security Proof of Short Schnorr Signature]</ref>
# A Schnorr signature is a digital signature produced by the Schnorr signature algorithm.<ref name="ref_02d5a128">[https://www.bitstamp.net/learn/blockchain/what-are-schnorr-signatures/ What are Schnorr Signatures?]</ref>
# Another advantage of Schnorr signatures is increased privacy in terms of securing your bitcoins.<ref name="ref_02d5a128" />
# By reducing the amount of signature data stored on the blockchain, Schnorr signatures free up block storage space.<ref name="ref_02d5a128" />
# But scaling is not the only way Schnorr signatures can improve the Bitcoin protocol.<ref name="ref_02d5a128" />
# The Schnorr signatures (Schnorr, n.d.) have been known before ECDSA signatures, yet they were not so widely used due to the patent which expired in the year 2008.<ref name="ref_a00e3450">[https://mareknarozniak.com/2021/05/25/schnorr-signature/ Schnorr Signature]</ref>
# One of the advantages is the existence of proof that breaking the Schnorr signature is equivalent to breaking the discrete logarithm problem.<ref name="ref_a00e3450" />
# If you like to know more, I based this tutorial on what the heck is Schnorr medium article and cryptography fandom Schnorr signature page.<ref name="ref_a00e3450" />
# We have implemented Schnorr signatures on Bitcoin.<ref name="ref_2b5e61bd">[https://coingeek.com/schnorr-signatures-on-bitcoin/ Schnorr signatures on Bitcoin]</ref>
# FROST is a threshold Schnorr signature protocol that contains two important components.<ref name="ref_adf8fce0">[https://blog.coinbase.com/frost-flexible-round-optimized-schnorr-threshold-signatures-b2e950164ee1 FROST: Flexible Round-Optimized Schnorr Threshold Signatures]</ref>
# Afterwards, any t-out-of-n participants can run a threshold signing protocol to collaboratively generate a valid Schnorr signature.<ref name="ref_adf8fce0" />
# In addition, FROST also requires each participant to demonstrate knowledge of their own secret by sending to other participants a zero-knowledge proof, which itself is a Schnorr signature.<ref name="ref_adf8fce0" />
# To create a valid Schnorr signature, any t participants work together to execute this round.<ref name="ref_adf8fce0" />
# In 2005, when elliptic curve cryptography was being standardized people built on top of DSA rather than Schnorr signatures that had advantages.<ref name="ref_3fd754e6">[https://diyhpl.us/wiki/transcripts/scalingbitcoin/milan/schnorr-signatures/ schnorr-signatures]</ref>
# What I want you to take away from this is Schnorr signatures are not an established standard.<ref name="ref_3fd754e6" />
# The security proof of Schnorr signatures says that they are existentially unforgeable under the assumptions I mentioned before.<ref name="ref_3fd754e6" />
# It turns out if you take Schnorr signatures naively and apply it to an elliptic curve group it has a really annoying interaction with BIP 32 when used with public derivation.<ref name="ref_3fd754e6" />
# But Schnorr signatures can add a new advantage to CoinJoin.<ref name="ref_626bfe1f">[https://bitcoinmagazine.com/culture/the-power-of-schnorr-the-signature-algorithm-to-increase-bitcoin-s-scale-and-privacy-1460642496 The Power of Schnorr: The Signature Algorithm to Increase Bitcoin's Scale and Privacy]</ref>
# Note: The process of implementing Schnorr signatures in Bitcoin is still in the concept phase.<ref name="ref_626bfe1f" />
# Schnorr signatures can be proved secure in the random oracle model (ROM) under the discrete logarithm assumption (DL) by rewinding the adversary; but this security proof is loose.<ref name="ref_d7c091b0">[https://www.semanticscholar.org/paper/Blind-Schnorr-Signatures-in-the-Algebraic-Group-Fuchsbauer-Plouviez/abfbac3d8b2de10803b9df6fe6625090feddb991 PDF Blind Schnorr Signatures in the Algebraic Group Model]</ref>
# The written specication for Schnorr signatures should fully describe the algorithm.<ref name="ref_b8368f56">[https://courses.csail.mit.edu/6.857/2020/projects/4-Elbahrawy-Lovejoy-Ouyang-Perez.pdf Analysis of bitcoin improvement proposal 340]</ref>
# In the Bitcoin specication of Schnorr signatures, the public key Q is 32 bytes, and it can be converted from existing generated public keys by dropping the rst byte (the prex).<ref name="ref_b8368f56" />
# The Schnorr signature scheme is constructed by applying the Fiat-Shamir heuristic to Schnorrs identication protocol.<ref name="ref_b8368f56" />
# Schnorr signature is an alternative algorithm to Bitcoin’s original ECDSA.<ref name="ref_ee937991">[https://www.telemediaonline.co.uk/schnorr-signatures-role-in-bitcoin-transactions/ Schnorr Signatures Role in Bitcoin Transactions]</ref>
# Schnorr signatures are the second type of signatures scheme introduced with the Taproot upgrade to address some of the flaws of the ECDSA protocol.<ref name="ref_ee937991" />
# Schnorr signatures offer that advantage, allowing the Bitcoin network to optimize payment processing and data storage.<ref name="ref_ee937991" />
# That makes it impossible for chain analysis to distinguish between multi-sig and single-sign Bitcoin transactions with Schnorr signatures, ensuring enhanced privacy.<ref name="ref_ee937991" />
# In this blog post we will explain one of the main advantages of Schnorr signatures’: its native support for Multi-Signatures (MultiSig).<ref name="ref_d65743ce">[https://hackernoon.com/a-brief-intro-to-bitcoin-schnorr-multi-signatures-b9ef052374c5 A brief intro to Bitcoin Schnorr Multi-signatures]</ref>
# Package schnorr implements the vanilla Schnorr signature scheme.<ref name="ref_a63ab04b">[https://pkg.go.dev/go.dedis.ch/kyber/sign/schnorr go.dedis.ch/kyber/sign/schnorr]</ref>
# We provide two necessary conditions on hash functions for the Schnorr signature scheme to be secure, assuming compact group rep- resentations such as those which occur in elliptic curve groups.<ref name="ref_6d25ea82">[http://www.neven.org/papers/schnorr.pdf Hash function requirements]</ref>
# First, since security does not rely on the hash function being collision resistant, Schnorr signatures can still be securely instantiated with SHA-1/SHA- 256, unlike DSA signatures.<ref name="ref_6d25ea82" />
# Apart from instantiation candidates for the hash function, our results have a number of other important implications for the eciency and security of Schnorr signatures.<ref name="ref_6d25ea82" />
# Our work uses Schnorr signatures and leverages Bitcoin recent Taproot upgrade, allowing us to create a checkpointing transaction of constant size.<ref name="ref_0d6bbf83">[https://arxiv.org/pdf/2208.05408 Pikachu: securing pos blockchains from long-range attacks by]</ref>
# To overcome these weaknesses in the Ma-Chen scheme, we propose a new scheme based on the Schnorr signature.<ref name="ref_cf714458">[https://arxiv.org/pdf/cs/0504019 5]</ref>
# == r = (H(m) + r.x) (H(m) + r.x)1 (cid:0)k1(cid:1)1 G = k G = R 2.1.2 Schnorr The Schnorr signature variant over ECC has multiple standards.<ref name="ref_398d6469">[https://arxiv.org/pdf/2110.00274 1]</ref>
# This property allows Schnorr signatures to be aggregated easily to construct a multi-party signature.<ref name="ref_398d6469" />
===소스===
<references />

== 메타데이터 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q1465057 Q1465057]
===Spacy 패턴 목록===
* [{'LOWER': 'schnorr'}, {'LEMMA': 'signature'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'digital'}, {'LOWER': 'signature'}, {'LEMMA': 'scheme'}]
* [{'LOWER': 'schnorr'}, {'LOWER': 'signature'}, {'LEMMA': 'algorithm'}]
* [{'LOWER': 'sdsa'}]

슈노르 서명

2022-09-16T03:19:55Z

Pythagoras0: /* 노트 */ 새 문단

슈노르 서명

2022-09-16T02:27:11Z

Pythagoras0: /* 메타데이터 */ 새 문단