# 노트 디버깅

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## 노트

### 말뭉치

- A finite field with 256 elements would be written as GF(2^8) .
^{[1]} - You can’t have a finite field with 12 elements since you’d have to write it as 2^2 * 3 which breaks the convention of p^m .
^{[1]} - The notation GF(p) means we have a finite field with the integers {0, … , p-1} .
^{[1]} - It seemed like our finite field was coming along perfectly until we came across the identity for 0.
^{[1]} - But it may be surprising that there are finite fields.
^{[2]} - All finite fields have pn elements where p is prime and n is an integer at least 1.
^{[2]} - The field with pn elements is sometimes called the Galois field with that many elements, written GF(pn).
^{[2]} - The Galois fields of order GF(p) are simply the integers mod p. For n > 1, the elements of GF(pn) are polynomials of degree n-1 with coefficients coming from GF(p).
^{[2]} - These are based on linear recurring sequences in finite fields, and in most practical implementations on maximal period sequences (compare with Section 6).
^{[3]} - Multidimensional analogs of pseudorandom numbers are pseudorandom vectors, which can also be generated by means of finite fields.
^{[3]} - In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
^{[4]} - As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
^{[4]} - The most common examples of finite fields are given by the integers mod p when p is a prime number.
^{[4]} - Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.
^{[4]} - A finite field must be a finite dimensional vector space, so all finite fields have degrees.
^{[5]} - The previous result does not prove the existence of finite fields of these sizes.
^{[5]} - Constructing Finite Fields There are several ways to represent the elements of a finite field.
^{[5]} - Another idea that can be used as a basis for a representation is the fact that the non-zero elements of a finite field can all be written as powers of a primitive element.
^{[5]} - Finite fields as splitting fields Each nite eld is a splitting eld of a polynomial depending only on the elds size.
^{[6]} - A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field.
^{[7]} - For each prime power, there exists exactly one (with the usual caveat that "exactly one" means "exactly one up to an isomorphism") finite field GF( ), often written as in current usage.
^{[7]} - Note, however, that in the ring of residues modulo 4, so 2 has no reciprocal, and the ring of residues modulo 4 is distinct from the finite field with four elements.
^{[7]} - Finite fields are therefore denoted GF( ), instead of GF( ), where , for clarity.
^{[7]} - To understand IDEA, AES, and some other modern cryptosystems, it is necessary to understand a bit about finite fields.
^{[8]} - The fields that we commonly used in mathematics courses ( For cryptological purposes, finite fields are useful.
^{[8]} - Q R , Finite field of p elements Recall that the integers mod 26 do not form a field.
^{[8]} - Two basic finite fields are:Create a finite field with q = p^n elements usingThis creates the ring of characteristic 3, having 3^4 = 81 elements.
^{[9]} - Finite fields can be used as base rings for polynomial rings.
^{[9]} - In general, to make a finite field withelements, we use GF The generator of the field can be obtained as usual.
^{[9]} - If you have a quotient ring that you know is a finite field, then you can convert it to ring that is known by the system to be a finite field.
^{[9]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Learning Cryptography, Part 1: Finite Fields - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Finite fields (Galois fields) - ↑
^{3.0}^{3.1}Finite Field - an overview - ↑
^{4.0}^{4.1}^{4.2}^{4.3}Finite field - ↑
^{5.0}^{5.1}^{5.2}^{5.3}Finite fields - ↑ Finite fields
- ↑
^{7.0}^{7.1}^{7.2}^{7.3}Finite Field -- from Wolfram MathWorld - ↑
^{8.0}^{8.1}^{8.2}Introduction to finite fields - ↑
^{9.0}^{9.1}^{9.2}^{9.3}finite fields

## 메모

- 30번 문장