Schoof's algorithm

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1. The algorithm is an extension of Schoof's algorithm by Noam Elkies and A. O. L. Atkin to significantly improve its efficiency (under heuristic assumptions).
2. The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes S = { l 1 , … , l s } {\displaystyle S=\{l_{1},\ldots ,l_{s}\}} considered to primes of a certain kind.
3. For Elkies primes, this allows one to compute the number of points on E {\displaystyle E} modulo l {\displaystyle l} more efficiently than in Schoof's algorithm.
4. This is an implementation of Schoof's algorithm for counting the points on elliptic curves over finite fields (Schoof, René.
5. René Schoof's algorithm for counting the points on an elliptic curve over a finite field is the foundation for the (asymptotically) fastest Schoof–Elkies–Atkin counting algorithm.
6. Schoof's algorithm uses arithmetic on elliptic curves, finite fields, rings of polynomials, and quotient rings.
7. Before Schoof's algorithm, approaches to counting points on elliptic curves such as the naive and baby-step giant-step algorithms were, for the most part, tedious and had an exponential running time.
8. In any case Schoof's algorithm is most frequently used in addressing the case since there are more efficient, so called adic algorithms for small characteristic fields.
9. Schoof's algorithm stores the values of in a variable for each prime considered.
10. Given that this computation needs to be carried out for each of the primes, the total complexity of Schoof's algorithm turns out to be .
11. Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.
12. We define the curve, and compute the order, just to know which is the number we try to exhibit using Schoof's algorithm.
13. We also present an implementation of Schoof's algorithm as a collection of Mathematica functions.
14. Chapter 4 describes some methods for computing the elliptic curve group order, and includes an introduction to Schoof's algorithm.
15. We present the details of Schoof's algorithm in chapter 5.
16. I know Schoof's algorithm is mostly used over large prime fields.
17. Schoof's algorithm, first described by R. Schoof in 1985 , allows one to calculate the number of points on an elliptic curve over a finite field and is used mostly in elliptic curve cryptography .
18. mid-90s: lots of speed-ups, characteristic-2 algorithms note: basic Schoof algorithm also applicable for hyperelliptic curves; see Eric Schosts talk next week at ECC p.3 1.
19. In todays lecture, we study Schoof algorithm to compute the number of rational points of an elliptic curve E/Fq over a nite eld.

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• [{'LOWER': 'schoof'}, {'LEMMA': 'algorithm'}]