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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Schoof's algorithm ： ウィキペディア英語版 Schoof's algorithm Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the number of points to judge the difficulty of solving the discrete logarithm problem in the group of points on an elliptic curve. The algorithm was published by René Schoof in 1985 and it was a theoretical breakthrough, as it was the first deterministic polynomial time algorithm for counting points on elliptic curves. 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. This article explains Schoof's approach, laying emphasis on the mathematical ideas underlying the structure of the algorithm. ==Introduction== Let $E$ be an elliptic curve defined over the finite field $\backslash mathbb\_q$, where $q=p^n$ for $p$ a prime and $n$ an integer $\backslash geq\; 1$. Over a field of characteristic $\backslash neq\; 2,\; 3$ an elliptic curve can be given by a (short) Weierstrass equation : $$ y^2 = x^3 + Ax + B \, with $A,B\backslash in\; \backslash mathbb\_$. The set of points defined over $\backslash mathbb\_$ consists of the solutions $(a,b)\backslash in\backslash mathbb\_^2$ satisfying the curve equation and a point at infinity $O$. Using the group law on elliptic curves restricted to this set one can see that this set $E(\backslash mathbb\_)$ forms an abelian group, with $O$ acting as the zero element. In order to count points on an elliptic curve, we compute the cardinality of $E(\backslash mathbb\_)$. Schoof's approach to computing the cardinality $\backslash sharp\; E(\backslash mathbb\_)$ makes use of Hasse's theorem on elliptic curves along with the Chinese remainder theorem and division polynomials. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schoof's algorithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.