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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 Block Wiedemann algorithm ： ウィキペディア英語版 Block Wiedemann algorithm The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalisation of an algorithm due to Don Coppersmith. == Coppersmith's algorithm == Let M be an $n\backslash times\; n$ square matrix over some finite field F, let $x\_\}$. Consider the sequence of vectors $S\; =\; \backslash left(Mx,\; M^2x,\; \backslash ldots\backslash right)$ obtained by repeatedly multiplying the vector by the matrix M; let y be any other vector of length n, and consider the sequence of finite-field elements $S\_y\; =\; \backslash left(\backslash cdot\; x,\; y\; \backslash cdot\; Mx,\; y\; \backslash cdot\; M^2x\; \backslash ldots\backslash right)$ We know that the matrix M has a minimal polynomial; by the Cayley–Hamilton theorem we know that this polynomial is of degree (which we will call $n\_0$) no more than n. Say $\backslash sum\_^\; p\_rM^r\; =\; 0$. Then $\backslash sum\_^\; y\; \backslash cdot\; (p\_r\; (M^r\; x))\; =\; 0$; so the minimal polynomial of the matrix annihilates the sequence $S$ and hence $S\_y$. But the Berlekamp–Massey algorithm allows us to calculate relatively efficiently some sequence $q\_0\; \backslash ldots\; q\_L$ with $\backslash sum\_^L\; q\_i\; S\_y()=0\; \backslash forall\; r$. Our hope is that this sequence, which by construction annihilates $y\; \backslash cdot\; S$, actually annihilates $S$; so we have $\backslash sum\_^L\; q\_i\; M^i\; x\; =\; 0$. We then take advantage of the initial definition of $x$ to say $M\; \backslash sum\_^L\; q\_i\; M^i\; x\_^L\; q\_i\; M^i\; x\_\{\backslash mathrm\; \{base\}\}$ is a hopefully non-zero kernel vector of $M$. 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「Block Wiedemann 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.