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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 Zech's logarithm ： ウィキペディア英語版 Zech's logarithm Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator $\backslash alpha$. Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms,〔 〕 after C. G. J. Jacobi who used them for number theoretic investigations (C. G. J. Jacoby, "Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie", in Gesammelte Werke, Vol.6, pp. 254–274). ==Definition== If $\backslash alpha$ is a primitive element of a finite field, then the Zech logarithm relative to the base $\backslash alpha$ is defined by the equation :$Z\_\backslash alpha(n)\; =\; \backslash log\_\backslash alpha(1\; +\; \backslash alpha^n),$ or equivalently by :$\backslash alpha^\; =\; 1\; +\; \backslash alpha^n.$ The choice of base $\backslash alpha$ is usually dropped from the notation when it's clear from context. To be more precise, $Z\_\backslash alpha$ is a function on the integers modulo the multiplicative order of $\backslash alpha$, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol $-\backslash infty$, along with the definitions :$\backslash alpha^\; =\; 0$ :$n\; +\; (-\backslash infty)\; =\; -\backslash infty$ :$Z\_\backslash alpha(-\backslash infty)\; =\; 0$ :$Z\_\backslash alpha(e)\; =\; -\backslash infty$ where $e$ is an integer satisfying $\backslash alpha^e\; =\; -1$, that is $e=0$ for a field of characteristic 2, and $e=\backslash frac$ for a field of odd characteristic with $q$ elements. Using the Zech logarithm, finite field arithmetic can be done in the exponential representation: :$\backslash alpha^m\; +\; \backslash alpha^n\; =\; \backslash alpha^m\; \backslash cdot\; (1\; +\; \backslash alpha^)\; =\; \backslash alpha^m\; \backslash cdot\; \backslash alpha^\; =\; \backslash alpha^$ :$-\backslash alpha^n\; =\; (-1)\; \backslash cdot\; \backslash alpha^n\; =\; \backslash alpha^e\; \backslash cdot\; \backslash alpha^n\; =\; \backslash alpha^$ :$\backslash alpha^m\; -\; \backslash alpha^n\; =\; \backslash alpha^m\; +\; (-\backslash alpha^n)\; =\; \backslash alpha^$ :$\backslash alpha^m\; \backslash cdot\; \backslash alpha^n\; =\; \backslash alpha^$ :$\backslash left(\; \backslash alpha^m\; \backslash right)^\; =\; \backslash alpha^$ :$\backslash alpha^m\; /\; \backslash alpha^n\; =\; \backslash alpha^m\; \backslash cdot\; \backslash left(\; \backslash alpha^n\; \backslash right)^\; =\; \backslash alpha^$ These formulas remain true with our conventions with the symbol $-\backslash infty$, with the caveat that subtraction of $-\backslash infty$ is undefined. In particular, the addition and subtraction formulas need to treat $m\; =\; -\backslash infty$ as a special case. This can be extended to arithmetic of the projective line by introducing another symbol $+\backslash infty$ satisfying $\backslash alpha^\; =\; \backslash infty$ and other rules as appropriate. Notice that for fields of characteristic two, :$Z\_\backslash alpha(n)\; =\; m$ ⇔ $Z\_\backslash alpha(m)\; =\; n$. 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「Zech's logarithm」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.