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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 learning with errors ： ウィキペディア英語版 learning with errors Learning with errors (LWE) is a problem in machine learning that is conjectured to be hard to solve. Introduced〔 by Oded Regev in 2005, it is a generalization of the parity learning problem. Regev showed, furthermore, that the LWE problem is as hard to solve as several worst-case lattice problems. The LWE problem has recently〔Oded Regev, “On lattices, learning with errors, random linear codes, and cryptography,” in Proceedings of the thirty-seventh annual ACM symposium on Theory of computing (Baltimore, MD, USA: ACM, 2005), 84-93, http://portal.acm.org/citation.cfm?id=1060590.1060603.〕〔Chris Peikert, “Public-key cryptosystems from the worst-case shortest vector problem: extended abstract,” in Proceedings of the 41st annual ACM symposium on Theory of computing (Bethesda, MD, USA: ACM, 2009), 333-342, http://portal.acm.org/citation.cfm?id=1536414.1536461.〕 been used as a hardness assumption to create public-key cryptosystems. such as the ring learning with errors key exchange by Peikert. An algorithm is said to solve the LWE problem if, when given access to samples $(x,y)$ where $x\backslash in\; \backslash mathbb\_q^n$ and $y\; \backslash in\; \backslash mathbb\_q$, with the assurance, for some fixed linear function $f:\backslash mathbb\_q^n\; \backslash rightarrow\; \backslash mathbb\_q,$ that $y=f(x)$ with high probability and deviates from it according to some known noise model, the algorithm can recreate $f$ or some close approximation of it with high probability. == Definition == Denote by $\backslash mathbb=\backslash mathbb/\backslash mathbb$ the additive group on reals modulo one. Denote by $A\_$ the distribution on $\backslash mathbb\_q^n\; \backslash times\; \backslash mathbb$ obtained by choosing a vector $\backslash mathbf\backslash in\; \backslash mathbb\_q^n$ uniformly at random, choosing $e$ according to a probability distribution $\backslash phi$ on $\backslash mathbb$ and outputting $(\backslash mathbf,\backslash langle\; \backslash mathbf,\backslash mathbf\; \backslash rangle\; /q\; +\; e)$ for some fixed vector $\backslash mathbf\; \backslash in\; \backslash mathbb\_q^n$. Here $\backslash textstyle\; \backslash langle\; \backslash mathbf,\backslash mathbf\; \backslash rangle\; =\; \backslash sum\_^n\; a\_i\; s\_i$ is the standard inner product $\backslash mathbb\_q^n\; \backslash times\; \backslash mathbb\_q^n\; \backslash longrightarrow\; \backslash mathbb\_q$, the division is done in the field of reals (or more formally, this "division by $q$" is notation for the group homomorphism $\backslash mathbb\_q\; \backslash longrightarrow\; \backslash mathbb$ mapping $1\; \backslash in\; \backslash mathbb\_q$ to $1/q\; +\; \backslash mathbb\; \backslash in\; \backslash mathbb$), and the final addition is in $\backslash mathbb$. The learning with errors problem $\backslash mathrm\_$ is to find $\backslash mathbf\; \backslash in\; \backslash mathbb\_q^n$, given access to polynomially many samples of choice from $A\_$. For every $\backslash alpha\; >\; 0$, denote by $D\_\backslash alpha$ the one-dimensional Gaussian with density function $D\_\backslash alpha(x)=\backslash rho\_\backslash alpha(x)/\backslash alpha$ where $\backslash rho\_\backslash alpha(x)=e^$, and let $\backslash Psi\_\backslash alpha$ be the distribution on $\backslash mathbb$ obtained by considering $D\_\backslash alpha$ modulo one. The version of LWE considered in most of the results would be $\backslash mathrm\_$ 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「learning with errors」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.