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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 Hidden Field Equations ： ウィキペディア英語版 Hidden Field Equations Hidden Fields Equations (HFE) is a public key cryptosystem which was introduced at Eurocrypt in 1996 and proposed by Jacques Patarin following the idea of the Matsumoto and Imai system. HFE is also known as HFE trapdoor function. It is based on polynomials over finite fields $\backslash mathbb\_q$ of different size to disguise the relationship between the private key and public key. HFE is in fact a family which consists of basic HFE and combinatorial versions of HFE. The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate quadratic equations (the so-called MQ problem) since it uses private affine transformations to hide the extension field and the private polynomials. Hidden Field Equations also have been used to construct digital signature schemes, e.g. Quartz and Sflash.〔(Christopher Wolf and Bart Preneel, Asymmetric Cryptography: Hidden Field Equations )〕 == Mathematical background == One of the central notions to understand how Hidden Field Equations work is to see that for two extension fields $\backslash mathbb\_$ $\backslash mathbb\_$ over the same base field $\backslash mathbb\_q$ one can interpret a system of $m$ multivariate polynomials in $n$ variables over $\backslash mathbb\_q$ as a function $\backslash mathbb\_\; \backslash to\; \backslash mathbb\_$ by using a suitable basis of $\backslash mathbb\_$ over $\backslash mathbb\_q$. In almost all applications the polynomials are quadratic, i.e. they have degree 2.〔(Nicolas T. Courtois On Multivariate Signature-only public key cryptosystems )〕 We start with the simplest kind of polynomials, namely monomials, and show how they lead to quadratic systems of equations. Let us consider a finite field $\backslash mathbb\_q$, where $q$ is a power of 2, and an extension field $K$. Let $\backslash beta\_1,...,\backslash beta\_n$ to be a basis of $K$ as an $\backslash mathbb\_q$ vector space. Let :$v=u^\; u\; \backslash \; \backslash \; \backslash \; \backslash \; (1)$ The condition gcd$(h,q^n-1)\; =1$ is equivalent to requiring that the map $u\; \backslash to\; u^h$ on $K$ is one to one and its inverse is the map $u\; \backslash to\; u^$ where $h\text{'}$ is the multiplicative inverse of $h\; \backslash \; \backslash bmod\; q^n-1$. Choose two secret affine transformation, i.e. two invertible $n\backslash times\; n$ matrices $S=\backslash$ and $T=\backslash$ with entries in $\backslash mathbb\_q$ and two vectors $c=(c\_1,...,c\_n)$ and $d=(d\_1,...,d\_n)$ of length $n$ over $\backslash mathbb\_q$ and define $x$ and $y$ via: :$u=Sx+c\; \backslash \; \backslash \; \backslash \; \backslash \; v=Ty+d\; \backslash \; \backslash \; \backslash \; \backslash \; (2)$ Let $A^=\}$ be the matrix of linear transformation in the basis $\backslash beta\_1,...,\backslash beta\_n$ such that :$\backslash beta\_^=\backslash sum\_^\; a\_^\backslash beta\_,\backslash \; \backslash \; a\_^\backslash in\backslash mathbb\_q$ for $1\backslash le\; i,k\backslash le\; n$. Write all products of basis elements in terms of the basis, i.e.: :$\backslash beta\_i\backslash beta\_j=\backslash sum\_^m\_\backslash beta\_,\backslash \; \backslash \; m\_\backslash in\backslash mathbb\_q$ for each $1\backslash le\; i,j\backslash le\; n$. The system of $n$ equations which is explicit in the $v\_i$ and quadratic in the $u\_j$ can be obtain by expanding (1) and equating to zero the coefficients of the $\backslash beta\_i$. By using the affine relations in (2) to replace the $u\_j,\; v\_i$ with $x\_k,y\_l$, the system of $n$ equations is linear in the $y\_l$ and of degree 2 in the $x\_k$. Applying linear algebra it will give $n$ explicit equations, one for each $y\_l$ as polynomials of degree 2 in the $x\_k$.〔(Ilia Toli Hidden Polynomial Cryptosystems )〕 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「Hidden Field Equations」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.