Abstract

Systems of polynomial equations are one of the most universal mathematical objects. Almost all problems of cryptographic analysis can be reduced to solving systems of polynomial equations. The corresponding direction of research is called algebraic cryptanalysis. In terms of computational complexity, systems of polynomial equations cover the entire range of possible variants, from the algorithmic insolubility of Diophantine equations to well-known efficient methods for solving linear systems. Buchberger’s method [5] brings the system of algebraic equations to a system of a special type defined by the Gröbner original system of equations, which enables the elimination of dependent variables. The Gröbner basis is determined based on an admissible ordering on a set of terms. The set of admissible orderings on the set of terms is infinite and even continual. The most time-consuming step in finding the Gröbner basis by using Buchberger’s algorithm is to prove that all S-polynomials represent a system of generators of K[X]-module S-polynomials. Thus, a natural problem of finding this minimal system of generators arises. The existence of this system follows from Nakayama’s lemma. In this paper, we propose an algorithm for constructing this basis for any ordering.

中文翻译：

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前导词的Syzygy模块的最小基础

摘要

多项式方程组是最通用的数学对象之一。密码分析的几乎所有问题都可以简化为多项式方程组的求解。相应的研究方向称为代数密码分析。在计算复杂度方面，多项式方程组涵盖了可能的变体的整个范围，从Diophantine方程的算法不溶性到著名的求解线性系统的有效方法。Buchberger的方法[5]将代数方程组带到由Gröbner原始方程组定义的特殊类型的系统，该系统可以消除因变量。Gröbner基础是根据一组条件下的可接受顺序确定的。一组术语上的一组可接受的顺序是无限的，甚至是连续的。使用Buchberger算法找到Gröbner基础的最耗时的步骤是证明所有S-多项式表示K [ X ]-模块S-多项式的生成器的系统。因此，出现了寻找这种最小的发电机系统的自然问题。该系统的存在源于中山引理。在本文中，我们提出了一种用于为任何排序构建此基础的算法。