Minimal Basis of the Syzygy Module of Leading Terms

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.