Minimal basis of the syzygies module of leading terms

Abstract

Systems of polynomial equations are one of the most universal mathematical objects. Almost all the problems of cryptographic analysis can be reduced to finding solutions to 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 options, from algorithmic insolubility of Diophantine equations to well-known efficient methods for solving linear systems. The method of Buchberger [ 5] brings a system of algebraic equations to the system of a special type defined by the Gröbner original system of equations, allowing the use of the exception of the dependent variables. The basis for determining the Groebner basis is the permissible ordering on the set of terms. The set of admissible orderings on the set of terms is infinite and even continuum. The most time-consuming step in finding the Groebner basis using the Buchberger algorithm is to prove that all S-polynomials representing a system of generators of K[X]-module S-polynomials. There is a natural problem of finding such a minimal system of generators. The existence of such a system of generators follows from Nakayama's theorem. An algorithm for constructing such a basis for any ordering is proposed.