The complexity of solving a system of equations of the same degree

TL;DR

This paper establishes upper bounds on the complexity of solving systems of polynomial equations of the same degree, which are common in cryptography, by analyzing their degree of regularity and solving degree using Gr"obner bases.

Contribution

It provides new theoretical bounds on the solving degree of such systems, considering the number of equations, variables, and degree, enhancing understanding of their computational complexity.

Findings

Derived upper bounds on the degree of regularity.

Translated bounds into solving degree estimates.

Applied results to systems with and without field equations.

Abstract

Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gr\"obner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.