Solving multi-variate polynomial equations in a finite field

Abstract

Solving large systems of multivariate polynomial equations is an active area of mathematical research, as these polynomials are used in many fields of science. The objective of this research is to advance the development of algebraic methods to attack the mathematical foundations of modern-day encryption methods, which can be modeled as a system of multivariate polynomial equations over a finite field. Our techniques overcome the limitations of previous methods. Additionally, a model is proposed to estimate the time required to solve large systems with our methods. All of these elements were tested successfully on AES and its predecessor, Square. The results showed our techniques to be comparable with a brute force technique. To the best of our knowledge, no other purely algebraic attack on AES has been shown to be this efficient.