Paper 2021/1639

A Simple Deterministic Algorithm for Systems of Quadratic Polynomials over $\mathbb{F}_2$

Charles Bouillaguet, Claire Delaplace, and Monika Trimoska

Abstract

This article discusses a simple deterministic algorithm for solving quadratic Boolean systems which is essentially a special case of more sophisticated methods. The main idea fits in a single sentence: guess enough variables so that the remaining quadratic equations can be solved by linearization (i.e. by considering each remaining monomial as an independent variable and solving the resulting linear system) and restart until the solution is found. Under strong heuristic assumptions, this finds all the solutions of $m$ quadratic polynomials in $n$ variables with $\mathcal{\tilde O}({2^{n-\sqrt{2m}}})$ operations. Although the best known algorithms require exponentially less time, the present technique has the advantage of being simpler to describe and easy to implement. In strong contrast with the state-of-the-art, it is also quite efficient in practice.