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

*Charles Bouillaguet and 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.

**Category / Keywords: **public-key cryptography / Boolean quadratic polynomials, exhaustive search, linear algebra

**Original Publication**** (in the same form): **SOSA 2022

**Date: **received 15 Dec 2021

**Contact author: **monika trimoska at ru nl, charles bouillaguet at lip6 fr, claire delaplace at u-picardie fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20211217:142616 (All versions of this report)

**Short URL: **ia.cr/2021/1639

[ Cryptology ePrint archive ]