### 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.

Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. SOSA 2022
Keywords
Contact author(s)
monika trimoska @ ru nl
charles bouillaguet @ lip6 fr
claire delaplace @ u-picardie fr
History
Short URL
https://ia.cr/2021/1639

CC BY

BibTeX

@misc{cryptoeprint:2021/1639,
author = {Charles Bouillaguet and Claire Delaplace and Monika Trimoska},
title = {A Simple Deterministic Algorithm for Systems of Quadratic Polynomials over $\mathbb{F}_2$},
howpublished = {Cryptology ePrint Archive, Paper 2021/1639},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1639}},
url = {https://eprint.iacr.org/2021/1639}
}

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