Cryptology ePrint Archive: Report 2021/1639

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)

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