Cryptology ePrint Archive: Report 2017/372

A crossbred algorithm for solving Boolean polynomial systems

Antoine Joux and Vanessa Vitse

Abstract: We consider the problem of solving multivariate systems of Boolean polynomial equations: starting from a system of $m$ polynomials of degree at most $d$ in $n$ variables, we want to find its solutions over $\F_2$. Except for $d=1$, the problem is known to be NP-hard, and its hardness has been used to create public cryptosystems; this motivates the search for faster algorithms to solve this problem. After reviewing the state of the art, we describe a new algorithm and show that it outperforms previously known methods in a wide range of relevant parameters. In particular, the first named author has been able to solve all the Fukuoka Type~I MQ challenges, culminating with the resolution of a system of 148 quadratic equations in 74 variables in less than a day (and with a lot of luck).

Category / Keywords: multivariate polynomial systems, Groebner basis, XL, multivariate cryptography, algebraic cryptanalysis

Date: received 25 Apr 2017, last revised 25 Apr 2017

Contact author: vanessa vitse at univ-grenoble-alpes fr

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Version: 20170428:165226 (All versions of this report)

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