Algorithm for Solving Massively Underdefined Systems of Multivariate Quadratic Equations over Finite Fields

Solving systems of m multivariate quadratic equations in n variables (MQ-problem) over finite fields is NP-hard. The security of many cryptographic systems is based on this problem. Up to now, the best algorithm for solving the underdefined MQ-problem is Hiroyuki Miura et al.'s algorithm, which is a polynomial-time algorithm when $n \ge m(m + 3)/2$ and the characteristic of the field is even. In order to get a wider applicable range, we reduce the underdefined MQ-problem to the problem of finding square roots over finite field, and then combine with the guess and determine method. In this way, the applicable range is extended to $n \ge m(m + 1)/2$, which is the widest range until now. Theory analysis indicates that the complexity of our algorithm is $O(q{n^ω}m{(\log {\kern 1pt} {\kern 1pt} q)^2}){\kern 1pt}$ when characteristic of the field is even and $O(q{2^m}{n^ω}m{(\log {\kern 1pt} {\kern 1pt} q)^2})$ when characteristic of the field is odd, where $2 \le ω\le 3$ is the complexity of Gaussian elimination.

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