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

TL;DR

This paper introduces an extended algorithm for solving underdefined multivariate quadratic equations over finite fields, broadening the applicable range and analyzing its complexity for cryptographic applications.

Contribution

It reduces the MQ-problem to square root finding and combines it with guess-and-determine, extending the solvable range to n ≥ m(m+1)/2, the widest so far.

Findings

Applicable range extended to n ≥ m(m+1)/2

Algorithm complexity analyzed for even and odd characteristic fields

Provides a more efficient method for cryptographic system security analysis

Abstract

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 \[O(q{n^\omega }m{(\log {\kern 1pt} {\kern 1pt} q)^2}){\kern 1pt} \] when characteristic of the field is even and \[O(q{2^m}{n^\omega }m{(\log {\kern 1pt} {\kern 1pt} q)^2})\] when characteristic of the field is odd, where \[2 \le \omega \le 3\] is the complexity of Gaussian elimination.