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

Multivariate Quadratic Equations over Finite Fields Heliang Huang, Wansu Bao* Zhengzhou Information Science and Technology Institute, Zhengzhou 450000, China 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 ( 3) / 2 n m m   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 ( 1) / 2 n m m   , which is the widest range until now. Theory analysis indicates that the complexity of our algorithm is