BMQE system: A MQ equations system based on ergodic matrix

In this paper, we propose a multivariate quadratic (MQ) equation system based on ergodic matrix (EM) over a finite field with q elements (denoted as F
^q
). The system actually implicates a problem which is equivalent to the famous Graph Coloring problem, and therefore is NP complete for attackers. The complexity of bisectional multivariate quadratic equation (BMQE) system is determined by the number of the variables, of the equations and of the elements of F
^q
, which is denoted as n, m, and q, respectively. The paper shows that, if the number of the equations is larger or equal to twice the number of the variables, and q
ⁿ
is large enough, the system is complicated enough to prevent attacks from most of the existing attacking schemes.