An Efficient $$F_4$$ -style Based Algorithm to Solve MQ Problems

The multivariate public key cryptosystem (MPKC) is a potential post-quantum cryptosystem. Its safety depends on the hardness of solving systems of algebraic equations over finite fields. In particular, the multivariate quadratic (MQ) problem is that of solving such a system consisting of quadratic polynomials and is regarded as an important research subject in cryptography. In the Fukuoka MQ challenge project, the hardness of the MQ problem is discussed, and the algorithms used for the MQ problem and the computational results obtained by these algorithms are reported. The algorithms to compute Grobner basis for the polynomial set given by the MQ problem are for solving the MQ problem. For example, the \(F_4\) algorithm and M4GB algorithm have succeeded in solving several MQ problems provided by the project. In this paper, based on the \(F_4\)-style algorithm, we present an efficient algorithm to solve the MQ problems with dense polynomials generated in the Fukuoka MQ challenge project. We experimentally show that our algorithm requires less computational time and memory for these MQ problems than the \(F_4\) algorithm and M4GB algorithm. We succeeded in solving Type II problems using our algorithm when the numbers of variables are 36 and 37.