Multivariate Quadratic Solver in GF(31) with Grover's Algorithm

The development of quantum computers has led to the possibility of solving problems intractable to deal with in clas-sical computers. That threatens the current standard of public-key cryptography based on factoring or discrete-log problem. As a result, it is necessary to use a secure cryptosystem even in a quantum computing environment. NIST has been conducting a Post-Quantum Cryptography (PQC) standardization process. Recently, there have been attempting to design binary MQ solver with the Grover's algorithm to solve the binary Multivariate Quadratic (MQ) problem. However, most PQC candidates based on the MQ problem do not select a small security parameter and are over GF(16) and above. In this paper, we design the Grover's oracle circuit to solve the MQ problem over GF(31) that can be used to analyze the quantum security of PQC candidates such as Rainbow and MQDSS.