Miyaji Laboratory

We are currently using RSA cryptosystem based on the hardness of integer factoring problem and Elliptic- curve cryptography based on the hardness of the discrete logarithm problem. However, Shor's algorithm reported that these problems can be decoded in polynomial time by a quantum computer. For this reason, we research cryptography, which is difficult to crack even with a large-scale quantum computer such as Learning with Errors(LWE) problem and Ring-LWE problem. Module-LWE problem is a version of Ring-LWE problem ,having module rank. We can compare the hardness of Ring-LWE problem and Module-LWE problem by reducing Module-LWE problem to Ring-LWE problem. So far, it has been proved that search Module-LWE problem can be reduced to search Ring-LWE problem in power-of-two cyclotomic field. It is also shown that decision Module-LWE problem can be reduced to the decision Ring- LWE problem in arbitrary cyclotomic field. Thus, the purpose of this study is to search for algebraic field other than cyclotomic field that can be reduced Module-LWE problem to Ring-LWE problem.