Is it necessary to protect MQ-based Cryptosystems from side-channel attacks?

MQ-based cryptosystems have the potential to resist future quantum computer attacks, and can be efficiently implemented. We investigate the side channel attacks against MQ-based cryptosystems in this paper. For MPKCs, which are based on solving multivariate quadratic system, there isn't side-channel leakage found for their major computation components. For Cryptosystems based on calculating multivariate quadratic system, our research first found that the serial implementations would leak the secret information when computing monomials and restoring the results to the register.