Compressed M-SIDH: an instance of compressed SIDH-like schemes with isogenies of highly composite degrees

Recently, SIDH was broken by a series of attacks. To avoid the attacks, several new countermeasures, such as M-SIDH and binSIDH, have been developed. Different from SIDH, the new SIDH-like schemes have relatively large public key sizes. Besides, the orders of the torsion groups considered in new SIDH-like schemes are the products of many prime factors. Therefore, the key compression techniques in SIDH can not be directly applied to these schemes. It remains an open problem to compress the public key in new SIDH-like schemes. This paper takes M-SIDH as an instance to explore how to compress the public key in new SIDH-like schemes efficiently. We propose compressed M-SIDH, which is reminiscent of compressed SIDH. We also show that our approach to compress the public key of M-SIDH is valid and prove that compressed M-SIDH is secure as long as M-SIDH is secure. In addition, new algorithms to accelerate the performance of public-key compression in M-SIDH are presented in this paper. We provide a proof-of-concept implementation of compressed M-SIDH in SageMath. Experimental results show that our approach fits well with compressed M-SIDH. The techniques proposed in this work also benefit public-key compression in other SIDH-like protocols, such as binSIDH and terSIDH. Besides, our method for torsion basis generation has the potential to improve the performance of SQALE and dCSIDH.