Exploring SIDH-Based Signature Parameters

Isogeny-based cryptography is an instance of post-quantum cryptography whose fundamental problem consists of finding an isogeny between two (isogenous) elliptic curves E and \(E'\) . This problem is closely related to that of computing the endomorphism ring of an elliptic curve. Therefore, many isogeny-based protocols require the endomorphism ring of at least one of the curves involved to be unknown. In this paper, we explore the design of isogeny based protocols in a scenario where one assumes that the endomorphism ring of all the curves are public. In particular, we identify digital signatures based on proof of isogeny knowledge from SIDH squares as such a candidate. We explore the design choices for such constructions and propose two variants with practical instantiations. We analyze their security according to three lines, the first consists of attacks based on KLPT with both polynomial and superpolynomial adversary, the second consists of attacks derived from the SIDH attacks and finally we study the zero-knowledge property of the underlying proof of knowledge.