Biblio

The Supersingular Isogeny Diffie-Hellman protocol (SIDH) has recently been the subject of increased attention in the cryptography community. Conjecturally quantum-resistant, SIDH has the feature that it shares the same data flow as ordinary Diffie-Hellman: two parties exchange a pair of public keys, each generated from a private key, and combine them to form a shared secret. To create a potentially quantum-resistant scheme, SIDH depends on a new family of computational assumptions involving isogenies between supersingular elliptic curves which replace both the discrete logarithm problem and the computational and decisional Diffie-Hellman problems. As in the case of ordinary Diffie-Hellman, one is interested in knowing if these problems are related. In fact, more is true: there is a rich network of reductions between the isogeny problems securing the private keys of the participants in the SIDH protocol, the computational and decisional SIDH problems, and the problem of validating SIDH public keys. In this article we explain these relationships, which do not appear elsewhere in the literature, in hopes of providing a clearer picture of the SIDH problem landscape to the cryptography community at large.

In this work, we analyze the feasibility of a physically secure implementation of the quantum-resistant supersingular isogeny Diffie-Hellman (SIDH) protocol. Notably, we analyze the defense against timing attacks, simple power analysis, differential power analysis, and fault attacks. Luckily, the SIDH protocol closely resembles its predecessor, the elliptic curve Diffie-Hellman (ECDH) key exchange. As such, much of the extensive literature in side-channel analysis can also apply to SIDH. In particular, we focus on a hardware implementation that features a true random number generator, ALU, and controller. SIDH is composed of two rounds containing a double-point multiplication to generate a secret kernel point and an isogeny over that kernel to arrive at a new elliptic curve isomorphism. To protect against simple power analysis and timing attacks, we recommend a constant-time implementation with Fermat's little theorem inversion. Differential power analysis targets the power output of the SIDH core over many runs. As such, we recommend scaling the base points by secret scalars so that each iteration has a unique power signature. Further, based on recent oracle attacks on SIDH, we cannot recommend the use of static keys from both parties. The goal of this paper is to analyze the tradeoffs in elliptic curve theory to produce a cryptographically and physically secure implementation of SIDH.