On hybrid SIDH schemes using Edwards and Montgomery curve arithmetic
Supersingular isogeny Diffie-Hellman (SIDH) is a proposal for a quantumresistant key exchange. The state-of-the-art implementation works entirely with Montgomery curves and basically can be divided into elliptic curve arithmetic and isogeny arithmetic. It is well known that twisted Edwards curves can provide a more efficient elliptic curve arithmetic. Therefore it was hinted by Costello and Hisil, that by using only Edwards curves for isogeny and curve arithmetic, or a hybrid scheme, that uses Edwards curve arithmetic and switches between the models whenever needed, a speedup in the computation may be gained. Following the latter case, we investigated how to efficiently switch between Montgomery and twisted Edwards curves in SIDH, and how to insert Edwards arithmetic in the current state-of-the-art implementation. We did not gain a speedup compared to the results of Costello, Longa, and Naehrig, but in some cases the performance of Edwards arithmetic is almost equally fast. Thus, we suppose that a hybrid scheme does not improve the performance of SIDH, but still can be interesting for platforms having special hardware acceleration for Edwards curves. However, a full Edwards SIDH version may give a speedup, if fast Edwards isogeny formulas can be found.