Abstract
In ASIACRYPT 2017, Costello and Hisil proposed two methods of evaluating ℓ-isogenous curves: the usage of Vélu formulas and the recovery from special isogenous points. In isogeny-based cryptography, such as the supersingular isogeny Diffie–Hellman (SIDH)/supersingular isogeny key encapsulation protocol, small-degree isogenies are typically prioritized, because the costs of evaluating points and curves through isogenies using Vélu's formulas will increase with the increase of isogeny degrees. Compared with the use of Vélu formulas, the recovery from special isogenous points is independent from ℓ and runs in constant time under certain conditions. Therefore, it could be efficient in the evaluation of the ℓ-isogenous curve in some variants of SIDH (such as SIDH using twisted torsion (B-SIDH)). In this study, we optimize the evaluation of the ℓ-isogenous curve when three differential isogenous points are provided in advance. We also propose the optimized evaluation in projective coordinates on different elliptic curve models such as the Montgomery, Edwards, Jacobi, and Huff models. Finally, we propose an acceleration of B-SIDH by applying our new function to evaluate the ℓ-isogenous curve in the Montgomery model.
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