Multiprime Strategies for Serial Evaluation of eSIDH-Like Isogenies

We present new results and speedups for the large-degree isogeny computations within the extended supersingular isogeny Diffie-Hellman (eSIDH) key agreement framework. As proposed by Cervantes-V ' azquez, Ochoa-Jim ' enez, and Rodr ' iguez-Henr ' iquez, eSIDH is an extension to SIDH and fourth round NIST post-quantum cryptographic standardization candidate SIKE. By utilizing multiprime large-degree isogenies, eSIDH and eSIKE are faster than the standard SIDH/SIKE and amenable to parallelization techniques that can noticeably increase their speed with multiple cores. Here, we investigate the use of multiprime isogeny strategies to speed up eSIDH and eSIKE in serial implementations. These strategies have been investigated for other isogeny schemes such as CSIDH. We apply them to the eSIDH/eSIKE scenario to speed up the multiprime strategy by about 10%. When applied to eSIDH, we achieve a 7-8% speedup for Bob's shared key agreement operation. When applied to eSIKE, we achieve a 3-4% speedup for key decapsulation. Historically, SIDH and SIKE have been considerably slower than its competitors in the NIST PQC standardization process. These results continue to highlight the various speedups achievable with the eSIKE framework to alleviate these speed concerns. Though eSIDH and eSIKE are susceptible to the recent devastating attacks on SIKE, our analysis applies to smooth degree isogeny computations in general, and isogenybased signature schemes which use isogenies of smooth (not necessarily powersmooth) degree.