### Computing supersingular isogenies on Kummer surfaces

Craig Costello

##### Abstract

We apply Scholten's construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $GF(p^2)$ and corresponding abelian surfaces over $GF(p)$. Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography.

Available format(s)
Publication info
Keywords
Supersingular isogeniesSIDHKummer surfaceRichelot isogenyScholten's construction
Contact author(s)
craigco @ microsoft com
History
2018-10-09: revised
See all versions
Short URL
https://ia.cr/2018/850

CC BY

BibTeX

@misc{cryptoeprint:2018/850,
author = {Craig Costello},
title = {Computing supersingular isogenies on Kummer surfaces},
howpublished = {Cryptology ePrint Archive, Paper 2018/850},
year = {2018},
note = {\url{https://eprint.iacr.org/2018/850}},
url = {https://eprint.iacr.org/2018/850}
}

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