### Rational isogenies from irrational endomorphisms

Wouter Castryck, Lorenz Panny, and Frederik Vercauteren

##### Abstract

In this paper, we introduce a polynomial-time algorithm to compute a connecting $\mathcal{O}$-ideal between two supersingular elliptic curves over $\mathbb{F}_p$ with common $\mathbb{F}_p$-endomorphism ring $\mathcal{O}$, given a description of their full endomorphism rings. This algorithm provides a reduction of the security of the CSIDH cryptosystem to the problem of computing endomorphism rings of supersingular elliptic curves. A similar reduction for SIDH appeared at Asiacrypt 2016, but relies on totally different techniques. Furthermore, we also show that any supersingular elliptic curve constructed using the complex-multiplication method can be located precisely in the supersingular isogeny graph by explicitly deriving a path to a known base curve. This result prohibits the use of such curves as a building block for a hash function into the supersingular isogeny graph.

Available format(s)
Category
Public-key cryptography
Publication info
Keywords
Isogeny-based cryptographyendomorphism ringsCSIDH
Contact author(s)
wouter castryck @ esat kuleuven be
lorenz @ yx7 cc
frederik vercauteren @ esat kuleuven be
History
2020-03-09: revised
See all versions
Short URL
https://ia.cr/2019/1202

CC BY

BibTeX

@misc{cryptoeprint:2019/1202,
author = {Wouter Castryck and Lorenz Panny and Frederik Vercauteren},
title = {Rational isogenies from irrational endomorphisms},
howpublished = {Cryptology ePrint Archive, Paper 2019/1202},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/1202}},
url = {https://eprint.iacr.org/2019/1202}
}

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