### Radical Isogenies on Montgomery Curves

Hiroshi Onuki and Tomoki Moriya

##### Abstract

We work on some open problems in radical isogenies. Radical isogenies are formulas to compute chains of $N$-isogenies for small $N$ and proposed by Castryck, Decru, and Vercauteren in Asiacrypt 2020. These formulas do not need to generate a point of order $N$ generating the kernel and accelerate some isogeny-based cryptosystems like CSIDH. On the other hand, since these formulas use Tate normal forms, these need to transform Tate normal forms to curves with efficient arithmetic, e.g., Montgomery curves. In this paper, we propose radical-isogeny formulas of degrees 3 and 4 on Montgomery curves. Our formulas compute some values determining Montgomery curves, from which one can efficiently recover Montgomery coefficients. And our formulas are more efficient for some cryptosystems than the original radical isogenies. In addition, we prove a conjecture left open by Castryck et al. that relates to radical isogenies of degree 4.

Available format(s)
Category
Public-key cryptography
Publication info
A minor revision of an IACR publication in PKC 2022
Keywords
Contact author(s)
onuki @ mist i u-tokyo ac jp
History
2021-12-25: last of 2 revisions
See all versions
Short URL
https://ia.cr/2021/699

CC BY

BibTeX

@misc{cryptoeprint:2021/699,
author = {Hiroshi Onuki and Tomoki Moriya},
title = {Radical Isogenies on Montgomery Curves},
howpublished = {Cryptology ePrint Archive, Paper 2021/699},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/699}},
url = {https://eprint.iacr.org/2021/699}
}

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