### CSIDH on Other Form of Elliptic Curves

Xuejun Fan, Song Tian, Bao Li, and Xiu Xu

##### Abstract

Isogenies on elliptic curves are of great interest in post-quantum cryptography and appeal to more and more researchers. Many protocols have been proposed such as OIDH, SIDH and CSIDH with their own advantages. We now focus on the CSIDH which based on the Montgomery curves in finite fields Fp with p=3 mod 8 whose endomorphism ring is O. We try to change the form of elliptic curves into y^2=x^3+Ax^2-x and the characteristic of the prime field into p=7 mod 8 , which induce the endomorphism ring becomes O_K. Moreover, many propositions&#65292;including the formula of isogenies between elliptic curves of the special form and the unique of the representation of Fp-isomorphism class, are given to illustrate the rationality of our idea. An important point to notice that the efficiency can't be reduced because the only difference between our formula of isogenies and that of CSIDH is the sign of some items. Furthermore, we also give a proposition that the protocol based on our case can avoid the collision proposed in [17].

Available format(s)
Category
Public-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
CSIDHMontogomery CurvesEndomorphism RingCollision.
Contact author(s)
fanxuejun @ iie ac cn
History
2020-07-26: last of 3 revisions
See all versions
Short URL
https://ia.cr/2019/1417

CC BY

BibTeX

@misc{cryptoeprint:2019/1417,
author = {Xuejun Fan and Song Tian and Bao Li and Xiu Xu},
title = {CSIDH on Other Form of Elliptic Curves},
howpublished = {Cryptology ePrint Archive, Paper 2019/1417},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/1417}},
url = {https://eprint.iacr.org/2019/1417}
}

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