Paper 2019/1417

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，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].