On correctness of conditions for the CSIDH algorithm implementation on Edwards curves
DOI:
https://doi.org/10.30837/rt.2022.1.208.02Keywords:
curve in generalized Edwards form, twisted Edwards curve, quadratic Edwards curve, curve order, point order, isomorphism, isogeny, w-coordinates, square, non squareюAbstract
Incorrect formulation and incorrect solution of the problem of the CSIDH algorithm implementation on Edwards curves was revealed in one of the famous works. The purpose of this paper is to present a detailed critique of such concept with a proof of its inconsistency. Specific properties of three non-isomorphic classes of super singular curves in the generalized Edwards form are considered: full, quadratic, and twisted Edwards curves. Conditions for existence of curves of all 3 classes with order of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees is based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be built both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems that state that there is a solution within one class of curves with a parameter which is a square. The critical analysis of theorems, lemmas, erroneous statements in this work is carried out. Theorem 2 on quadratic twist in classes of Edwards curves is proved. The CSIDH algorithm modification based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered for.
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