How to construct CSIDH on quadratic and twisted Edwards curves

Бессалов, Анатолій Володимирович (2022) How to construct CSIDH on quadratic and twisted Edwards curves Кібербезпека: освіта, наука, техника, 3 (15). pp. 148-163. ISSN 2663-4023

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Abstract

In one of the famous works, an incorrect formulation and an incorrect solution of the implementation problem of the CSIDH algorithm on Edwards curves is discovered. A detailed critique of this work with a proof of the fallacy of its concept is given. Specific properties of three non-isomorphic classes of supersingular curves in the generalized Edwards form is considered: complete, quadratic, and twisted Edwards curves. Conditions for the existence of curves of all classes with the order of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be construct 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 with statement about existing a solution within one class of curves with a parameter that is a square. The critical analysis of theorems, lemmas, and erroneous statements in this work is given. Theorem 2 on quadratic twist in classes of Edwards curves is proved. A modification of the CSIDH algorithm 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.

Item Type: Article
Additional Information: https://doi.org/10.28925/2663-4023.2022.15.148163
Uncontrolled Keywords: curve in generalized Edwards form; complete Edwards curve; twisted Edwards curve; quadratic Edwards curve; curve order; point order; isomorphism; isogeny; w-coordinates; square.
Subjects: Статті у наукометричних базах > Index Copernicus
Статті у журналах > Фахові (входять до переліку фахових, затверджений МОН)
Divisions: Факультет інформаційних технологій та математики > Кафедра інформаційної та кібернетичної безпеки імені професора Володимира Бурячка
Depositing User: професор Анатолій Володимирович Бессалов
Date Deposited: 08 Nov 2022 07:49
Last Modified: 08 Nov 2022 08:08
URI: https://elibrary.kubg.edu.ua/id/eprint/42382

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