Conference Paper

Anatoly Bessalov , Volodymyr Sokolov , Pavlo Skladannyi

An analysis is made of the properties and conditions for the existence of 3-and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of minimal odd 3 and 5 degrees, which allows bypassing the problem of singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A review of the main properties of the classes of complete, quadratic and twisted Edwards curves over a simple field is given. Formulas for the isogeny of odd degrees are reduced to a form adapted to curves in Weierstrass form. To do this, the modified law of addition of curve points in the generalized Edwards form is used, which preserves the horizontal symmetry of the curve's return points. Examples of the calculation of 3-and 5-isogenies of complete Edwards supersingular curves over small simple fields are given, and the properties of the isogeny composition for computing isogenies with large-order kernels are discussed. Formulas of upper bounds for the complexity of computing isogeny of odd degrees 3 and 5 in the classes of complete and quadratic Edwards curves in projective coordinates are obtained. Algorithms for calculating 3-and 5-isogenies of Edwards curves with complexity and 12M+5S, respectively, are constructed. The conditions for the existence of supersingular complete and quadratic Edwards curves of the order $$4•3^m•5^n$$ and $$83^m5^n$$ are found. Some parameters of the cryptosystem were determined during the implementation of the SIDH algorithm at the quantum security level of 128 bits.

Complete Edwards Curve; Curve Order; Degree Of Isogeny; Generalized Edwards Curve; Isogeny; Isomorphism; Kernel Of Isogeny; Point Order; Quadratic Edwards Curve; Quadratic Non-Residue; Quadratic Residue; Twisted Edwards Curve

Elliptic Curve; Scalar Multiplication; Public-Key Cryptography

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2020 2nd International Workshop on Modern Machine Learning Technologies and Data Science (MoMLeT+DS)

2–3 June 2020 Lviv-Shatsk, Ukraine

First Online: 3 July 2020

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• ISSN: 1613-0073

• EID: 2-s2.0-85088879005

• WOS: 000664104800003

• URN: urn:nbn:de:0074-2631-6

• DBLP: conf/momlet/BessalovSS20

• KUBG: 31627

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Bessalov, A., Sokolov, V., & Skladannyi, P. (2020). Modeling of 3- and 5-Isogenies of Supersingular Edwards Curves. In 2nd International Workshop on Modern Machine Learning Tech-nologies and Data Science (Vol. 2631, no. I, pp. 30–39).

A. Bessalov, V. Sokolov, and P. Skladannyi, “Modeling of 3- and 5-Isogenies of Supersingular Edwards Curves,” 2nd International Workshop on Modern Machine Learning Tech-nologies and Data Science, vol. 2631, no. I, pp. 30–39, 2020.

A. Bessalov, V. Sokolov, P. Skladannyi, Modeling of 3- and 5-Isogenies of Supersingular Edwards Curves, in: 2nd International Workshop on Modern Machine Learning Technologies and Data Science, no. I, vol. 2631 (2020) 30–39.