Conference Paper

Anatoly Bessalov , Serhii Abramov , Volodymyr Sokolov , Pavlo Skladannyi , Oleksii Zhyltsov

A multifunctional cryptosystem RCNIE on isogenies of non-supersingular Edwards curves is proposed, which solves the problems of Diffie-Hellman secret sharing, digital signature, and public key encryption. The problems of choosing the parameters of non-supersingular Edwards curves forming pairs of quadratic twist with orders $$p + 1 ± t ≡ 0 mod 8$$ over a prime field $$F_p$$ are considered. Encryption algorithms with mutual authentication of Alice and Bob based on the sharing of their secrets are given, while the length of the key and the size of the digital signature are minimally short and do not exceed the size of the field $$F_p$$ element. An illustration is given of the operation of the cryptosystem model on 4 degrees of isogenies {3,5,7,37} over the field $$F_{863}$$ for a pair of quadratic twist curves with orders 840 and 888. It is shown that for non-supersingular curves there are main and dual cryptosystems, each of which has also an isomorphic cryptosystem. This allows you to perform parallel computing and speed up algorithms. A comparative evaluation of the arithmetic and properties of CSIDH and RCNIE is given. It is noted that we have not found strong arguments for the slow implementation of the CRS scheme in comparison with CSIDH. Taking into account the peculiarities of each of them, both schemes are certainly promising.

complete curve; Curve in generalized Edwards form; curve order; isogeny; isomorphism; non-supersingular curve; point order; quadratic curve; twisted curve

Quantum Cryptography; Elliptic Curve; Finite Field

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2023 Classic, Quantum, and Post-Quantum Cryptography (CQPC)

1 August 2023 Kyiv, Ukraine

First Online: 13 October 2023

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• ISSN: 1613-0073

• EID: 2-s2.0-85175724530

• URN: urn:nbn:de:0074-3504-7

• DBLP: conf/cqpc/BessalovASSZ23

• KUBG: 46394

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Bessalov, A., Abramov, S., Sokolov, V., Skladannyi, P., & Zhyltsov, O. (2023). Multifunctional CRS Encryption Scheme on Isogenies of Non-Supersingular Edwards Curves. In Classic, Quantum, and Post-Quantum Cryptography (Vol. 3504, pp. 12–25).

A. Bessalov, S. Abramov, V. Sokolov, P. Skladannyi, and O. Zhyltsov, “Multifunctional CRS Encryption Scheme on Isogenies of Non-Supersingular Edwards Curves,” Classic, Quantum, and Post-Quantum Cryptography, vol. 3504, pp. 12–25, 2023.

A. Bessalov, et al., Multifunctional CRS Encryption Scheme on Isogenies of Non-Supersingular Edwards Curves, in: Classic, Quantum, and Post-Quantum Cryptography, vol. 3504 (2023) 12–25.