How to compute an isogeny on the extended Jacobi quartic curves?

• Autorzy: Dzierzkowski Łukasz , Wroński Michał

Identyfikatory

DOI

10.24425/ijet.2022.139890

• Języki publikacji: EN

Abstrakty

EN

Computing isogenies between elliptic curves is a significant part of post-quantum cryptography with many practical applications (for example, in SIDH, SIKE, B-SIDH, or CSIDH algorithms). Comparing to other post-quantum algorithms, the main advantages of these protocols are smaller keys, the similar idea as in the ECDH, and a large basis of expertise about elliptic curves. The main disadvantage of the isogeny-based cryptosystems is their computational efficiency - they are slower than other post-quantum algorithms (e.g., lattice-based). That is why so much effort has been put into improving the hitherto known methods of computing isogenies between elliptic curves. In this paper, we present new formulas for computing isogenies between elliptic curves in the extended Jacobi quartic form with two methods: by transforming such curves into the short Weierstrass model, computing an isogeny in this form and then transforming back into an initial model or by computing an isogeny directly between two extended Jacobi quartics.

Słowa kluczowe

EN

cryptology; post-quantum; elliptic curves; Jacobi quartics; isogenies

• Wydawca: Polish Academy of Sciences, Committee of Electronics and Telecommunication
• Czasopismo: International Journal of Electronics and Telecommunications
• Rocznik: 2022
• Tom: Vol. 68. No. 3
• Strony: 463--468
• Opis fizyczny: Bibliogr. 9 poz., schem., tab.

Twórcy

autor

Dzierzkowski Łukasz

• Faculty of Cybernetics, Military University of Technology, Warsaw, Poland

autor

Wroński Michał

• Faculty of Cybernetics, Military University of Technology, Warsaw, Poland

Bibliografia

• [1] J. Velu, “Isogenies entre courbes elliptiques,” C. R. Acad. Sci. Paris Ser. A-B, vol. 273, 1971.
• [2] D. Moody and D. Shumow, “Analogues of Velu’s formulas for isogenies on alternate models of elliptic curves,” Mathematics of Computation, vol. 85, no. 300, pp. 1929–1951, 2016.
• [3] X. Xu, W. Yu, K. Wang, and X. He, “Constructing Isogenies on Extended Jacobi Quartic Curves,” in Information Security and Cryptology, K. Chen, D. Lin, and M. Yung, Eds. Cham: Springer International Publishing, pp. 416–427.
• [4] Z. Hu, Z. Liu, L. Wang, and Z. Zhou, “Simplified isogeny formulas on twisted Jacobi quartic curves,” Finite Fields and Their Applications, vol. 78, p. 101981, 2022. [Online]. Available: https://doi.org/10.1016/j.ffa.2021.101981
• [5] O. Billet and M. Joye, “The Jacobi Model of an Elliptic Curve and Side-Channel Analysis,” in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, M. Fossorier, T. Høholdt, and A. Poli, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003, pp. 34–42. [Online]. Available: https://doi.org/10.1007/3-540-44828-4_5
• [6] I. Semaev, “Summation polynomials and the discrete logarithm problem on elliptic curves,” Cryptology ePrint Archive, Report 2004/031, 2004.
• [7] L. Dzierzkowski, “Analysis of the possibility of hardware implementation of SIDH key exchange scheme,” Military University of Technology in Warsaw, 2020.
• [8] W. Castryck, T. Lange, C. Martindale, L. Panny, and J. Renes, “CSIDH: an efficient post-quantum commutative group action,” in International Conference on the Theory and Application of Cryptology and Information Security. Springer, 2018, pp. 395–427. [Online]. Available: https://doi.org/10.1007/978-3-030-03332-3_15
• [9] H. Hisil, K. K.-H. Wong, G. Carter, and E. Dawson, “Jacobi Quartic Curves Revisited,” in Information Security and Privacy, C. Boyd and J. Gonzalez Nieto, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009, pp. 452–468. [Online]. Available: https://doi.org/10.1007/978-3-642-02620-1_31

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).