Abstract
Methods of generating supersingular and ordinary elliptic curves for isogeny-based cryptosystems have been studied. The influence of the class field polynomial on the time of generating ordinary elliptic curves has been analyzed and the comparative time of generating curves using Weber and Hilbert polynomials have been presented. Parameters that influence on the cryptographic security of isogeny-based cryptosystems have been considered.
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References
Silverman, J.H., The Arithmetic of Elliptic Curves, Springer, 1986, 2nd ed.
Rostovtsev, A.G., Ellipticheskie krivye v kriptografii. Teoriya i vychislitel’nye algoritmy (Elliptic Curves in Cryptography. Theory and Computational Algorithms), St. Petersburg: NPO Professional, 2010.
Moody, D. and Shumov, D., Analogues of Velu’s formulas for isogenies on alternate models of elliptic curves, Cryptology ePrint archive. Report 2011/430, 2011. https://eprint.iacr.org/2011/430.pdf.
Childs, A., Jao, D., and Soukharey, V., Constructing elliptic curve isogenies in quantum subexponential time, J. Math. Cryptol., 2014, vol. 8, no. 1, pp. 1–29.
Rostovtsev, A. and Stolbunov, A., Public-key cryptosystem based on isogenies, Cryptology ePrint Archive. Report 2006/145, 2006. http://eprint.iacr.org/2006.145.pdf.
Rostovtsev, A.G. and Makhovenko, E.B., Hilbert polynomials and j-invariants of elliptic curves, Probl. Inf. Bezop., Komp’yut. Sist., 2004, no. 4, pp. 32–38.
Yui, N. and Zagier, D., On the singular values of Weber modular functions, Math. Comput., 1997, vol. 66, no. 220, pp. 1645–1662.
Stein, W.A., et al., Sage Mathematics Software (Version 4.3). The Sage Development Team, 2009. http://www.sagemath.org.
Konstantinou, E. and Kontogeorgis, A., Introducing Ramanujan’s Class Polynomials in the Generation of Prime Order Elliptic Curves. https://arxiv.org/pdf/0804.1652v1.pdf.
De Feo, L., Jao, D., and Plut, J., Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies, Post-Quantum Cryptogr., 2011, vol. 4341, pp. 193–210.
Bröker, R., Constructing supersingular elliptic curves, J. Comb. Number Theory, 2009, vol. 1, no. 3, pp. 269–273.
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Original Russian Text © E.B. Aleksandrova, A.A. Shtyrkina, A.V. Yarmak, 2017, published in Problemy Informatsionnoi Bezopasnosti, Komp’yuternye Sistemy.
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Aleksandrova, E.B., Shtyrkina, A.A. & Yarmak, A.V. Elliptic curves generation for isogeny-based cryptosystems. Aut. Control Comp. Sci. 51, 928–935 (2017). https://doi.org/10.3103/S0146411617080028
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DOI: https://doi.org/10.3103/S0146411617080028