<div>
	Building cryptographic schemes based on elliptic curves over rational numbers</div>

Davydov Vadim Valerievich, Zhan-Mishel N. Dakuo, Ivan D. Ioganson , Altana F. Khutsaeva

2022 , VOLUME 22, NUMBER 4 ( July-August )

ISSN 2226-1494 (print), ISSN 2500-0373 (online)

Publications

Editor-in-Chief

Nikiforov
Vladimir O.
D.Sc., Prof.

Partners

doi: 10.17586/2226-1494-2022-22-4-674-680

Building cryptographic schemes based on elliptic curves over rational numbers

V. V. Davydov, Z. N. Dakuo, I. D. Ioganson, A. F. Khutsaeva

Read the full article

Article in Russian

For citation:

Davydov V.V., Dakuo Z.-M.N., Ioganson I.D., Khutsaeva A.F. Building cryptographic schemes based on elliptic curves over rational numbers. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2022, vol. 22, no. 4, pp. 674–680 (in Russian). doi: 10.17586/2226-1494-2022-22-4-674-680

Abstract

The possibility of using elliptic curves over the rational field of non-zero ranks in cryptographic schemes is studied. For the first time, the construction of cryptosystems is proposed the security of which is based on the complexity of solving the knapsack problem on elliptic curves over rational numbers of non-zero ranks. A new approach to the use of elliptic curves for cryptographic schemes is proposed. A few experiments have been carried out to estimate the heights characteristic of points on elliptic curves of infinite order. A model of a cryptosystem resistant to computations on a quantum computer and based on rational points of an infinite order curve is proposed. A study of the security and effectiveness of the proposed scheme has been carried out. An attack on the secret search in such a cryptosystem is implemented and it is shown that the complexity of the attack is exponential. The proposed solution can be applied in the construction of real cryptographic schemes as well as cryptographic protocols.

Keywords: elliptic curves, rational numbers, curve rank, asymmetric encryption, knapsack problem

Acknowledgements. This research was supported by Priority 2030 Federal Academic Leadership Program.

References

Koblitz N., Menezes A., Vanstone S. The state of elliptic curve cryptography. Designs, Codes and Cryptography, 2000, vol. 19, no. 2-3, pp. 173–193. https://doi.org/10.1023/A:1008354106356
Koblitz N. Elliptic curve cryptosystems. Mathematics of Computation, 1987, vol. 48, no. 177, pp. 203–209. https://doi.org/10.1090/S0025-5718-1987-0866109-5
Washington L.C. Elliptic Curves: Number Theory and Cryptography. Chapman and Hall/CRC, 2008, 536 p.
Silverman J.H. The Arithmetic of Elliptic Curves. New York, Springer, 2009, 513 p. Graduate Texts in Mathematics, vol. 106.
Mordell L. On the rational solutions of the indeterminate equation of the third and fourth degree. Proceedings Cambridge Philosophical Society, 1922, vol. 21, pp. 179–192.
Weil A. L'arithmétique sur les courbes algébriques. Acta Mathematica, 1929, vol. 52, no. 1, pp. 281–315. https://doi.org/10.1007/BF02592688
Silverman J.H. Heights and elliptic curves.Arithmetic Geometry. New York, NY, Springer, 1986, pp. 253–265. https://doi.org/10.1007/978-1-4613-8655-1_10
Klagsbrun Z., Sherman T., Weigandt J. The Elkies curve has rank 28 subject only to GRH. Mathematics of Computation, 2019, vol. 88, no. 316, pp. 837–846. https://doi.org/10.1090/mcom/3348
Menezes A.J., Van Oorschot P.C., Vanstone S.A. Handbook of Applied Cryptography. CRC press, 2018, 810 p.
Danzig T. Numbers: The Language of Science. Revised. Now York,Macmillan, 1933.
Mathews G.B. On the partition of numbers. Proceedings of the London Mathematical Society, 1896, vol. 1, no. 1, pp. 486–490. https://doi.org/10.1112/plms/s1-28.1.486
Diffie W., Hellman M. New directions in cryptography. IEEE Transactions on Information Theory, 1976, vol. 22, no. 6, pp. 644–654. https://doi.org/10.1109/TIT.1976.1055638
Noro K., Kobayashi K. Knapsack cryptosystem on elliptic curves. Cryptology ePrint Archive, 2009, pp. 2009/091.
Mazur B., Goldfeld D. Rational isogenies of prime degree. Inventiones Mathematicae, 1978, vol. 44, no. 2, pp. 129–162. https://doi.org/10.1007/BF01390348
Lozano-Robledo Á. Elliptic curves, modular forms, and their L-functions. The Student Mathematical Library, 2011, vol. 58. http://dx.doi.org/10.1090/stml/058

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License