Building cryptographic schemes based on elliptic curves over rational numbers
https://doi.org/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.
Supporting agencies: This research was supported by Priority 2030 Federal Academic Leadership Program.
About the Authors
V. V. Davydov
ITMO University
Russian Federation
Russian Federation
Vadim V. Davydov — Lecturer
sc 57203909696
Saint Petersburg, 197101
J.-M. N. Dakuo
ITMO University
Russian Federation
Russian Federation
Jean-Michelle N. Dakuo — Student
Saint Petersburg, 197101
I. D. Ioganson
ITMO University
Russian Federation
Russian Federation
Ivan D. Ioganson — Engineer
Saint Petersburg, 197101
A. F. Khutsaeva
ITMO University
Russian Federation
Russian Federation
Altana F. Khutsaeva — Engineer
Saint Petersburg, 197101
References
1. 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
2. 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
3. Washington L.C. Elliptic Curves: Number Theory and Cryptography. Chapman and Hall/CRC, 2008, 536 p.
4. Silverman J.H. The Arithmetic of Elliptic Curves. New York, Springer, 2009, 513 p. Graduate Texts in Mathematics, vol. 106.
5. 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.
6. Weil A. L'arithmétique sur les courbes algébriques. Acta Mathematica, 1929, vol. 52, no. 1, pp. 281–315. https://doi.org/10.1007/BF02592688
7. 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
8. 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
9. Menezes A.J., Van Oorschot P.C., Vanstone S.A. Handbook of Applied Cryptography. CRC press, 2018, 810 p.
10. Danzig T. Numbers: The Language of Science. Revised. Now York, Macmillan, 1933.
11. 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
12. 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
13. Noro K., Kobayashi K. Knapsack cryptosystem on elliptic curves. Cryptology ePrint Archive, 2009, pp. 2009/091.
14. 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
15. 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
Review
For citations:
Davydov V.V., Dakuo J.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;22(4):674-680. (In Russ.) https://doi.org/10.17586/2226-1494-2022-22-4-674-680
Views:
8
Email this article 