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Journal of Discrete Mathematical Sciences and Cryptography cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

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Open Access Research Article

On a Vernam cipher scheme based on elliptic curve L-functions

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pp. 151–174Vol. 27Issue 1January 2024DOI: 10.47974/JDMSC-1638 Crossmark XML
Received:
09 Mar 2022
Accepted:
25 May 2022
Published Online:
24 Jan 2024
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-1638
Pages:
151–174

Abstract

In this paper, we present a new Vernam-like cryptosystem based on the L-functions attached to elliptic curves. Actually, we show that for infinitely many elliptic curves, the coefficients of the L-series can be considered as a secure cryptographic cipher key when combined with linear and classical pseudo-random number generators. Hence, fast and widely known classical generators, like linear congruential generators, should be reintroduced into our list of cryptographic design choices.

Keywords

Subject Classifications

(2010) 94A6011G0511T7114G5014H52

References

[1] Abdalla, M., Bellare, M., Rogaway, P.: The oracle Diffie-Hellman assumptions and an analysis of DHIES. In: Topics in Cryptology-
CT-RSA 2001, Lecture Notes in Computer Science, vol. 2020, pp. 143–158 (2001).
[2] Anshel, M., Goldfeld, D.: Zeta functions, one-way functions, and pseudorandom number generators. Duke Mathematical Journal 88(2), 371–390 (1997).
[3] Anyanwu, M.N., Deng, L.Y., Dasgupta, D.: Design of cryptographically strong generator by transforming linearly generated sequences. International Journal of Computer Science and Security 3(3), 186–200 (2009).
[4] Becker, A., Gama, N., Joux, A.: Solving shortest and closest vector problems: The decomposition approach. IACR Cryptology ePrint Archive (2013). Available at http://eprint.iacr.org/2013/685.
[5] Bellare, M., Goldwasser, S., Micciancio, D.: Pseudo-random generators within cryptographic applications: the DSS case. In: Advances in Cryptology, Lecture Notes in Computer Science, vol. 1294, pp. 277–291. Springer (1997).
[6] Boyar, J.: Inferring sequences produced by pseudo-random number generators. Journal of the ACM 36(1), 129–141 (1989).
[7] Brown, D., Antipa, A., Campagna, M., Struik, R.: Ecoh: the elliptic curve only hash. Submission to NIST (2008).
[8] Canetti, R., Dwork, C., Naor, M., Ostrovsky, R.: Deniable encryption. In: CRYPTO, Lecture Notes in Computer Science, vol. 1294, pp. 90–104. Springer (1997).
[9] Cremona, J.E.: Algorithms for modular elliptic curves, second edn. Cambridge University Press, Cambridge (1997).
[10] Dale, H.: Elliptic Curves, Graduate Texts in Mathematics, vol. 111. Springer (1987).
[11] Deng, L.Y., Xu, H.: A system of high-dimensional, efficient, long-cycle and portable uniform random number generators. ACM Trans. Model. Comput. Simul. 13(4), 299–309 (2003).
[12] Frieze, A.M., Hastad, J., Kannan, R., Lagarias, J.C., Shamir, A.: Reconstructing truncated integer variables satisfying linear congruences. SIAM J. Comput. 17(2), 262–280 (1988).
[13] Krawczyk, H.: How to predict congruential generators. J. Algorithms 13(4), 527–545 (1992).
[14] Lehmer, D.H.: Mathematical methods in large-scale computing units. In: Proc. 2nd Symp. on Large-Scale Digital Calculating Machinery, pp. 141–146. Harvard University Press (1951).
[15] Lenstra, H.: Factoring integers with elliptic curves. Annals of Mathematics 126(2), 649–673 (1987).
[16] Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: a cryptographic perspective, The Kluwer International Series in Engineering and Computer Science, vol. 671. Kluwer Academic Publishers (2002).
[17] Micciancio, D., Regev, O.: Lattice-based cryptography. In: Post-Quantum Cryptography, pp. 147–191. Springer (2009)19.
[18] Omar, S., Ouni, R., Bouanani, S.: Hashing with elliptic curve L-functions. In: F. Ozbudak, F. Rodriguez-Henriquez (eds.) WAIFI, Lecture Notes in Computer Science, vol. 7369, pp.196–207. Springer (2012).
[19] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3 edn. Cambridge University Press (2007).
[20] Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Heckert, A., Dray, J., Vo, S.: A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications (2010). Available at http://csrc.nist.gov/publications/nistpubs/800-22-rev1a/SP800-22rev1a.pdf.
[21] Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod p. Mathematics of Computation 44, 483–494 (1985).
[22] Shannon, C.: Communication theory of secrecy systems. Bell Systems Techn. Journal 28, 656–719 (1949).
[23] Shoup, V.: A proposal for an iso standard for public key encryption. IACR Cryptology ePrint Archive (2001). Available at http://eprint.iacr.org/2001/112.
[24] Silverman, J.H.: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106. Springer (1986).
[25] Stern, J.: Secret linear congruential generators are not cryptographically secure. In: 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, 27-29 October 1987, pp. 421–426 (1987).
[26] Wagner, D.: A generalized birthday problem. In: Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology, CRYPTO, pp. 288–303. Springer (2002).
[27] Zheng, Y.: Digital signcryption or how to achieve cost(signature & encryption) cost(signature) + cost(encryption). In: Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology, CRYPTO, pp. 165–179. Springer (1997).

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