Public Key Cryptosystem and Binary Edwards Curves on the Ring
Abstract
Let be a finite ring of characteristic 2, where e2 = e and n is a positive integer. Let (a, d) 2 ( )2, such that a and d + a2 + a are invertible in , we study the binary Edwards curve over this ring, denoted by and we give a bijection between this curve and produces two binary Edwards curves defined on the finite field . Afterthat we study the addition law of binary Edwards curves over the ring . We end this work with cryptography applications, ElGamal twisted Edwards curve cryptosystem and Cramer-Shoup twisted Edwards curve cryptosystem.
References
[1] Harold M. Edwards. (2007). Normal form for elliptic curves. Bulletin of the American Mathematical Society, 44 (3) 393-423, April.
[2] Bernstein, D. J., Birkner, P., Joye, M., Lange, T., Peters, C. (2008). Twisted Edwards curves. In: Progress in Cryptology — AFRICACRYPT 2008, First International Conference on Cryptology in Africa, Casablanca, Morocco, June 11-14, 2008. vol. 5023 of Lecture Notes in Computer Science, 389-405. Springer Verlag.
[3] Maher Boudabra., Abderrahmane Nitaj. (2019). A New Public Key Cryptosystem Based on Edwards Curves., Journal of Applied Mathematics and Computing, Springer, 2019, 10.1007/s12190-019-01257-y. hal-02321013.
[4] Bernstein, D. J., Lange, T., Rezaeian Farashahi, R. (2008). Binary Edwards Curves. In: Oswald E., Rohatgi P. (eds) Cryptographic Hardware and Embedded Systems - CHES 2008. Lecture Notes in Computer Science, vol 5154. Springer, Berlin, Heidelberg. https://doi.org/10.1007/ 978- 3- 540-85053-3-16.
[5] Ben Taleb, E.M., Chillali, A., El Fadil, L. (2020).Twisted Hessian curves over the Ring Fq[e]; e2 = e. Bol. Soc. Paran, in press, doi:10.52699/bspm.15867, (2020).
[6] Boulbot, A., Chillali, A., Mouhib, A. (2020). Elliptic Curves Over the Ring R, Bol. Soc. Paran. (2020), 38 (3) 193—201.
[7] Boulbot, A., Chillali, A., Mouhib, A. ELLIPTIC CURVES OVER THE RING Fq[e]; e3 = e2, Gulf Journal of Mathematics, 4 (4) 123–129.
[8] Huseyin Hisil., Kenneth Koon-Ho Wong., Gary Carter., and Ed Dawson. (2008). Twisted Edwards Curves Revisited J. Pieprzyk (Ed.) In: - ASIACRYPT 2008, LNCS 5350, p. 326–343.
[9] ElGamal, Taher. (1985). A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Proceedings of CRYPTO 84 on Advances in cryptology, 10-18. Springer-Verlag New York, Inc,1985.
[10] Cramer, Ronald., Shoup, Victor. (1998). A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. Advances in Cryptology - CRYPTO ’98, Springer Berlin Heidelberg, 13-25.
[2] Bernstein, D. J., Birkner, P., Joye, M., Lange, T., Peters, C. (2008). Twisted Edwards curves. In: Progress in Cryptology — AFRICACRYPT 2008, First International Conference on Cryptology in Africa, Casablanca, Morocco, June 11-14, 2008. vol. 5023 of Lecture Notes in Computer Science, 389-405. Springer Verlag.
[3] Maher Boudabra., Abderrahmane Nitaj. (2019). A New Public Key Cryptosystem Based on Edwards Curves., Journal of Applied Mathematics and Computing, Springer, 2019, 10.1007/s12190-019-01257-y. hal-02321013.
[4] Bernstein, D. J., Lange, T., Rezaeian Farashahi, R. (2008). Binary Edwards Curves. In: Oswald E., Rohatgi P. (eds) Cryptographic Hardware and Embedded Systems - CHES 2008. Lecture Notes in Computer Science, vol 5154. Springer, Berlin, Heidelberg. https://doi.org/10.1007/ 978- 3- 540-85053-3-16.
[5] Ben Taleb, E.M., Chillali, A., El Fadil, L. (2020).Twisted Hessian curves over the Ring Fq[e]; e2 = e. Bol. Soc. Paran, in press, doi:10.52699/bspm.15867, (2020).
[6] Boulbot, A., Chillali, A., Mouhib, A. (2020). Elliptic Curves Over the Ring R, Bol. Soc. Paran. (2020), 38 (3) 193—201.
[7] Boulbot, A., Chillali, A., Mouhib, A. ELLIPTIC CURVES OVER THE RING Fq[e]; e3 = e2, Gulf Journal of Mathematics, 4 (4) 123–129.
[8] Huseyin Hisil., Kenneth Koon-Ho Wong., Gary Carter., and Ed Dawson. (2008). Twisted Edwards Curves Revisited J. Pieprzyk (Ed.) In: - ASIACRYPT 2008, LNCS 5350, p. 326–343.
[9] ElGamal, Taher. (1985). A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Proceedings of CRYPTO 84 on Advances in cryptology, 10-18. Springer-Verlag New York, Inc,1985.
[10] Cramer, Ronald., Shoup, Victor. (1998). A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. Advances in Cryptology - CRYPTO ’98, Springer Berlin Heidelberg, 13-25.
Published
2024-03-28
How to Cite
ELHAMAM, Moha ben taleb; CHILLALI, Abdelhakim; FADIL, Lhoussain El.
Public Key Cryptosystem and Binary Edwards Curves on the Ring.
Journal of Digital Information Management(JDIM), [S.l.], v. 20, n. 1, p. 26-30, mar. 2024.
ISSN 0972-7272.
Available at: <https://www.dline.info/ojs/index.php/jdim/article/view/38>. Date accessed: 21 apr. 2026.
doi: https://doi.org/10.6025/jdim/2022/20/1/25-30.
Section
Research
