Classification the elements of the twisted Hessian curves in the ring $\mathbb{F}_{q}[e], e^{3}=e^2$

Moha ben taleb El Hamam; Abdelhakim  Chillali; Lhoussain  El Fadil

doi:10.5269/bspm.62308

Moha ben taleb El Hamam Sidi Mohamed Ben Abdellah University https://orcid.org/0000-0003-4003-5542
Abdelhakim Chillali Sidi Mohamed Ben Abdellah University https://orcid.org/0000-0002-2033-0280
Lhoussain El Fadil Sidi Mohamed Ben Abdellah University https://orcid.org/0000-0003-4175-8064

Résumé

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.

Téléchargements

Les données sur le téléchargement ne sont pas encore disponible.

Références

D. J. Bernstein, C. Chuengsatiansup, D. Kohel, and T. Lange, Twisted Hessian Curves, In: Lauter K., Rodrguez-Henrques F. (eds) Progress in Cryptology – LATINCRYPT 2015, Lecture Note in Computer Science, vol. 9230, pp 269–294. Springer, Cham(2015).

D. J. Bernstein and T. Lange. Faster addition and doubling on elliptic curves. In Asiacrypt 2007 37, pages 29?50, 2007. http://cr.yp.to/newelliptic/newelliptic-20070906.pdf.

A. Boulbot, A. Chillali and A. Mouhib, ELLIPTIC CURVES OVER THE RING Fq[e]; e3 = e2, Gulf Journal of Mathematics, Vol. 4, Issue 4, pp 123–129, (2016).

A. Boulbot, A. Chillali and A. Mouhib, Elliptic Curves Over the Ring R, Bol. Soc. Paran(2020), v. 38 3, pp 193–201, (2020).

M. Joye and J. Quisquater, Hessian elliptic curves and sidechannel attacks Cryptographic Hardware and Embedded Systems - CHES 2001’, vol. 2162 of Lecture Notes in Computer Science, pp. 402–410, Springer-Verlag, 2001.

A. Grini, A. Chillali and H. Mouanis, Twisted Hessian curves over the Ring Fq[e], e2 = 0. International Journal of Computer Aided Engineering and Technology, (2020, to appear).

M. B. T. El Hamam, A. Chillali and L. El Fadil, Twisted Hessian curves over the Ring Fq[e], e2 = e. Bol. Soc. Paran, (3s.) v. 2022 (40) ISSN-0037-8712, (2022)

M. B. T. El Hamam, A. Chillali and L. El Fadil, A New Addition Law in Twisted Edwards Curves on Non Local Ring. In: Nitaj, A., Zkik, K. (eds) Cryptography, Codes and Cyber Security. I4CS 2022. Communications in Computer and Information Science, vol 1747. Springer, Cham. https://doi.org/10.1007/978-3-031-23201-5_3, 2022.

M. B. T. El Hamam, A. Chillali and L. El Fadil, TWISTED EDWARDS CURVE OVER THE RING Fq[e], e2 = 0. In: Tatra Mt. Math. Publ. 83 (2023), 43–50. DOI: 10.2478/tmmp-2023-0004, 2023.

M.B. T. El Hamam, A. Chillali and L. El Fadil, Public key cryptosystem and binary Edwards curves on the ring F2n[e], e2 = e for data management. In: 2nd International Conference on Innovative Research in Applied Science, Engineering and Technology (IRASET), pp. 1-4, doi: 10.1109/IRASET52964.2022.9738249, 2022.

A. Grini, A. Chillali and H. Mouanis, A new cryptosystem based on a twisted Hessian curve H4 a,d. J. Appl. Math. Comput, 2021.

C. Chuengsatianup, C. Martindale, Pairing-Friendly Twisted Hessian curves In: Chakraboty D., Iwata T. (eds) Progress in Cryptology INDOCRYPT 2018. Lecture Notes in Computer Science, vol 11356. Springer, Cham (2018).

M. Joye, J. Quisquater, Hessian elliptic curves and sidechannel attacks. Cryptographic Hardware and Embedded Systems-CHES 2001, Lecture Notes in Computer Science, vol. 2162, Springer, pp. 402–410, (2001).

M. Joye and J.J. Quisquater, Hessian elliptic curves and side-channel attacks. In CHES 2001 13, pages 402–410, (2001). http://joye.site88.net/.

H. W. Lenstra, Eliptic Curves and Number-Theoretic Algorithms Processing of the International Congress of Mathematicians, Berkely, California, USA(1986).

H. M. Edwards. A normal form for elliptic curves. Bulletin of the American Mathematical Society, 44: pp. 393–422, (2007). http://www.ams.org/bull/2007-44-03/S0273-0979-07-01153-6/home.html.