Elliptic curves over a finite ring

Zakariae Cheddour, Abdelhakim Chillali, Ali Mouhib

Abstract


Let Fq be a finite field, where q is a power of a prime p such that p>5. Let \alpha be a root of a monic polynomial of a minimal degree over Fq. In this paper, we will study elliptic curves over (Fq[\alpha],+,*), where + is the usual addition and * represent a non-standard product law over Fq[\alpha]. Elliptic curves over this ring can be used to create a new type of cryptography. The method is fast, simple, and secure.

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References


A. Amadori, F. Pintore, M. Sala, On the discrete logarithm problem for prime-field elliptic curves, Finite Fields and Their Applicationsx 51 (2018), 168-182.

I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, 1999.

W. Bosma, H.W. Lenstra, Complete System of Two Addition Laws for Elliptic Curves, Journal of Number Theory 53 (1995), 229-240.

A. Boulbot, A. Chillali, A. Mouhib, Elliptic curves over the ring Fq[e], e3 = e2, Gulf J. Math 4 (2016), no. 4, 123-129.

A. Boulbot, A. Chillali, A. Mouhib, Elliptic Curves Over the Ring R*, Boletim da Sociedade Paranaense de Matematica 38 (2020), no. 3, 193-201.

Z. Cheddour, A. Chillali, A. Mouhib, The "Elliptic" matrices and a new kind of cryptography, Boletim da Sociedade Paranaense de Matematica 41 (2023).

R. Cramer and V. Shoup, A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In: Krawczyk, H. (eds) Advances in Cryptology| CRYPTO '98, CRYPTO 1998, Lecture Notes in Computer Science 1462 (1998), Springer, Berlin, Heidelberg.

R. Cramer, V. Shoup, Signature Schemes Based on the Strong RSA Assumption, ACM Trans. Inf. Syst. Security 3 (2000), no. 3, 161-185.

A. Chillali, Ellipic curve over ring, International Mathematical Forum 4 (2011), 1501-1505.

T. ElGamal, A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms, In: Advances in Cryptology, CRYPTO 1984, Lecture Notes in Computer Science 196 (1984), Springer, Berlin, Heidelberg.

M.H. Hassib, A. Chillali, M.A. Elomary, Elliptic curve over a chain ring of characteristic 3, Journal of Taibah University for Science 9 (2015), no. 3, 276-287.

M. Gotaishi, S. Tsujii, Organizational Cryptography for Access Control, IACR Cryptology ePrint Archive (2018). Available at https://eprint.iacr.org/2018/1120.pdf

A. Grini, A. Chillali, H. Mouanis, Cryptography over twisted Hessian curves of the ring Fq[e], e2 = 0, Adv. Math. Sci. J. 10 (2021), no. 1, 235-243.

A. Grini, A. Chillali, H. Mouanis, The Binary Operations Calculus in H2 a;d, Boletim da Sociedade Paranaense de Matematica 40 (2022).

A. Grini, A. Chillali, H. Mouanis, A new cryptosystem based on a twisted Hessian curve H_(a,d)^4, Journal of Applied Mathematics and Computing 68 (2022), no. 4, 235-243.

N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation 48 (1987), no. 177, 203-209.

N. Koblitz, A. Menezes, S. Vanstone, The State of Elliptic Curve Cryptography,Designs, Codes and Cryptography 19 (2000), 173-193.

H.W. Lenstra, Elliptic Curves and Number-Theoretic Algorithms, Proceedings of the International Congress of Mathematicians, Berkely, California, USA, 1986, 156-163.

M. Ciet, J.J. Quisquater, F. Sica, Compact elliptic curve representations, J. Math. Cryptol 5 (2011), 89-100.

V. Miller, Use of elliptic curves in cryptography, In: Williams, H.C. (eds) Advances in Cryptology | CRYPTO '85 Proceedings. CRYPTO 1985, Lecture Notes in Computer Science 218 (1986), Springer, Berlin, 417-426.

A. Menezes, T. Okamoto, S. Vanstone, Reducing elliptic curve logarithms to a finite field, IEEE Transactions on Information Theory 39 (1993), no. 5, 1639-1646.

A. Odlyzko, Discrete logarithms: the past and the future, Designs, Codes and Cryptography 19 (2000), 129-145.

R. Schoof, Elliptic Curves Over finite Fields and the Computation of Square Roots mod p, Mathematics of Computation 44(1985), no. 170, 483-494.

D.R. Stinson, Cryptography Theory And Practice, 3rd edition, Chapman Hall/CRC, New York, 2006.

A. Tadmori, A. Chillali, M. Ziane, Cryptography over the elliptic curve Ea;b, Journal of Taibah University for Science 9 (2015), no. 3, 326-331.

A. Tadmori, A. Chillali, M. Ziane, Elliptic curve over ring A4, Applied Mathematical Sciences 9 (2015), no. 35, 1721-1733.

A. Tadmori, A. Chillali, M. Ziane, Elliptic Curves Over a non-local ring F2d [ε], ε2 = ε, Asian-European Journal of Mathematics 19 (2021), no 3, 2250046.

T.V. Deursen, S. Radomirovic, Insider Attacks and Privacy of RFID Protocols, In: Petkova-Nikova, S., Pashalidis, A., Pernul, G. (eds) Public Key Infrastructures, Services and Applications. EuroPKI 2011, Lecture Notes in Computer Science 7163 (2011), Springer, Berlin, Heidelberg, 91-105.

M. Virat, Courbe elliptique sur un anneau et applications cryptographiques, Thése de Doctorat, Université Nice Sophia Antipolis, 2009.




DOI: https://doi.org/10.52846/ami.v50i2.1689