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Cryptography Over Elliptic Curve Of The Ring Fq[e], e4 = 0

Authors: Chillali Abdelhakim


Groups where the discrete logarithm problem (DLP) is believed to be intractable have proved to be inestimable building blocks for cryptographic applications. They are at the heart of numerous protocols such as key agreements, public-key cryptosystems, digital signatures, identification schemes, publicly verifiable secret sharings, hash functions and bit commitments. The search for new groups with intractable DLP is therefore of great importance.The goal of this article is to study elliptic curves over the ring Fq[], with Fq a finite field of order q and with the relation n = 0, n ≥ 3. The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic Curve Cryptosystems . In a first time, we describe these curves defined over a ring. Then, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. In anther article we study their cryptographic properties, an attack of the elliptic discrete logarithm problem, a new cryptosystem over these curves.

Keywords: discrete logarithm problem, Elliptic Curve Over Ring

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[1] A. Chillali, Ellipic cuvre over ring, International Mathematical Forum, Vol. 6, no . 31, 1501-1505, 2011.
[2] T. El Gamal, A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms,In IEEE Transactions on Information Theory, volume IT-31, no. 4, pages 469-472, july 1985.
[3] S. Vanstone and R. Zuccherato, Elliptic Curve Cryptosystem Using Curves of Smooth Order Over the Ring Zn, IEEE Transaction on Information Theory, IT-43, 1997.
[4] M. Virat,Courbe elliptique sur un anneau et applications cryptographiques, Thse Docteur en Sciences, Nice-Sophia Antipolis 2009.
[5] N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, In Studies in Advenced Mathematics, pages 21-76. American mathematical society and international press edition, Based on a talk given at the conference in honor of A.O.L. Atkin, 1998.