Abstract— In this work, we study the elliptic curve over the ring ; ; where d is a positive integer. More precisely in cryptography applications, we will give many various explicit formulas describing the binary operations calculus in . The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic Curve Cryptosystems. Keywords— Elliptic Curves, Finite Ring, Cryptography.. I. INTRODUCTION ET d be an integer, we consider the quotient ring A = where is the finite field of order . Then the ring A is identified to the ring with ; ie: A = { + | ; }, See, [3] and, [5]. We consider the elliptic curve over the ring A which is given by equation where a, b, c are in A and is inv ertible in A ; but we can take c = 1; see, [4]. • Notation Let a, b A such that b is invertible in A and c = 1: So, We denote the elliptic curve over A by and we write: = { [X : Y : Z] | } if and , we also write: = { [X : Y : Z] | }. II. CLSSIFICATION OF ELEMENTS OF Let [X :Y :Z] , where X, Y and Z are in A. We have two cases for Z: * Z invertible: then [X : Y : Z] = [X : Y : 1]; hence we take just [X: Y: 1]. * Z non invertible: So Z = ; see [3] in this cases we have two cases for Y. This work was supported by the Department of Mathematics in the university of Mohammed First, Oujda MOROCCO. Abdelhamid Tadmori Author is with the Department of Mathematics FSO UMF Oujda MOROCCO; (e-mail: atadmori@yahoo.fr). Abdelhakim Chillali Author is the Department of FST USMBA, FEZ, MOROCCO; (e-mail: chil2007@voila.fr) M’hammed Ziane. Author is with the Department of Mathematics FSO UMF Oujda MOROCCO; (e-mail: ziane20011@yahoo.fr). - Y invertible: Then [X : Y : Z] = [X : 1 : Z ]; so we just take [X : 1 : ] , then is verified the equation of . so we can write: a = b = X = We have: Which implies that : Then : Since, is a base of the vector space A over then so X = and hence : 1 :0]. - Y non invertible: then we have ; so is invertible so we take ; thus, which is absurd. Proposition 1: Every element of , is of the form or ; where and we write [X : Y : 1] | } . III. EXPLICIT FORMULAS We consider the canonical projection defined by : We have is a morphism of ring. * Let the mapping defined by : The mapping is a surjective homomorphism of groups. Theorem 1 : • If then : The Binary Operations Calculus in Abdelhamid Tadmori, Abdelhakim Chillali, M'hammed Ziane L INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 2015 ISSN: 1998-0140 171