Applied Mathematical Sciences, Vol. 9, 2015, no. 35, 1721 - 1733 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.5147 Elliptic Curve over Ring  ૝ = ૛  [];  ૝ = Abdelhamid Tadmori Department of Mathematics and Computer Sciences Faculty of Sciences, University of Mohamed First. BP.717. 60000, Oujda, Morocco Abdelhakim Chillali USMBA, FPT, Taza, Morocco M’hammed Ziane UMPO, Faculty of Sciences, Morocco Copyright © 2015 Abdelhamid Tadmori, Abdelhakim Chillali and M’hammed Ziane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this work, we study the elliptic curves over the ring A ସ =  2 d [X] ሺX 4 ሻ ; precisely we will establish the short exact sequence, which gives the extension group ܧ ௔,௕ ሺ ସ ሻ, and we give the cryptography application. Keywords: Elliptic curves over ring, Finite ring, Cryptography 1. Introduction Elliptic curve cryptography (ECC) is a public key cryptography. In public key cryptography each user or the device taking part in the communication generally have a pair of keys, a public key and a private key, and a set of operations associated with the keys to do the cryptography operation. Only the particular user knows the private key, whereas the public key is distributed to all user taking part in the communication. In this work we will study the elliptic curves over the ring  ସ = ଶ  [] ;  ସ = Ͳ, and we define the mathematical operations of ECC; ܧ ௔,௕ ሺ ସ ሻ :  ଶ  +  =  ଷ +  ଶ  +  ଷ which provides