El Gamal Cryptosystem on a Montgomery Curves Over Non Local Ring

Let Fq be the finite field of q elements, where q is a prime power. In this paper, we study the Montgomery curves over the ring Fq[X]/(X^2−X), denoted by MA,B(Fq[X]/(X^2−X) ); (A,B) ∈ (Fq[X]/(X^2−X))^2. Using the Montgomery equation, we define the Montgomery curves MA,B(Fq[X]/X^2−X) and we give a bijection between this curve and product of two Montgomery curves defined on Fq. Furthermore, we study the addition law of Montgomery curves over the ring Fq[X]/X^2−X. We close this paper by introducing a public key cryptosystem which is a variant of the ElGamal cryptosystem on a Montgomery curves over the same ring.