Journal of Digital Information Management Moha ben taleb Elhamam, Abdelhakim Chillali, Lhoussain El Fadil 20 1 2022 https://doi.org/10.6025/jdim/2022/20/1/25-30 https://www.dline.info/fpaper/jdim/v20i1/jdimv20i1_3.pdf Let be a finite ring of characteristic 2, where e2 = e and n is a positive integer. Let (a, d) 2 ( )2, such that a and d + a2 + a are invertible in , we study the binary Edwards curve over this ring, denoted by and we give a bijection between this curve and produces two binary Edwards curves defined on the finite field . Afterthat we study the addition law of binary Edwards curves over the ring . We end this work with cryptography applications, ElGamal twisted Edwards curve cryptosystem and Cramer-Shoup twisted Edwards curve cryptosystem.