Elliptic curves over a chain ring of characteristic 3

This paper proposes the generalization of our previous work to the ring A, = F-3d[X]/(X-n). All results found before in A2, A3 and A(4) [1-3] hold in Ati; but the approach here is clearly different, and has given more interesting results, specially when 3 does not divide #E-a0,b0(1) ; the elliptic curve over the ring An is a direct sum of the elliptic curve over the field F-3d and, unexpectedly its own subgroup of elements with the third projective coordinate not invertible, instead of F-3d(n) as it was thought in the earlier works. Other results are deduced from, we cite the equivalence of the Discrete Logarithm Problem (DLP) on the elliptic curve over the ring A(n) and the field F-3d, which is beneficial for cryptanalysts and cryptographers as well, and we will set the theoretic foundations to build a cryptosystem similar to the one in [4] with more benefits, which will be specified later. (C) 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).