Elliptic curves over a chain ring of characteristic 3

This paper proposes the generalization of our previous work to the ring An=F3d[X]/(Xn). All results found before in A2, A3 and A4[1], [2] and [3] hold in An; but the approach here is clearly different, and has given more interesting results, specially when 3 does not divide #Ea0,b01; the elliptic curve over the ring An is a direct sum of the elliptic curve over the field F3d and, unexpectedly its own subgroup of elements with the third projective coordinate not invertible, instead of F3dn 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 An and the field F3d, 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.