In this paper, we study the elliptic curve \(E_{a,b}(A_P)\), with \(A_P\) the localization of the ring \(A=F _p[e_1,...,e_n]\) where \(e_ie_i=e_i\) and \(e_ie_j=0\) if \(i\neq j\), in the maximal ideal \(P=(e_1,...,e_n)\). Finally we show that \( \text{Card}(E_{a,b}(A_P)) \geq (\text{Card}(E_{a,b}(F_p)) - 3)^n + \text{Card}(E_{a,b}(F_p)) \) and the execution time to solve the problem of discrete logarithm in \(E_{a,b}(A_P)\)…

In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. In this work, we proposed a new problem applicable to the public key cryptography, based on the Matrices, called “Matrix discrete logarithm problem”, it uses certain elements formed by matrices whose coefficients are elements in a finite field. We have…