On Some Novel Results about Split-Complex Numbers, the Diagonalization Problem, and Applications to Public Key Asymmetric Cryptography

Abstract

In this paper, we present some of the foundational concepts of split-complex number theory such as split-complex divison, gcd, and congruencies. Also, we prove that Euler’s theorem is still true in the case of split-complex integers, and we use this theorem to present a split-complex version of the RSA algorithm which is harder to be broken than the classical version. On the other hand, we study some algebraic properties of split-complex matrices, where we present the formula of computing the exponent of a split-complex matrix $e^X$ with a novel algorithm to represent a split-complex matrix $X$ by a split-complex diagonal matrix, which is known as the diagonalization problem. In addition, many examples were illustrated to clarify the validity of our work.