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

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...

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Bibliographic Details
Published in:Journal of mathematics (Hidawi) 2023, Vol.2023, p.1-12
Main Authors: Merkepci, Mehmet, Abobala, Mohammad
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Complex numbers
Cryptography
Mathematics
Multimedia
Number theory
Prime numbers
Theorems
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Summary: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 eX 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.
ISSN:2314-4629
2314-4785
DOI:10.1155/2023/4481016