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

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Introduction and Preliminaries
Te feld of real numbers plays an important role in the history of mathematics and computer science. Te feld of real numbers was extended by the feld of complex numbers C � a + bi; i 2 � − 1 [1].
In [2,3], we fnd other extensions of real numbers for algebraic or geometric purposes such as dual numbers D � a + bJ; J 2 � 0 , neutrosophic numbers N � a + bJ; { J 2 � J}, and split-complex numbers S(J) � a + bJ, J 2 � 1 . For the defnitions and properties of split-complex numbers, see [1]. Split-complex numbers have been studied widely, especially their applications in physics, where this class of numbers is useful in representing Dirac brackets [4], and they have been used in defning generalized version of vector spaces, see [5]. We assert that split-complex numbers represent an expansion of the real numbers in two dimensions and possess many interesting geometric properties similar to the complex numbers.
Te main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse, where we fnd some applications of Catalan numbers and other algebraic numbers in cryptology in [6][7][8]. In addition, several kinds of attacks on some cryptosystems were discussed widely [9,10]. Computer scientists have used number theory to protect messages and multimedia from attackers by providing concepts such as symmetric and asymmetric cryptoalgorithms, where some powerful algorithms such as El-Gamal algorithm and RSA algorithm were built over the algebraic properties of integers especially congruencies and relatively prime numbers [11][12][13]. Te usage of split-complex numbers in cryptography has been suggested recently in [14].
Tis motivates us to introduce a novel version of RSA cryptoalgorithm depending on the algebraic properties of split-complex numbers, where we provide a brief discussion for the foundations of split-complex number theory, and we use these algebraic properties to present the algorithm in a similar way of the classical one which has more complexity compared to the classical version of RSA cryptosystem. In addition, we illustrate some examples to clarify the validity of our results.
In addition, we study square matrices with split-complex entries where the diagonalization problem [15] of splitcomplex square matrices will be handled in details. Te mai result is to present a novel algorithm to represent any diagonalizable split-complex square matrix by a diagonal matrix, which means the possibility of writing the splitcomplex matrix X by the formula X � A − 1 DA with A as an invertible split-complex matrix and D is a diagonal splitcomplex matrix, and in connection with this subtraction, we have determined the necessary and sufcient condition for the diagonalization of a matrix of the mentioned type.
Te fundamental advance that this study makes is to close a research gap related to the diagonalization of any square matrix of the type of split-complex square matrices. Also, on the other hand, for the frst time, an efcient algorithm was presented that is useful for calculating the exponent of a matrix of this type in addition to calculating the eigenvalues and related eigenvectors.
Tis paper consists of three main sections. Section 2 discusses the foundations of split-complex number theory such as division, relatively primes, and split-complex congruencies; Section 3 concerns with the applications of splitcomplex number theory in generalizing RSA cryptoalgorithm. Section 4 discusses the conditions of diagonalizing a square split-complex matrix, as well as, it introduces an easy algorithm to solve this problem.

Split-Complex Number Theory and Integers
Te converse holds by a similar argument.
According to the classical Euler's theorem, we can write On the other hand, we have the following equation: Journal of Mathematics 3 □ Remark 10. We defne the raising of a split-complex integer to a split-complex integer power as follows: Tis concept were used in the previous theorem while computing Y φ S (X) .

Applications to Cryptography
Now, we are suggesting the split-complex version of RSA cryptoalgorithm by using the foundational concepts of splitcomplex number theory we have established.
3.1. RSA Algorithm. Assume that we have two sides X, Y: the frst side X is a sender, the second Y is a receiver.
Consider that X has decided to send the text M � m 1 , X and Y should follow these steps: Step 1. Y should pick two large positive prime integers P, Q, then Y computes Step 2. Y should compute: Step Te public key is (N, E). Y computes the secret key E − 1 as follows: Step 4. X gets the cipher text as follows: For the second side Y, it decrypts the message as follows: 3.2. Split-Complex RSA. Assume that we have two sides X, Y: the frst side X is a sender, the second Y is a receiver.
Consider that X has decided to send the text M � m 1 + m 2 J (denoted as a split-complex integer), X and Y should follow these steps: Step 1. Y should pick two large positive split-complex Step 2. Y should compute: Step Te public key is (N, E).
Y computes the secret key E − 1 as follows: Step 4. X gets the cipher text as follows: For the second side Y, it decrypts the message as follows: (Y) generates the public key as follows: Te secret key is E − 1 � 9 + 8J. Assume that (X) has decided to send the message M � 8 + 3J to (Y).
(X) decrypts the message as follows: which is the cipher text.
(Y) decrypts the message as follows: which is the plain text.

Split-Complex Matrices and Teir Diagonalizations
Defnition 11. Let A � (a ij ) be an n-square matrix with split-complex entries a ij � x ij + y ij J ; x ij , y ij ∈ R, J 2 � 1.
We call A a split-complex square matrix.

Remark 12.
Te split-complex matrix can be written as follows: A � A 1 + A 2 J and A 1 , A 2 are two square matrices over R.

Journal of Mathematics
Example 7. Consider the following 3 × 3 split-complex matrix: It is exactly equal to Tus, proof is complete.
We have the following expression: Tus,  Journal of Mathematics Also,

Proof. Suppose that
where U n×n is the n-unit matrix, so that X is invertible.
For the converse, we assume that X � A + BJ is an invertible n-square split-complex matrix, then there exists By adding (29) to (30), we obtain AC + BD + AD + BC � U n×n , thus (A + B)(C + D) � U n×n , which implies that A + B is invertible matrix.  For n � 1, it is clear. We assume that it is true for n � k. We must prove it for n � k + 1. (31)

Journal of Mathematics
By induction, we obtain the desired formula.
On the other hand, we have the following expression: Te diagonalization problem is as follows. An n-square matrix A is diagonalizable if and only if there exists an invertible matrix S and a diagonal matrix D such that A � S − 1 DS.
In the literature, we do not have an algorithm to diagonalize a split-complex matrix X � A + BJ.
In the following, we present an easy algorithm to check if a split-complex matrix is diagonalizable, as to fnd its formula of diagonalization.

□ Theorem 16. Let X � A + BJ be an n-square split-complex matrix, then X is diagonalizable if and only if
Proof. Assume that X is diagonalizable, then there exists to a diagonal split-complex matrix D and an invertible splitcomplex matrix P such that PDP − 1 � X, where Te equation PDP − 1 � X, we obtain the following expression: So that 8

Journal of Mathematics
(35) For the converse, we assume that A + B, A − B are diagonalizable, so there exists two invertible matrices P 1 , P 2 and two diagonal matrices D 1 , D 2 such that: We put that P � 1/2(P 1 + Now, let us check the matrices product PDP − 1 .
We put Journal of Mathematics 9 On the other hand, we put So that By a similar discussion, we put So that (44)

Algorithm for the Diagonalization of a Split-Complex Matrix
Let X � A + BJ be a split-complex matrix with A + B, A − B are diagonalizable matrices.

Journal of Mathematics 11
It is easy to see that PDP − 1 � X.

Conclusion
In this paper, we have found many novel properties and applications of split-complex numbers, where we have introduced an algorithm to diagonalize a split-complex real matrix and raising a split-complex matrix to powers. We have shown that a diagonalizable square split-complex matrix X � A + BJ is diagonalizable if and only if A + B, A − B are diagonalizable. Also, a formula of computing the splitcomplex matrix powers was obtained and proved. Also, we have presented a generalization of RSA cryptosystem by using split-complex integers with more complexity. In addition, the foundations of split-complex number theory were established and clarifed in terms of theorems and examples.
As a future research direction, we aim to use the number theoretical foundations of split-complex integers in generalizing El-Gamal cryptosystem with a novel split-complex version. In addition, the problem of representing splitcomplex matrices by split-complex linear transformations should be discussed, we recommend researchers to study this important problem to answer the following research question: How can we represent a square split-complex matrix by split-complex linear function defned between two splitcomplex vector spaces defned in [5]?

Data Availability
Te data used to support the study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that there are no conficts of interest.