The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

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<abstract><p>In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking <inline-formula><tex-math id="M1">\begin{document}$ x = 1 $\end{document}</tex-math></inline-formula>. We call this coding theory the k-order Gaussian Fibonacci coding theory. This coding method is based on the <inline-formula><tex-math id="M2">\begin{document}$ {Q_k}, {R_k} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ E_n^{(k)} $\end{document}</tex-math></inline-formula> matrices. In this respect, it differs from the classical encryption method. Unlike classical algebraic coding methods, this method theoretically allows for the correction of matrix elements that can be infinite integers. Error detection criterion is examined for the case of <inline-formula><tex-math id="M4">\begin{document}$ k = 2 $\end{document}</tex-math></inline-formula> and this method is generalized to <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> and error correction method is given. In the simplest case, for <inline-formula><tex-math id="M6">\begin{document}$ k = 2 $\end{document}</tex-math></inline-formula>, the correct capability of the method is essentially equal to 93.33%, exceeding all well-known correction codes. It appears that for a sufficiently large value of <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula>, the probability of decoding error is almost zero.</p></abstract>

In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.