Features of Generation Sequences of Refined Fibonacci Polynomial Matrices

The features of generating the $n$th sequence of refined Fibonacci polynomial matrices of the $m$th order, the elements of which are Fibonacci polynomials of no higher than $(m+n–2)th$ number, are considered. The obtained Fibonacci matrices allow finding both their determinants and inverse matrices suitable for matrix encryption of block data. It was found that even over the last decade a significant number of publications have been published, each of which substantiates different approaches to generating sequences of Fibonacci polynomial matrices and proves the feasibility of their use for encrypting block data. It was found that the use of such matrices as a separate procedure for protecting block data in the theory and practice of cryptography is extremely rare. The Fibonacci polynomial matrix sequences from the $2nd$ to the $5th$ order are analyzed, according to which shortcomings in the traditional approach to forming the structure of the elements of such matrices are revealed, first of all, the number $(k)$ of its different elements, which are Fibonacci polynomials of the not higher than $(n–1)$th power. It is established that for Fibonacci matrices of any order $(m)$ of such different Fibonacci polynomials there will be only $k = 3$, the structure of which will depend only on the number $(n)$ of the sequence of the Fibonacci polynomial matrix. Such a small number is not only uninformative and easily analyzable by a cryptanalyst but also vulnerable to cryptanalysis. The structure of the elements of Fibonacci polynomial matrices is specified, the number of which already depends on the order of the $matrix (m)$ and is $k = m+1$.
A method for generating a sequence of refined Fibonacci polynomial matrices has been developed, which consists in using a recurrent matrix relation, according to which the next polynomial matrix is formed by multiplying the variable x sequentially by the elements of the current matrix, adding the elements of the newly generated matrix to those of the previous matrix, after which all similar terms are grouped in the formed elements. The mechanisms for forming a sequence of $8$ polynomial refined Fibonacci matrices from the $2nd$ to the $4th$ order, the elements of which are Fibonacci polynomials of no higher than $(m+n–2)$th power, have been presented, which made it possible to analyze not only the features of their construction, but also to understand the corresponding procedures for finding their determinants and inverse matrices.
It was found that the proposed structure of the elements of the nth sequence of Fibonacci polynomial matrices of the mth order has an interesting property, according to which it is possible to avoid the use of the recurrent matrix relation, and to generate the corresponding Fibonacci polynomial matrices only by the numbers $(m+n–2–j)$th sequence of Fibonacci polynomials, the specific values of which depend on their location in the matrix and the number of its column, namely $\forall j \in[0 \div(m-1)]$. Software has been developed that allows generating both sequences of refined Fibonacci polynomial matrices of the mth order, and finding their determinants and inverse polynomial matrices of a similar order. An example of the application of the matrix method of encrypting block data with a refined Fibonacci polynomial matrix is given, which allows the interested reader to understand the basic principle of encrypting both the initial message and decrypting the encrypted message.