Confusion and diffusion features are two fundamental needs of encoded text or images. These features have been used in various encryption algorithms such as Advanced Encryption Standard (AES) and Data Encryption Standard (DES). The AES adopts the Sbox table formed with irreducible polynomials, while the DES employs the Feistel and Sbox structures. This study proposes a new encryption algorithm based on Fibonacci polynomials and matrices, which meets the fundamental needs of image encryption and provides an alternative to other encryption algorithms. The success of the proposed method was tested on three different images, as evidenced by the histogram analysis results of the sample images, together with the number of changing pixel rate (NPCR) and the unified averaged changed intensity (UACI). In addition, the root mean squared error (RMSE) suggests that the decoded images are consistent with the original images. It can therefore be summarized that the proposed encryption algorithm is suitable for image encryption.

Galois field, has an important position in cryptology. Advanced Encryption Standard (AES) also used in polynomial operations. In this paper, we consider the polynomial operations on the Galois fields, the Fibonacci polynomial sequences. Using a certain irreducible polynomial, we redefine the elements of Fibonacci polynomial sequences to use in our cryptology algorithm. So, we find the classical AES-like cryptology via the Fibonacci polynomial matrix. Successful results were achieved with the method used.

In this paper, we consider Pell numbers. We define the gell numbers which generalize the Pell numbers. Moreover, we derive Binet-like formula, generating function and exponential generating function for the gell sequence. Also, we obtain the gell series and some important identities for the gell sequence.

In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions' properties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.