In the present paper, we consider a nonself-adjoint fourth-order differential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions.

In the present work the -adic gamma function has been considered. The Volkenborn integral of the -adic gamma function by using its Mahler expansion has been derived. Moreover, a new representation for the -adic Euler constant has been given.

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.

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.

No abstract