Cryptography using Fibonacci–Mersenne and Fibonacci-balancing p-sequences with a self-invertible matrix and the Affine–Hill cipher

In this paper, we define two new sequences using the Fibonacci p-numbers, the generalized Mersenne numbers, and m-balancing numbers. These sequences are obtained from the corresponding characteristic polynomials. The determinants and combinatorial and exponential representations of these new sequences are given. As an application, we provide two algorithms using these new sequences with self-invertible matrices and the Affine–Hill cipher.