The rapid development of communication in the last decade has heightened the necessity to create a secure platform for transferring data, including images, more than in previous years. One of the methods of secure image transmission is the encryption method. In this work, an encryption algorithm for multiple images is introduced. In the first step of the proposed algorithm, a key generation algorithm based on a chaotic system and wavelet transform is introduced, and in the next step, the encryption algorithm is developed by introducing rearrange and shift functions based on a chaotic system. One of the most important tools used in the proposed algorithm is the hybrid chaotic system, which is obtained by fractional derivatives and the Cat map. Different types of tests used to study the behavior of this system demonstrate the efficiency of the proposed hybrid system. In the last step of the proposed method, various statistical and security tests, including histogram analysis, correlation coefficient analysis, data loss and noise attack simulations, have been performed on the proposed algorithm. The results show that the proposed algorithm performs well in secure transmission.

This article presents the general solution f : G → V of the following functional equation: f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G , where ( G , + ) is an abelian group and V is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.