Basic principles for constructing mixing functions based on the simplest linear and nonlinear mappings

Abstract

This article is dedicated to exploring the possibility of using the simplest principles of nonlinear discrete dynamic systems theory in computer cryptography, which are characterized by their chaotic behavior. The main problem of using chaotic systems in computer calculations is that the number of possible states in a computer is finite. Therefore, computer models of chaos are only an approximation of the true chaotic behavior, and each trajectory of the approximated system is periodic. From a mathematical point of view, encryption in information systems involves transforming the space of finite messages, which is similar to the phase space in the theory of dynamical systems. The mixing function specifies such encryption. The main requirements for the mixing function are the absence of collisions, i.e., bijectivity of the mapping, good diffusion properties, and, in addition, the inverse transformation should not be more complicated than the direct one. The article demonstrates that it is possible to utilize the diffusion properties of nonlinear dynamical systems in spaces with a finite number of states by using the simplest nonlinear mapping, Tent. To enhance the diffusion properties, a superposition of the nonlinear Tent map and the linear permutation map (in the more general case of the Hill map) was used. The main advantages of the constructed functions are their simplicity of implementation, speed of calculations in mixing problems, and strong cryptographic persistence. Correlation analysis, sensitivity analysis, and analysis of the lengths of cycle periods that divide the space into non-overlapping subsets have been conducted for these functions. As a result, the expected good diffusion properties of these mixing functions are confirmed. The possibility of applying these functions to image encryption is demonstrated.