A new method to improve cryptographic properties of chaotic discrete dynamical systems

In this paper, we propose a new method, based on composition of chaotic maps of same type, to ameliorate some shortcomings of using chaotic discrete dynamical systems in cryptographic applications. Such well-known drawbacks are a small keyspace and high predictability of the values generated by the orbits of a single dynamical system. We prove, using some specific tools from chaos theory (such as Lyapunov exponent), that the dynamical system obtained by the proposed method has a chaotic behavior for a large parameters´ values space. Also, using Birkhoff´s theorem in conjunction with Kolmogorov-Smirnov test, we prove the ergodicity property. Finally, we present an analysis of the randomness of the bitstreams obtained by applying two computational methods for discretization of the real values generated using the proposed method, performed with the NIST statistical battery. Theoretical arguments and the very good statistical results confirm viability of the proposed method, recommending it for secure cryptographic applications.