Digitized Chaos for Pseudo-random Number Generation in Cryptography

In this work we propose to take certain chaotic systems as a reference for the design of PRNGs based on nonlinear congruences. In detail, in Section 2 we report a brief comparison between linear and nonlinear PRNGs. Since our aim is to derive nonlinear congruential generators from certain chaotic maps, in Section 3 we overview some theoretical fundamentals about TRNGs based on statistically stable mixing dynamical systems, focusing on the family of the Rényi maps. In Section 4 we discuss the link that exists between the dynamics of chaotic and pseudo-chaotic systems: to explain how the two dynamics are related it is necessary to project some results achieved within the Ergodic Theory (valid for chaotic systems) on the world of digital pseudo-chaos. To this aim, we have proposed a weaker and more general interpretation of the Shadowing Theory proposed by Coomes et al., focusing on probability measures, rather than on single chaotic trajectories. In Section 5 we study how to digitize the R´enyi maps, discussing how to set a minimum period length of the digitized trajectories. In Section 6 we present two alternative methods for the design of PRNGs based on nonlinear recurrences derived from the Renyi map, reporting the results of the NIST SP800.22 standard statistical test suite.