A Class of Maximum-Period Nonlinear Congruential Generators Derived From the Rényi Chaotic Map

In this paper, a family of nonlinear congruential generators (NLCGs) based on the digitized Rényi map is considered for the definition of hardware-efficient pseudorandom number generators (PRNGs), and a theoretical framework for their study is presented. The authors investigate how the nonlinear structure of these systems eliminates some of the statistical regularities spoiling the randomness of sequences generated with linear techniques. In detail, in this paper, a necessary condition that the considered NLCGs must satisfy to have maximum period length is given, and a list of such maximum period PRNGs for period lengths up to 2 ^(31)-1 is provided. Referring to the NIST800-22 statistical test suite, two PRNG examples are presented and compared to well-known PRNGs based on linear recurrencies requiring a similar amount of resources for their implementation.