Randomizing nonlinear maps via symbolic dynamics

Abstract

Pseudo Random Number Generators (PRNG) have attracted intense attention due to their obvious importance for many branches of science and technology. A randomizing technique is a procedure designed to improve the PRNG randomness degree according the specific requirements. It is obviously important to quantify its effectiveness. In order to classify randomizing techniques based on a symbolic dynamics' approach, we advance a novel, physically motivated representation based on the statistical properties of chaotic systems. Recourse is made to a plane that has as coordinates (i) the Shannon entropy and (ii) a form of the statistical complexity measure. Each statistical quantifier incorporates a different probability distribution function, generating thus a representation that (i) sheds insight into just how each randomizing technique operates and also (ii) quantifies its effectiveness. Using the Logistic Map and the Three Way Bernoulli Map as typical examples of chaotic dynamics it is shown that our methodology allows for choosing the more convenient randomizing technique in each instance. Comparison with measures of complexity based on diagonal lines on the recurrence plots [N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438 (2007) 237] support the main conclusions of this paper