Chaos Based Generation of True Random Bits.

In this work, referring to some major results achieved in Ergodic Theory, we discuss a theoretical approach for studying how to generate random bits using piecewise linear chaotic maps, even taking into account the possible dispersion of system parameters. In detail, referring to the work developed by Boyarsky and Gora, we discuss the ergodic theory developed for piecewise affine expanding transformations, with a special reference to the family of the Sawtooth maps for truly random number generation. Ergodic Theory classifies dynamical systems depending on the complexity degree of their dynamics. Referring to this theoretical framework, we identify a region within the parameter space in which the Sawtooth map preserves its exactness property. Even if this work is particularly focused on this special family of maps, we stress that the proposed approach can be generalized for studying other piecewise affine maps proposed in literature for ICT applications. Furthermore, we discuss the link that there exists between the theoretical forms of entropy developed in Ergodic Theory (e.g., the Kolmogorov-Sinai entropy) and the classical Shannon entropy for information sources. This work is organized as in the following: in the next two subsections some formal definitions about true random number generators are given. In Section 2 we introduce the concept of symbolic dynamics, presenting the family of Sawtooth maps. Moreover, we prove the exactness property of these systems, considering a parameter space which covers all those values of practical interest in true random bit generation. Section 3 is entirely devoted to the analysis of chaos-based symbolic dynamics, with a discussion on the link existing between the theoretical forms of entropy developed in the Ergodic Theory (e.g., the Kolmogorov-Sinai entropy) and the classical Shannon entropy for information sources. Conclusions and references close this work.