Parametrically perturbed logistic map - a new approach based on the least significant bits in the state variable’s representation

Abstract

Cryptography involves controlled randomization and de-randomization of digital data. In the essential cryptographic infrastructure, the role of Pseudo Random Number Generators (PRNGs) becomes significant. The present work focuses on a new method of implementing PRNGs using chaotic maps. Using this approach, the classical chaotic maps, which otherwise are found to be cryptographically inadequate, can be enhanced to meet the necessary statistical requirements. In this new approach, one of the parameters of a chaotic map can be varied using the lower bits of the floating-point representation of the map’s state variable. This methodology is demonstrated using one of the most commonly discussed chaotic maps - the logistic map. The modified logistic map is shown to have excellent chaotic characteristics across the entire range of the only remaining parameter - the map’s initial state. The resulting chaotic map is named ’Parametrically Perturbed Logistic Map (PPLM)’. The PPLM is used to implement a new Pseudo Random Bit Generator (PRBG) - the PPLM-PRBG. An extensive set of simulations is carried out on the PPLM-PRBG. Using a standard set of parametric studies, including the ’NIST test suite’, the new PRBG is found to have excellent statistical and cryptographic properties.