Chaotification of One-Dimensional Maps Based on Remainder Operator Addition

In this work, a chaotification technique is proposed that can be used to enhance the complexity of any one-dimensional map by adding the remainder operator to it. It is shown that by an appropriate parameter choice, the resulting map can achieve a higher Lyapunov exponent compared to its seed map, and all periodic orbits of any period will be unstable, leading to robust chaos. The technique is tested on several maps from the literature, yielding increased chaotic behavior in all cases, as indicated by comparison of the bifurcation and Lyapunov exponent diagrams of the original and resulting maps. Moreover, the effect of the proposed technique in the problem of pseudo-random bit generation is studied. Using a standard bit generation technique, it is shown that the proposed maps demonstrate increased statistical randomness compared to their seed ones, when used as a source for the bit generator. This study illustrates that the proposed method is an efficient chaotification technique for maps that can be used in chaos-based encryption and other relevant applications.