An improvement on the chaotic behavior of the Gauss Map for cryptography purposes using the Circle Map combination

Abstract

Chaos based cryptography has becoming an interesting topic lately, as it utilizes chaotic systems properties for secure key concealment. Many chaotic functions are discovered, constructed, and used time over time for this purpose, which will be our main aim here. Two well known maps that has been known for exhibiting chaotic behaviors are the Gauss Map and the Circle Map, where the Circle Map has unlimited chaos potential, while the Gauss Map's is much weaker and limited. In this paper, we investigate computationally using Python whether the Gauss Map can be improved by combining it with the Circle Map, allowing exploitation of greater chaotic behaviors. For this purpose, an improved version of the Gauss map is constructed, from which, we plot its bifurcation diagrams and Lyapunov exponents graphics, and show that it has a good potential to be a random number generator (RNG) using the NIST test, as these are the three main aspects of chaotic maps utilized in chaos based cryptography. The results obtained from this observation shows that composing the Circle Map into the Gauss Map, along with several manipulations, generates a significantly improved version of the Gauss Map, as it has a bifurcation diagram with much higher density, much higher Lyapunov exponents, and mostly better P-Values from the NIST tests, although it is still not fully suitable for a RNG. The manipulations done here, which aims to conserve the maps ranges to stay within the chaotic intervals and position the Circle Map to be the "variable"of the Gauss Map, allows the chaotic behaviors from the original maps to be bequeathed and strengthened in the new map.