# Some New Maximally Chaotic Discrete Maps

## TL;DR

This paper introduces a new discrete chaotic map that achieves maximum chaos, making it highly unpredictable and useful for applications like cryptography.

## Contribution

A new discrete chaotic map is proposed and proven to achieve asymptotically maximal chaos among permutation maps.

## Key findings

- The proposed map is a bijection for all control parameters.
- The discrete Lyapunov exponent of the map approaches the maximum possible value as m increases.
- Numerical experiments confirm high chaos through entropy and correlation analysis.

## Abstract

In this paper, we first prove (Theorem 1) that any two inputs producing the same output in a symmetric pair of discrete skew tent maps always have the same parity, meaning that they are either both even or both odd. Building on this property, we then propose (Definition 1) a new discrete chaotic map and prove that (Theorem 2) the proposed map is a bijection for all control parameters. We further prove that (Theorem 3) the discrete Lyapunov exponent (dLE) of the proposed map is not only positive but also approaches the maximum value among all permutation maps over the integers {0,1,…,2m−1} as m gets larger. In other words, (Corollary 1) the proposed map asymptotically achieves the highest possible chaotic divergence among the permutation maps over the integers {0,1,…,2m−1}. To provide some further evidence that the proposed map is highly chaotic, we present at the end some results from the numerical experiments. We calculate the approximation and permutation entropy of the output integer sequences. We also show the NIST SP800-22 tests results and correlation properties of some derived binary sequences.