A Topological Study of Chaotic Iterations. Application to Hash Functions

TL;DR

This paper provides a comprehensive topological analysis of chaotic iterations, demonstrating their chaotic properties and potential for creating truly chaotic computer programs, including novel hash functions with neural network integration.

Contribution

It offers a detailed topological characterization of chaotic iterations, establishing their chaotic nature and applying them to design new hash functions with neural network enhancement.

Findings

Chaotic iterations are proven to be topologically mixing and highly sensitive.

The study shows chaotic iterations can generate truly unpredictable computer programs.

A new chaotic hash function, including a neural network version, is proposed.

Abstract

Chaotic iterations, a tool formerly used in distributed computing, has recently revealed various interesting properties of disorder leading to its use in the computer science security field. In this paper, a comprehensive study of its topological behavior is proposed. It is stated that, in addition to being chaotic as defined in the Devaney's formulation, this tool possesses the property of topological mixing. Additionally, its level of sensibility, expansivity, and topological entropy are evaluated. All of these properties lead to a complete unpredictable behavior for the chaotic iterations. As it only manipulates binary digits or integers, we show that it is possible to use it to produce truly chaotic computer programs. As an application example, a truly chaotic hash function is proposed in two versions. In the second version, an artificial neural network is used, which can be stated as chaotic according to Devaney.