# Random Walk in a N-cube Without Hamiltonian Cycle to Chaotic Pseudorandom Number Generation: Theoretical and Practical Considerations

## TL;DR

This paper presents a novel chaotic pseudorandom number generator based on a walk in an N-cube missing a Hamiltonian cycle, with theoretical proofs and practical validation of its chaotic behavior and statistical quality.

## Contribution

It introduces a new PRNG construction using N-cube walks without Hamiltonian cycles, providing theoretical analysis and practical implementation details.

## Key findings

- Proven chaotic behavior of the PRNG.
- Constructed Hamiltonian cycle removal method.
- Successfully passes classical statistical tests.

## Abstract

Designing a pseudorandom number generator (PRNG) is a difficult and complex task. Many recent works have considered chaotic functions as the basis of built PRNGs: the quality of the output would indeed be an obvious consequence of some chaos properties. However, there is no direct reasoning that goes from chaotic functions to uniform distribution of the output. Moreover, embedding such kind of functions into a PRNG does not necessarily allow to get a chaotic output, which could be required for simulating some chaotic behaviors. In a previous work, some of the authors have proposed the idea of walking into a $\mathsf{N}$-cube where a balanced Hamiltonian cycle has been removed as the basis of a chaotic PRNG. In this article, all the difficult issues observed in the previous work have been tackled. The chaotic behavior of the whole PRNG is proven. The construction of the balanced Hamiltonian cycle is theoretically and practically solved. An upper bound of the expected length of the walk to obtain a uniform distribution is calculated. Finally practical experiments show that the generators successfully pass the classical statistical tests.