Applications of Combinatorics on Words with Symbolic Dynamics

TL;DR

This paper reviews how combinatorics on words and symbolic dynamics underpin various applications like data compression, cryptography, and pseudorandom generation, emphasizing theoretical foundations and practical algorithms.

Contribution

It provides a comprehensive overview of the theoretical concepts and algorithms connecting combinatorics on words with applications in information theory and cybersecurity.

Findings

Lyndon words count formula for fixed length and alphabet size

Role of de Bruijn sequences and topological entropy in data optimization

Insights into pseudorandomness and complexity measures in cryptography

Abstract

In this paper, we explore applications of combinatorics on words across various domains, including data compression, error detection, cryptographic protocols, and pseudorandom number generation. The examination of the theoretical foundations enabling these applications, emphasizing important concepts of mathematical relationships and algorithms. In data compression, we discuss the Lempel-Ziv family of algorithms and Lyndon factorization, with the number of Lyndon words of length \( n \) over an alphabet of size \( k \) given by \[ L(n,k) = \frac{1}{n} \sum_{d|n} \mu(d) k^{n/d}. \] We address cryptographic protocols and pseudorandom number generation, highlighting the role of pseudorandomness theory and complexity measures. Also, by explore de Bruijn sequences, topological entropy, and synchronizing words in their practical contexts, demonstrating their contributions to optimizing information storage, ensuring data integrity, and enhancing cybersecurity.