A Survey on Complexity Measures of Pseudo-Random Sequences

TL;DR

This survey reviews four decades of research on various complexity measures of pseudo-random sequences, highlighting their theoretical importance and practical applications in cryptography.

Contribution

It provides a comprehensive overview of different complexity measures and their interrelations, summarizing key developments over the past forty years.

Findings

Analyzed relations between linear, quadratic, and maximum-order complexities.

Reviewed connections with Lempel-Ziv, expansion, and 2-adic complexities.

Summarized recent advances in complexity measures for cryptographic applications.

Abstract

Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements in the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical computer science and of practical interest in cryptography. This survey reviews notable research from the past four decades on the linear, quadratic and maximum-order complexities of pseudo-random sequences and their relations with Lempel-Ziv complexity, expansion complexity, 2-adic complexity, and correlation measures.