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A Survey on Complexity Measures of Pseudo-Random Sequences

Chunlei Li

TL;DR

This survey synthesizes four decades of work on complexity measures for pseudo-random sequences generated by feedback shift registers (FSRs), with a focus on linear, quadratic, and maximum-order complexities and their relationships to Lempel–Ziv, expansion, 2-adic complexities, and correlation measures. It covers fundamental definitions, computational methods (e.g., Berlekamp–Massey, DAWG-based approaches), and probabilistic results for random sequences, as well as constructions of sequences with high maximum-order complexity. The work highlights key results such as the typical linear complexity behavior $L_n( extbf{s})\approx n/2$ for random sequences, the existence and characterization of maximum-order–length sequences, and the intricate connections among several complexity measures that inform both theoretical understanding and practical randomness testing. It also identifies open problems and the need for new techniques to advance the theory beyond linear and maximum-order metrics, with implications for cryptographic PRBG design and security analysis.

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.

A Survey on Complexity Measures of Pseudo-Random Sequences

TL;DR

This survey synthesizes four decades of work on complexity measures for pseudo-random sequences generated by feedback shift registers (FSRs), with a focus on linear, quadratic, and maximum-order complexities and their relationships to Lempel–Ziv, expansion, 2-adic complexities, and correlation measures. It covers fundamental definitions, computational methods (e.g., Berlekamp–Massey, DAWG-based approaches), and probabilistic results for random sequences, as well as constructions of sequences with high maximum-order complexity. The work highlights key results such as the typical linear complexity behavior for random sequences, the existence and characterization of maximum-order–length sequences, and the intricate connections among several complexity measures that inform both theoretical understanding and practical randomness testing. It also identifies open problems and the need for new techniques to advance the theory beyond linear and maximum-order metrics, with implications for cryptographic PRBG design and security analysis.

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.
Paper Structure (16 sections, 30 theorems, 92 equations, 3 figures, 1 algorithm)

This paper contains 16 sections, 30 theorems, 92 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Feedback Shift Registers
  3. Linear Complexity
  4. Quadratic Complexity
  5. Maximum-order Complexity
  6. Computation and Statistical Behavior
  7. Approximate statistical behavior
  8. Sequences with high maximum-order complexity
  9. Relations between Complexity Measures
  10. Relations between FSR-based complexities
  11. FSR-based complexities, Lempel-Ziv complexity, expansion complexity, and autocorrelation
  12. 2-adic complexity
  13. Lempel-Ziv complexity
  14. Expansion complexity
  15. $k$-order correlation measure
  16. ...and 1 more sections

Key Result

Proposition 1

The number of sequences of length $n$ and linear complexity $l$, denoted by $N_n(l)$, satisfies the following recursive relation and it, starting from $N_1(0) = N_1(1) =1$, can be given by

Figures (3)

  • Figure 1: An $m$-stage FSR with feedback function $f$
  • Figure 2: Recursive calculation of the quadratic complexity profile of $\mathbf{s}$
  • Figure 3: Direct Acyclic Word Graph of $\mathbf{s}=110100$Jansen1989

Theorems & Definitions (38)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 2
  • Definition 3
  • ...and 28 more