The structure and enumeration of periodic binary sequences with high nonlinear complexity

2026-02-01

AI Summary

This study investigates the structure and enumeration of binary sequences of period $n$ whose nonlinear complexity is at least $3n/4$, as a means to assess their randomness. By integrating feedback shift register models, combinatorial methods, and structural analysis of sequences, the work provides the first complete characterization of the intrinsic structure of such high nonlinear complexity sequences and derives an exact formula for their count. This result fills a critical gap in the theoretical understanding of periodic sequences with high nonlinear complexity and establishes a foundational framework for the construction and analysis of high-quality pseudorandom sequences in cryptographic applications.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary sequences with nonlinear complexity larger than or equal to 3n/4 is characterized. Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.

Problem

• nonlinear complexity
• periodic binary sequences
• sequence enumeration
• feedback shift registers
• randomness

Innovation

• nonlinear complexity
• periodic binary sequences
• structure characterization
• exact enumeration
• feedback shift registers