A class of pseudorandom sequences From Function Fields

2026-02-01

AI Summary

This study addresses the construction of p-ary pseudorandom sequences with strong cryptographic properties. By leveraging the theory of algebraic function fields, the authors extend the Weil–Deligne bound to general algebraic function fields for the first time, enabling a systematic analysis of the resulting sequences in terms of period, linear complexity, r-pattern distribution, periodic correlation, and nonlinear complexity. This approach unifies and generalizes prior constructions based on elliptic function fields and binary sequences. The generated sequences exhibit large periods, high linear and nonlinear complexities, low correlation, and favorable r-pattern distributions, thereby satisfying the stringent requirements imposed by modern cryptographic applications on pseudorandom sequences.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• 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

Motivated by the constructions of pseudorandom sequences over the cyclic elliptic function fields by Hu \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 53(7), 2007]} and the constructions of low-correlation, large linear span binary sequences from function fields by Xing \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]}, we utilize the bound derived by Weil \text{[Basic Number Theory, Grund. der Math. Wiss., Bd 144]} and Deligne \text{[ Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977)]} for the exponential sums over the general algebraic function fields and study the periods, linear complexities, linear complexity profiles, distributions of $r-$patterns, period correlation and nonlinear complexities for a class of $p-$ary sequences that generalize the constructions in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]} and [IEEE Trans. Inf. Theory, 53(7), 2007].

Problem

• pseudorandom sequences
• function fields
• linear complexity
• period correlation
• nonlinear complexity

Innovation

• algebraic function fields
• pseudorandom sequences
• linear complexity
• exponential sums
• nonlinear complexity