A class of pseudorandom sequences From Function Fields

TL;DR

This paper develops a new class of pseudorandom sequences over function fields, analyzing their properties such as period, linear complexity, and correlation, extending previous constructions with theoretical bounds from algebraic geometry.

Contribution

It introduces a generalized construction of pseudorandom sequences using algebraic function fields and analyzes their cryptographic properties with bounds from Weil and Deligne.

Findings

Sequences have large linear complexity and long periods.

Sequences exhibit low correlation properties.

Theoretical bounds confirm sequence pseudorandomness.

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].