A linear complexity analysis of quadratic residues and primitive roots spacings

TL;DR

This paper analyzes the linear complexities of sequences derived from quadratic residues and primitive roots modulo a prime, demonstrating high complexities through computational experiments, which has implications for cryptography.

Contribution

It provides a linear complexity analysis of sequences based on quadratic residues and primitive roots, revealing their high complexity and potential cryptographic strength.

Findings

Sequences have very good to perfect linear complexities.

Berlekamp-Massey algorithm confirms high complexity levels.

Results suggest strong cryptographic properties of these sequences.

Abstract

We investigate the linear complexities of the periodic 0-1 infinite sequences in which the periods are the sequence of the parities of the spacings between quadratic residues modulo a prime p, and the sequence of the parities of the spacings between primitive roots modulo p, respectively. In either case, the Berlekamp-Massey algorithm running on MAPLE computer algebra software shows very good to perfect linear complexities.