Linear complexity over ${\mathbb{F}_{{q}}}$ and 2-adic complexity of a class of binary generalized cyclotomic sequences with low-value autocorrelation

TL;DR

This paper constructs binary sequences using generalized cyclotomic classes and analyzes their linear and 2-adic complexities, showing they have optimal cryptographic properties depending on the prime modulus conditions.

Contribution

It determines the linear and 2-adic complexities of a new class of binary sequences using advanced algebraic methods, revealing conditions for maximum complexity.

Findings

Linear complexity attains maximum when p ≡ ±1 mod 8.

Linear complexity is p+1 when p ≡ ±3 mod 8.

Sequences have maximum 2-adic complexity, indicating strong cryptographic properties.

Abstract

A class of binary sequences with period $2p$ is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over ${\mathbb{F}_{{q}}}$ as well as 2-adic complexity are determined using Gauss period and group ring theory. The results show that the linear complexity of these sequences attains the maximum when $p\equiv \pm 1(\bmod~8)$ and is equal to {$p$+1} when $p\equiv \pm 3(\bmod~8)$ over extension field. Moreover, the 2-adic complexity of these sequences is maximum. According to Berlekamp-Massey(B-M) algorithm and the rational approximation algorithm(RAA), these sequences have quite good cryptographyic properties in the aspect of linear complexity and 2-adic complexity.