A further study on the linear complexity of new binary cyclotomic sequence of length $p^r$

TL;DR

This paper proves a conjecture on the linear complexity of a new class of binary cyclotomic sequences of length p^r, using Euler quotients, under certain number-theoretic conditions, and introduces a generic sequence construction.

Contribution

It provides a general proof of Xiao et al.'s conjecture for binary cyclotomic sequences using Euler quotients and introduces a flexible generic construction method.

Findings

The conjecture is proved under specified conditions.

A generic construction of sequences is proposed.

An efficient method for computing linear complexity is developed.

Abstract

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period $p^r$ was proposed by Z. Xiao et al. (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0408-7). Later, for the case $f$ being the form $2^r$ with $r\ge 1$, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, under the assumption of $2^{p-1} \not\equiv 1 \bmod p^2$ and $\gcd(\frac{p-1}{{\rm {ord}}_{p}(2)},f)=1$, the conjecture proposed by Z. Xiao et al. is proved for a general $f$ by using the Euler quotient. Actually, a generic construction of $p^r$-periodic binary sequence based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and includes Xiao's construction as a special case, and then an efficient method to compute the linear complexity of the sequence by the generic construction is presented, based on which the conjecture proposed by Z. Xiao et al. could be easily proved under the aforementioned assumption.