The $k$-error linear complexity distribution for $2^n$-periodic binary sequences

TL;DR

This paper analyzes the distribution of $k$-error linear complexity in $2^n$-periodic binary sequences, providing exact counts for various $k$ and linear complexity levels, which enhances understanding of sequence security measures.

Contribution

It characterizes the complete counting functions for $k$-error linear complexity of $2^n$-periodic binary sequences for specific $k$ and linear complexity cases, correcting previous inaccuracies.

Findings

Complete counting functions for $k=2,3$ with linear complexity less than $2^n$.

Complete counting functions for $k=3,4$ with linear complexity equal to $2^n$.

Validation and correction of a recent result on $k$-error linear complexity distribution.

Abstract

The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By studying the linear complexity of binary sequences with period $2^n$, one could convert the computation of $k$-error linear complexity into finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated in this paper. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic balanced binary sequences (with linear complexity less than $2^n$) are characterized. Second, for $k=3,4$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$ are presented. Third, as a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the $k$-error linear complexity for $k = 2$ and 3 are obtained. Further more, an important result in a recent paper is proved to be not completely correct.