Characterization of $2^n$-periodic binary sequences with fixed 3-error or 4-error linear complexity

TL;DR

This paper analyzes the distribution of k-error linear complexity in 2^n-periodic binary sequences, providing complete counting functions for specific error levels and linear complexities, which are crucial for assessing sequence security.

Contribution

It introduces new combinatorial methods to precisely count sequences with fixed 3-error or 4-error linear complexity, advancing understanding of sequence security measures.

Findings

Complete counting functions for 2-error and 3-error linear complexity with linear complexity less than 2^n.

Explicit counting functions for 3-error and 4-error linear complexity when linear complexity equals 2^n.

Derived counting functions for 4-error linear complexity of sequences with linear complexity less than 2^n.

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 using the sieve method of combinatorics, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated based on Games-Chan algorithm. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic 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, for $k=4,5$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity less than $2^n$ are derived. As a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the 3-error linear complexity are obtained, and the complete counting functions on the 4-error linear complexity of $2^n$-periodic binary sequences are obvious.