Periodic sequences with stable $k$-error linear complexity

TL;DR

This paper introduces the concept of stable $k$-error linear complexity for periodic sequences, using cube theory to construct sequences with maximum stability, and proves the maximum $k$-error linear complexity for binary sequences of period $2^n$.

Contribution

It presents a new framework called cube theory for analyzing and constructing sequences with high stable $k$-error linear complexity, and establishes the maximum $k$-error linear complexity for binary sequences with period $2^n$.

Findings

Introduces the concept of stable $k$-error linear complexity.

Uses cube theory to analyze sequences.

Proves maximum $k$-error linear complexity as $2^n-(2^l-1)$.

Abstract

The linear complexity of a sequence has been used as an important measure of keystream strength, hence designing a sequence which possesses high linear complexity and $k$-error linear complexity is a hot topic in cryptography and communication. Niederreiter first noticed many periodic sequences with high $k$-error linear complexity over GF(q). In this paper, the concept of stable $k$-error linear complexity is presented to study sequences with high $k$-error linear complexity. By studying linear complexity of binary sequences with period $2^n$, the method using cube theory to construct sequences with maximum stable $k$-error linear complexity is presented. It is proved that a binary sequence with period $2^n$ can be decomposed into some disjoint cubes. The cube theory is a new tool to study $k$-error linear complexity. Finally, it is proved that the maximum $k$-error linear complexity is $2^n-(2^l-1)$ over all $2^n$-periodic binary sequences, where $2^{l-1}\le k<2^{l}$.