[Paper Review] On the $k$-error linear complexity for $2^n$-periodic binary sequences via Cube Theory

Jianqin Zhou, Wanquan Liu|arXiv (Cornell University)|Sep 7, 2013

Coding theory and cryptography20 references3 citations

TL;DR

This paper introduces cube theory to analyze the $k$-error linear complexity of $2^n$-periodic binary sequences, enabling the construction of sequences with maximum stable $k$-error linear complexity. By decomposing sequences into disjoint cubes and deriving counting formulas, it establishes that the maximum $k$-error linear complexity is $2^n - (2^l - 1)$ when $2^{l-1} \leq k < 2^l$, and provides a framework for characterizing multi-cube sequences with fixed linear complexities.

ABSTRACT

The linear complexity and k-error linear complexity of a sequence have been used as important measures of keystream strength, hence designing a sequence with high linear complexity and $k$-error linear complexity is a popular research topic in cryptography. In this paper, the concept of stable $k$-error linear complexity is proposed to study sequences with stable and large $k$-error linear complexity. In order to study k-error linear complexity of binary sequences with period $2^n$, a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable $k$-error linear complexity. For such purpose, we first prove that a binary sequence with period $2^n$ can be decomposed into some disjoint cubes and further give a general decomposition approach. Second, it is proved that the maximum $k$-error linear complexity is $2^n-(2^l-1)$ over all $2^n$-periodic binary sequences, where $2^{l-1}\le k&lt;2^{l}$. Thirdly, a characterization is presented about the $t$th ($t&gt;1$) decrease in the $k$-error linear complexity for a $2^n$-periodic binary sequence $s$ and this is a continuation of Kurosawa et al. recent work for the first decrease of k-error linear complexity. Finally, A counting formula for $m$-cubes with the same linear complexity is derived, which is equivalent to the counting formula for $k$-error vectors. The counting formula of $2^n$-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al..

Motivation & Objective

• To address the instability of linear complexity under small perturbations by introducing the concept of stable $k$-error linear complexity.
• To develop a new analytical tool—cube theory—for studying $k$-error linear complexity in $2^n$-periodic binary sequences.
• To characterize sequences with multiple independent cubes and fixed linear complexities, extending prior work on sequences with only two $k$-error complexity values.
• To derive exact counting formulas for sequences decomposed into multiple cubes, generalizing results from Etzion et al.

Proposed method

• Proposes a decomposition of $2^n$-periodic binary sequences into disjoint cubes, enabling independent analysis of each cube's contribution to the overall linear complexity.
• Establishes a standard cube decomposition approach that ensures cubes are relatively independent, allowing modular construction of sequences.
• Uses cube theory to derive a counting formula for $m$-cubes with identical linear complexity, equivalent to counting $k$-error vectors.
• Applies the theory to characterize the $t$-th ($t>1$) decrease in $k$-error linear complexity, extending prior work on the first decrease.
• Introduces conditions for independent cube construction, such as $2^{j_1} > 2^t$, ensuring no interference between cubes during complexity computation.
• Derives explicit formulas for the number of sequences with $t=3$ independent cubes, each having a specified linear complexity.

Experimental results

Research questions

• RQ1 What is the maximum possible stable $k$-error linear complexity for $2^n$-periodic binary sequences, and how can it be achieved?
• RQ2 How can $2^n$-periodic binary sequences be systematically decomposed into disjoint cubes to facilitate analysis of $k$-error linear complexity?
• RQ3 What is the counting formula for sequences with $m$-cubes of equal linear complexity, and how does it relate to $k$-error vectors?
• RQ4 How can the $t$-th decrease in $k$-error linear complexity be characterized for $2^n$-periodic binary sequences, especially for $t>1$?
• RQ5 What conditions ensure that multiple cubes in a sequence can be constructed independently without affecting their individual linear complexities?

Key findings

• The maximum $k$-error linear complexity for $2^n$-periodic binary sequences is $2^n - (2^l - 1)$, where $2^{l-1} \leq k < 2^l$, which is the highest possible value under $k$-error perturbations.
• A general decomposition method is provided to split any $2^n$-periodic binary sequence into disjoint cubes, enabling modular analysis of linear complexity.
• For $t=3$, the number of $2^n$-periodic binary sequences with three independent cubes of specified linear complexities is given by a closed-form formula involving powers of two and Hamming weight conditions.
• The counting formula for $m$-cubes with the same linear complexity is derived and shown to be equivalent to the number of $k$-error vectors.
• The paper extends Etzion et al.'s result on sequences with only two $k$-error complexity values (either $L(s)$ or 0) to sequences with multiple cubes and fixed, non-zero linear complexities.
• The theory enables the construction of sequences with maximum stable $k$-error linear complexity by ensuring that no small number of changes can reduce the linear complexity below the original value.