Efficient Algorithm for the Linear Complexity of Sequences and Some Related Consequences

TL;DR

This paper introduces a simplified, efficient algorithm for calculating the linear complexity of sequences over finite fields, generalizing the Games-Chan Algorithm and analyzing its performance for various sequence families.

Contribution

It generalizes the Games-Chan Algorithm for broader sequence families and provides a detailed analysis of its efficiency and related properties.

Findings

Algorithm requires βN bit operations, with β being a small constant.

The algorithm also determines the shortest linear feedback shift-register for the sequence.

Analysis shows improved efficiency over previous algorithms.

Abstract

The linear complexity of a sequence $s$ is one of the measures of its predictability. It represents the smallest degree of a linear recursion which the sequence satisfies. There are several algorithms to find the linear complexity of a periodic sequence $s$ of length $N$ (where $N$ is of some given form) over a finite field $F_q$ in $O(N)$ symbol field operations. The first such algorithm is The Games-Chan Algorithm which considers binary sequences of period $2^n$, and is known for its extreme simplicity. We generalize this algorithm and apply it efficiently for several families of binary sequences. Our algorithm is very simple, it requires $\beta N$ bit operations for a small constant $\beta$, where $N$ is the period of the sequence. We make an analysis on the number of bit operations required by the algorithm and compare it with previous algorithms. In the process, the algorithm also finds the recursion for the shortest linear feedback shift-register which generates the sequence. Some other interesting properties related to shift-register sequences, which might not be too surprising but generally unnoted, are also consequences of our exposition.