Reducing the Computation of Linear Complexities of Periodic Sequences over $GF(p^m)$

TL;DR

This paper presents a method to simplify the calculation of linear complexities of periodic sequences over finite fields by reducing it to smaller, more manageable problems, under specific mathematical conditions.

Contribution

It introduces a reduction technique that simplifies the computation of linear complexities and minimal connection polynomials for sequences over GF(p^m).

Findings

Reduction of linear complexity computation to smaller sequences.

Conditions $u|p^m-1$ and $ ext{gcd}(n,p^m-1)=1$ are essential.

Applications include faster algorithms for sequence analysis.

Abstract

The linear complexity of a periodic sequence over $GF(p^m)$ plays an important role in cryptography and communication [12]. In this correspondence, we prove a result which reduces the computation of the linear complexity and minimal connection polynomial of a period $un$ sequence over $GF(p^m)$ to the computation of the linear complexities and minimal connection polynomials of $u$ period $n$ sequences. The conditions $u|p^m-1$ and $\gcd(n,p^m-1)=1$ are required for the result to hold. Some applications of this reduction in fast algorithms to determine the linear complexities and minimal connection polynomials of sequences over $GF(p^m)$ are presented.