Linear complexity of periodic sequences: A general theory

Abstract

The linear complexity of an N-periodic sequence with components in a field of characteristic p, where N = np ν and gcd(n, p) = 1, is characterized in terms of the n th roots of unity and their multiplicities as zeroes of the polynomial whose cofficients are the first N digits of the sequence. Hasse derivatives are then introduced to quantify these multiplicities and to define a new generalized discrete Fourier transform that can be applied to sequences of arbitrary lengthN with components in a field of characteristic p, regardless of whether or not gcd(N, p) = 1. This generalized discrete Fourier transform is used to give a simple proof of the validity of the well-known Games-Chan algorithm for finding the linear complexity of an N-periodic binary sequence with N = 2ν and to generalize this algorithm to apply to N-periodic sequences with components in a finite field of characteristic p when N = p ν. It is also shown how to use this new transform to study the linear complexity of Hadamard (i.e., component-wise) products of sequences.