Linear Complexity of Periodic Sequences: A General Theory

The linear complexity of an N -periodic sequence with components in a eld of characteristic p, where N = np and gcd(n; p) = 1, is characterized in terms of the n roots of unity and their multiplicities as zeroes of the polynomial whose coe cients are the rst N digits of the sequence. Hasse derivatives are then introduced to quantify these multiplicities and to de ne a new generalized discrete Fourier transform that can be applied to sequences of arbitrary length N with components in a eld 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 nding the linear complexity of an N -periodic binary sequence with N = 2 and to generalize this algorithm to apply to N -periodic sequences with components in a nite eld of characterisitic p when N = p . It is also shown how to use this new transform to study the linear complexity of Hadamard (i.e., component-wise) products of sequences.