Date: 06 Jul 2001

The Shortest Feedback Shift Register That Can Generate A Given Sequence

Abstract

In this paper the problem of finding the absolutely shortest (possibly nonlin- ear) feedback shift register, which can generate a given sequence with characters from some arbitrary finite alphabet, is considered. To this end, a new complex- ity measure is defined, called the maximum order complexity. A new theory of the nonlinear feedback shift register is developed, concerning elementary complexity properties of transposed and reciprocal sequences, and feedback functions of the maximum order feedback shift register equivalent. Moreover, Blumer’s algorithm is identified as a powerful tool for determining the maxi- mum order complexity profile of sequences, as well as their period, in linear time and memory. The typical behaviour of the maximum order complexity profile is shown and the consequences for the analysis of given sequences and the synthesis of feedback shift registers are discussed.