Aperiodic linear complexities of de bruijn sequences

Abstract

Binary de Bruijn sequences of period 2n bits have the property that all 2n distinct n-tuples occur once per period. To generate such a sequence with an n-stage shift-register requires the use of nonlinear feedback. These properties suggest that de Bruijn sequences may be useful in stream ciphers. However, any binary sequence can be generated using a linear-feedback shift register (LFSR) of sufficient length. Thus, the linear complexity of a sequence, defined as the length of the shortest LFSR which generates it, is often used as a measure of the unpredictability of the sequence. This is a useful measure, since a well-known algorithm[l] can be used to successfully predict all bits of any sequence with linear complexity C from a knowledge of 2C bits. As an example, an m-sequence of period 2n-1 has linear complexity C=n, which clearly indicates that m-sequences are highly predictable.