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[1] 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.Now, the widely used definition of linear complexity stated above is open to different interpretations. We distinguish here between the periodic linear complexity (PLC) the length of the shortest LFSR which generates the given sequence and then repeats it cyclically - and the aperiodic linear complexity (ALC) - the length of the shortest LFSR which generates the given sequence followed by any arbitrary sequence of bits. This distinction is not made in the literature on de Bruijn sequences, but it has important practical consequences. In a stream cipher, it is clearly undesirable for keystream sequences to be allowed to repeat. Consequently, no more than P bits (where P is the sequence period) will ever be used, which implies that it is the ALC, not the PLC, which is of red concern. Unfortunately, all of the published results on the linear compiexity of de Bruijn sequences (e.g [2] and [3]) relate only to the PLC not the ALC. The research described here goes some way towards addressing this imbalance.