On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

Abstract

We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial f (x). We study in detail the cycle structure of the set Ω ( f (x)) that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order n ≈ 20. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.Keywords Binary periodic sequence · LFSR · de Bruijn sequence · cycle structure · adjacency graph · cyclotomic number Mathematics Subject Classification (2010) 11B50 · 94A55 · 94A60 1 Introduction A binary de Bruijn sequence of order n has period N = 2 n in which each n-tuple occurs exactly once in each period. There are 2 2 n−1 −n of them [5]. Some of their earliest applications are in communication systems. They are generated in a deterministic way, yet satisfy the randomness criteria in [14, Ch. 5] and are balanced, containing the same number of 1s and 0s. In cryptography, they have been used as a source of pseudo-random numbers and in keysequence generators of stream ciphers [29, Sect. 6.3]. In computational molecular biology, one of the three assembly paradigms in DNA sequencing is the de Bruijn graph assemblers model [31, Box 2]. Some roles of de Bruijn sequences in robust positioning patterns are discussed by Bruckstein et al. in [4]. They have numerous applications, e.g., in robotics,

Binary de Bruijn Sequences via Zech’s Logarithms

“…To generate de Bruijn sequences, especially when their orders are large, efficient identification of conjugate pairs is crucial. A de Bruijn sequence generator was proposed in [7]. Its basic software implementation [16] demonstrates a decent performance up to order n ≈ 20 when the characteristic polynomials are products of distinct irreducible polynomials.…”

“…Its bit-by-bit complexity was 3n bits of storage and at most 4n FSR shifts. Numerous subsequent works, until very recently, are listed with the details on their input parameters and performance complexity in [7,Table 4].…”

“…464 ff.] or an alternative algorithm [7,Algorithm 5], due to Chang et al, to generate all of the spanning trees. The two algorithms share the same complexity, as detailed in [7,Section 8].…”

“…or an alternative algorithm [7,Algorithm 5], due to Chang et al, to generate all of the spanning trees. The two algorithms share the same complexity, as detailed in [7,Section 8]. If so desired, one can quickly choose a spanning tree at random by running Broder's Algorithm [4].…”

“…The approach via Zech's logarithms can be used in tandem with the one in [7] to generate de Bruijn sequences of even larger orders. To keep the exposition brief, we retain the notations from the said reference.…”