Cryptographic D-morphic Analysis and Fast Implementations of Composited De Bruijn Sequences

Recently, Mandal and Gong [23] refined and analyzed the recursive method by Lempel and Mykkeltveit et al. for generating de Bruijn sequences, where the recursive feedback function is the sum of a feedback function with k-th order composition and a sum of (k + 1) product-of-sum terms. In this paper we first determine the linear complexity of a composited de Bruijn sequence. We then conduct a profound analysis of the recursive construction by introducing the notion of the higher order D-morphism of a binary sequence. In the analysis, we consider both linearly and nonlinearly generated composited de Bruijn sequences and calculate the success probability of obtaining a k-th order D-morphic order n de Bruijn preimages ((n, k)-DMDP) of length (2 +k) and a k-th order D-morphic order n m-sequence preimages ((n, k)-DMMP) of length (2n + k) as one of (n, k)-DMMP and (n, k)-DMDP allows one to construct the starting de Bruijn sequence and to recover the feedback function. Moreover, we investigate the hardness of producing the whole composited de Bruijn sequence from a known (n, k)-DMDP of the composited de Bruijn sequence. Furthermore, we present a new iterative technique with its parallel extension for computing the product-of-sum terms of the feedback function where a productof-sum term is calculated in an iteration. In addition, we present three de Bruijn sequences of period 2 together with their software implementations and performances.