Feedback Reconstruction and Implementations of Pseudorandom Number Generators from Composited De Bruijn Sequences

A binary de Bruijn sequence of order $n$ is a sequence of zeros and ones of period $2^n$ that contains every binary $n$-tuple exactly once in a period of the sequence. A composited construction of a de Bruijn sequence is a construction of a nonlinear feedback shift register (NLFSR) that generates a de Bruijn sequence where the composited feedback function of the NLFSR is the sum of a feedback function with $k$th order composition and a sum of $(k+1)$ product-of-sum terms. The goals of this article are to perform a profound analysis of composited de Bruijn sequences for use in cryptography and find an efficient implementation of the composited feedback function. We first determine the lower bound of the linear complexity of a composited de Bruijn sequence and then conduct a profound analysis on the composited construction by introducing the notion of the higher order $D$-morphic preimages of a binary sequence. Our analysis aims at the reconstruction of a composited de Bruijn sequence from a segment known as $k$th order $D$-morphic order $n$ de Bruijn preimages ( $(n,k)$-DMDPs) of length $(2^n+k)$ and $k$th order $D$-morphic order $n$ $m$-sequence preimages ( $(n,k)$-DMMPs) of length $(2n+k)$ for a nonlinearly and linearly generated composited de Bruijn sequence, respectively. We also provide the success probability of finding an $(n,k)$-DMMP/DMDP from a composited de Bruijn sequence for the reconstruction. Furthermore, we develop a new iterative technique with its parallel extension for computing the feedback function and the new technique is faster than other known techniques for producing de Bruijn sequences of long period. In addition, we present three instances of composited de Bruijn sequences of period $2^{64}$ together with their software implementations and performances.