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.

[1]  I. Good.  Normal Recurring Decimals , 1946 .

[2]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[3]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[4]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

[5]  K. R. Dimond,et al.  Nonlinear product-feedback shift registers , 1970 .

[6]  Abraham Lempel,et al.  On a Homomorphism of the de Bruijn Graph and its Applications to the Design of Feedback Shift Registers , 1970, IEEE Transactions on Computers.

[7]  D. H. Green,et al.  Some polynomial compositions of nonlinear feedback shift registers and their sequence-domain consequences , 1970 .

[8]  Harold Fredricksen,et al.  A Class of Nonlinear de Bruijn Cycles , 1975, J. Comb. Theory, Ser. A.

[9]  K. Kjeldsen,et al.  On the Cycle Structure of a Set of Nonlinear Shift Registers with Symmetric Feedback Functions , 1976, J. Comb. Theory, Ser. A.

[10]  Solomon W. Golomb.  On the classification of balanced binary sequences of period 2n-1 (Corresp.) , 1980, IEEE Trans. Inf. Theory.

[11]  Solomon W. Golomb,et al.  Shift Register Sequences , 1981 .

[12]  Richard A. Games,et al.  On the Complexities of de Bruijn Sequences , 1982, J. Comb. Theory, Ser. A.

[13]  H. Fredricksen.  A Survey of Full Length Nonlinear Shift Register Cycle Algorithms , 1982 .

[14]  Tuvi Etzion,et al.  Construction of de Bruijn sequences of minimal complexity , 1984, IEEE Trans. Inf. Theory.

[15]  Richard A. Games,et al.  On the quadratic spans of DeBruijn sequences , 1990, IEEE Trans. Inf. Theory.

[16]  Cees J. A. Jansen,et al.  An efficient algorithm for the generation of DeBruijn cycles , 1991, IEEE Trans. Inf. Theory.

[17]  S. Golomb,et al.  Characterizations of generators for modified de Bruijn sequences , 1992 .

[18]  Fred S. Annexstein.  Generating De Bruijn Sequences: An Efficient Implementation , 1997, IEEE Trans. Computers.

[19]  Iickho Song,et al.  An Efficient Implementation of the D-Homomorphism for Generation of de Bruijn Sequences , 1999, IEEE Trans. Inf. Theory.

[20]  Amr M. Youssef,et al.  Cryptographic properties of the Welch-Gong transformation sequence generators , 2002, IEEE Trans. Inf. Theory.

[21]  Johanns MYmCELTWIT,et al.  On the Cycle Structure of Some Nonlinear Shift Register Sequences , 2004 .

[22]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[23]  Guang Gong.  Randomness and Representation of Span n Sequences , 2007, SSC.

[24]  Janusz Szmidt,et al.  Generation of nonlinear feedback shift registers with special-purpose hardware , 2012, 2012 Military Communications and Information Systems Conference (MCC).

[25]  Guang Gong,et al.  Cryptographically Strong de Bruijn Sequences with Large Periods , 2012, Selected Areas in Cryptography.

[26]  Guang Gong,et al.  Communication System Security , 2012 .

[27]  K. Mandal,et al.  Probabilistic Generation of Good Span n Sequences from Nonlinear Feedback Shift Registers , 2012 .