Abstract
A binary span n sequence generated by an n-stage nonlinear feedback shift register (NLFSR) is in a one-to-one correspondence with a de Bruijn sequence and has the following randomness properties: period 2n − 1, balance, and ideal n-tuple distribution. A span n sequence may have a high linear span. However, how to find a nonlinear feedback function that generates such a sequence constitutes a long-standing challenging problem for about 5 decades since Golomb’s pioneering book, Shift Register Sequences, published in the middle of the 1960s. In hopes of finding good span n sequences with large linear span, in this chapter we study the generation of span n sequences using orthogonal functions in polynomial representation as nonlinear feedback in a nonlinear feedback shift register. Our empirical study shows that the success probability of obtaining a span n sequence in this technique is better than that of obtaining a span n sequence in a random span n sequence generation method. Moreover, we analyze the linear span of new span n sequences, and the linear span of a new sequence lies between 2n − 2 − 3n (near optimal) and 2n − 2 (optimal). Two conjectures on the linear span of new sequences are presented and are valid for n ≤ 20.
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References
F.S. Annexstein, Generating de Bruijn sequences: an efficient implementation. IEEE Trans. Inf. Theory 46(2), 198–200 (1997)
E.R. Berlekamp, Algebraic Coding Theory, Ch. 7 (McGraw-Hill, New York, 1968)
A.H. Chan, R.A. Games, E.L. Key, On the complexities of de Bruijn sequences. J. Combin. Theory Ser. A 33(3) 233–246 (1982)
A.H. Chan, R.A. Games, J.J. Rushanan, On quadratic m-sequences, in IEEE International Symposium on Information Theory, vol. 364 (1994)
A.C. Chang, S.W. Golomb, G. Gong, P.V. Kumar, On the linear span of ideal autocorrelation sequences arising from the Segre hyperoval, in Sequences and their Applications—Proceedings of SETA’98, Discrete Mathematics and Theoretical Computer Science (Springer, London, 1999)
T. Chang, B. Park, Y.H. Kim, I. Song, An efficient implementation of the D-homomorphism for generation of de Bruijn sequences. IEEE Trans. Inf. Theory 45(4), 1280–1283 (1999)
C. De Canniére, O. Dunkelman, M. Knez̀ević, KATAN and KTANTAN—a family of small and efficient hardware-oriented block ciphers. in Proceedings of the 11th International Workshop on Cryptographic Hardware and Embedded Systems, LNCS, vol. 5747 (Springer, Heidelberg, 2009). pp. 272–288
J. Dillon, H. Dobbertin, New cyclic difference sets with singer parameters. Finite Fields Appl. 10(3), 342–389 (2004)
H. Dobbertin, Kasami power functions, permutation polynomials and cyclic difference sets, in Proceedings of the NATO-A.S.I. Workshop Difference Sets, Sequences and their Correlation Properties, (Kluwer, Bad Windsheim/Dordrecht, 1999), pp. 133–158
E. Dubrova, A list of maximum period NLFSRs. Report 2012/166, Cryptology ePrint Archive (2012), http://eprint.iacr.org/2012/166.pdf
eSTREAM: The ECRYPT stream cipher project. http://www.ecrypt.eu.org/stream/
T. Etzion, A. Lempel, Construction of de Bruijn sequences of minimal complexity. IEEE Trans. Inf. Theory 30(5), 705–709 (1984)
R. Evan, H.D.L. Hollman, C. Krattenthaler, Q. Xiang, Gauss sums, Jacobi sums and p-ranks of cyclic difference sets. J. Combin. Theory Ser. A, 87(1), 74–119 (1999)
H. Fredricksen, A class of nonlinear de Bruijn cycles. J. Combin. Theory Ser. A 19(2), 192–199 (1975)
H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24(2), 195–221 (1982)
H. Fredricksen, I. Kessler, Lexicographic compositions and de Bruijn sequences. J. Combin. Theory Ser. A 22, 17–30 (1977)
H. Fredricksen, J. Maiorana, Necklaces of beads in k colors and k-ary de Bruijn sequences. Discrete Math. 23(3), 207–210 (1978)
R.A. Games, A generalized recursive construction for de Bruijn sequences. IEEE Trans. Inf. Theory 29(6), 843–850 (1983)
B.M. Gammel, R. Göttfert, O. Kniffler, Achterbahn-128/80 (2006), http://www.ecrypt.eu.org/stream/p2ciphers/achterbahn/achterbahn_p2.pdf
S.W. Golomb, Shift Register Sequences (Aegean Park Press, Laguna Hills, 1981)
S.W. Golomb, On the classification of balanced binary sequences of period 2n − 1. IEEE Trans. Inf. Theory, 26(6), 730–732 (1980)
S.W. Golomb, G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography, and Radar (Cambridge University Press, New York, 2004)
G. Gong, Randomness and representation of span n sequences, in Proceedings of the 2007 International Conference on Sequences, Subsequences, and Consequences, SSC’07 (Springer, Heidelberg, 2007), pp. 192–203
E.R. Hauge, T. Helleseth, De Bruijn sequences, irreducible codes and cyclotomy. Discrete Math. 159(1–3), 143–154 (1996)
C.J.A. Jansen, W.G. Franx, D.E. Boekee, An efficient algorithm for the generation of de Bruijn cycles. IEEE Trans. Inf. Theory 37(5), 1475–1478 (1991)
A. Lempel, On a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registers. IEEE Trans. Comput. C-19(12), 1204–1209 (1970)
K. Mandal, Design and analysis of cryptographic pseudorandom number/sequence generators with applications in RFID. Ph.D. Thesis, University of Waterloo, 2013
K. Mandal, G. Gong, in Cryptographically Strong de Bruijn Sequences with Large Periods, ed. by L.R. Knudsen, H. Wu SAC 2012. LNCS, vol. 7707 (Springer, Heidelberg, 2012), pp. 104–118
K. Mandal, G. Gong, Cryptographic D-morphic analysis and fast implementations of composited De Bruijn sequences. Technical Report CACR 2012–27, University of Waterloo (2012)
K. Mandal, X. Fan, G. Gong, in Warbler: A Lightweight Pseudorandom Number Generator for EPC Class 1 Gen 2 RFID Tags, ed. by N.W. Lo, Y. Li. Cryptology and Information Security Series—The 2012 Workshop on RFID and IoT Security (RFIDsec’12 Asia), vol. 8 (IOS Press, Amsterdam, 2012), pp. 73–84
J.L. Massey, Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969)
G.L. Mayhew, Weight class distributions of de Bruijn sequences. Discrete Math. 126, 425–429 (1994)
G.L. Mayhew, Clues to the hidden nature of de Bruijn sequences. Comput. Math. Appl., 39(11), 57–65 (2000)
G.L. Mayhew, S.W. Golomb, Linear Spans of modified de Bruijn sequences. IEEE Trans. Inf. Theory 36(5), 1166–1167 (1990)
G.L. Mayhew, S.W. Golomb, Characterizations of generators for modified de Bruijn sequences. Adv. Appl. Math. 13, 454–461 (1992)
J. Mykkeltveit, M.-K. Siu, P. Tong, On the cycle structure of some nonlinear shift register sequences. Inf. Control 43(2), 202–215 (1979)
J.L.-F. Ng, Binary nonlinear feedback shift register sequence generator using the trace function, Master’s Thesis, University of Waterloo, 2005
J.S. No, S.W. Golomb, G. Gong, H.K. Lee, P. Gaal, New binary pseudorandom sequences of period 2n − 1 with ideal autocorrelation. IEEE Trans. Inf. Theory 44(2), 814–817 (1998)
T. Rachwalik, J. Szmidt, R. Wicik, J. Zablocki, Generation of nonlinear feedback shift registers with special-purpose hardware. Cryptology ePrint Archive, Report 2012/314 (2012), http://eprint.iacr.org/
J.-H. Yang, Z.-D. Dai, Construction of m-ary de Bruijn sequences (extended abstract), in Advances in Cryptology—AUSCRYPT’92, LNCS (Springer, Heidelberg, 1993), pp. 357–363
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Mandal, K., Gong, G. (2014). Generating Good Span n Sequences Using Orthogonal Functions in Nonlinear Feedback Shift Registers. In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_7
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