Abstract
The well-known Kolmogorov complexity of binary sequences is a beautiful theoretical concept, but it is impractical since it is not computable. This paper will focus on complexity measures that are not only of theoretical interest, but that can also be obtained in an algorithmic manner, such as the linear complexity, the linear complexity profile, and the tree complexity. A survey of such complexity measures will be given, with an emphasis on recent results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S.R. Blackburn, A generalisation of the discrete Fourier transform: determining the minimal polynomial of a periodic sequence, IEEE Trans. Inform. Theory 40 (1994), 1702–1704.
S.R. Blackburn, A generalisation of the discrete Fourier transform, in “Applications of Finite Fields” (D. Gollmann, ed.), pp. 111–116, Oxford University Press, Oxford, 1996.
S.R. Blackburn, Fast rational interpolation, Reed-Solomon decoding, and the linear complexity profiles of sequences, IEEE Trans. Inform. Theory 43 (1997), 537–548.
C. Calude, “Information and Randomness: An Algorithmic Perspective”, Springer-Verlag, Berlin, 1994.
G. Christol, T. Kamae, M. Mendès France, and G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), 401–419.
T.W. Cusick, C. Ding, and A. Renvall, “Stream Ciphers and Number Theory”, Elsevier, Amsterdam, 1998.
Z. Dai and K. Zeng, Continued fractions and the Berlekamp-Massey algorithm, in “Advances in Cryptology — AUSCRYPT ‘80” (J. Seberry and J. Pieprzyk, eds.), Lecture Notes in Computer Science, Vol. 453, pp. 24–31, Springer-Verlag, Berlin, 1990.
M. Dekking, M. Mendès France, and A. van der Poorten, FOLDS!, The Math. Intelligencer 4 (1983), 130–138,173–181,190–195.
C. Ding, G. Xiao, and W. Shan, “The Stability Theory of Stream Ciphers”, Lecture Notes in Computer Science, Vol. 561, Springer-Verlag, Berlin, 1991.
J.-L. Dornstetter, On the equivalence between Berlekamp’s and Euclid’s algorithms, IEEE Trans. Inform. Theory 33 (1987), 428–431.
R.A. Games and A.H. Chan, A fast algorithm for determining the complexity of a binary sequence with period 272, IEEE Trans. Inform. Theory 29 (1983), 144–146.
C.J.A. Jansen, “Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods”, Ph.D. Thesis, Technical University of Delft, 1989.
C.J.A. Jansen, The maximum order complexity of sequence ensembles, in “Advances in Cryptology — EUROCRYPT ‘81” (D.W. Davies, ed.), Lecture Notes in Computer Science, Vol. 547, pp. 153–159, Springer-Verlag, Berlin, 1991.
C.J.A. Jansen and D.E. Boekee, The shortest feedback shift register that can generate a given sequence, in “Advances in Cryptology — CRYPTO ‘89” (G. Brassard, ed.), Lecture Notes in Computer Science, Vol. 435, pp. 90–99, Springer-Verlag, Berlin, 1990.
C.J.A. Jansen and D.E. Boekee, On the significance of the directed acyclic word graph in cryptology, in “Advances in Cryptology — AUSCRYPT ‘80” (J. Seberry and J. Pieprzyk, eds.), Lecture Notes in Computer Science, Vol. 453, pp. 318–326, Springer-Verlag, Berlin, 1990.
T. Kaida, S. Uehara, and K. Imamura, A new algorithm for the k -error linear complexity of sequences over GF(p m ) with period pn, this volume.
A. Klapper and M. Goresky, 2-adic shift registers, in “Fast Software Encryption” (R. Anderson, ed.), Lecture Notes in Computer Science, Vol. 809, pp. 174–178, Springer-Verlag, Berlin, 1994.
A. Klapper and M. Goresky, Feedback shift registers, 2-adic span, and combiners with memory, J. Cryptology 10 (1997), 111–147.
A. Lempel and J. Ziv, On the complexity of finite sequences, IEEE Trans. Inform. Theory 22 (1976), 75–81.
R. Lidl and H. Niederreiter, “Finite Fields”, Addison-Wesley, Reading, MA, 1983; reprint, Cambridge University Press, Cambridge, 1997.
R. Lidl and H. Niederreiter, Finite fields and their applications, in “Handbook of Algebra” (M. Hazewinkel, ed.), Vol. 1, pp. 321–363, Elsevier, Amsterdam, 1996.
W.H. Mills, Continued fractions and linear recurrences, Math. Comp. 29 (1975), 173–180.
S. Mund, Ziv-Lempel complexity for periodic sequences and its cryptographic application, in “Advances in Cryptology — EUROCRYPT ‘81” (D. W. Davies, ed.), Lecture Notes in Computer Science, Vol. 547, pp. 114–126, Springer-Verlag, Berlin, 1991.
H. Niederreiter, Sequences with almost perfect linear complexity profile, in “Advances in Cryptology — EUROCRYPT ‘87” (D. Chaum and W.L. Price, eds.), Lecture Notes in Computer Science, Vol. 304, pp. 37–51, Springer-Verlag, Berlin, 1988.
H. Niederreiter, The probabilistic theory of linear complexity, in “Advances in Cryptology — EUROCRYPT ‘88” (C.G. Günther, ed.), Lecture Notes in Computer Science, Vol. 330, pp. 191–209, Springer-Verlag, Berlin, 1988.
H. Niederreiter, Keystream sequences with a good linear complexity pro-file for every starting point, in “Advances in Cryptology — EUROCRYPT ‘89” (J.-J. Quisquater and J. Vandewalle, eds.), Lecture Notes in Computer Science, Vol. 434, pp. 523–532, Springer-Verlag, Berlin, 1990.
H. Niederreiter, Finite fields and cryptology, in “Finite Fields, Coding Theory, and Advances in Communications and Computing” (G.L. Mullen and P.J.-S. Shiue, eds.), pp. 359–373, M. Dekker, New York, 1993.
H. Niederreiter and H. Paschinger, Counting functions and expected values in the stability theory of stream ciphers, this volume.
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity 12 (1996), 187–198.
H. Niederreiter and M. Vielhaber, Simultaneous shifted continued fraction expansions in quadratic time, Applicable Algebra Engrg. Comm. Comput. 9 (1998), 125–138.
H. Niederreiter and M. Vielhaber, An algorithm for shifted continued fraction expansions in parallel linear time, Theoretical Computer Science,to appear.
F. Piper, Stream ciphers, Elektrotechnik and Maschinenbau 104 (1987), 564–568.
R.A. Rueppel, Stream ciphers, in “Contemporary Cryptology: The Science of Information Integrity” (G.J. Simmons, ed.), pp. 65–134, IEEE Press, New York, 1992.
M. Stamp and C.F. Martin, An algorithm for the k-error linear complexity of binary sequences with period 2n, IEEE Trans. Inform. Theory 39 (1993), 1398–1401.
L.R. Welch and R.A. Scholtz, Continued fractions and Berlekamp’s algorithm, IEEE Trans. Inform. Theory 25 (1979), 19–27.
J. Ziv and A. Lempel, A universal algorithm for sequential data compression, IEEE Trans. Inform. Theory 23 (1977), 337–343.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag London
About this paper
Cite this paper
Niederreiter, H. (1999). Some Computable Complexity Measures for Binary Sequences. In: Ding, C., Helleseth, T., Niederreiter, H. (eds) Sequences and their Applications. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0551-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0551-0_5
Publisher Name: Springer, London
Print ISBN: 978-1-85233-196-2
Online ISBN: 978-1-4471-0551-0
eBook Packages: Springer Book Archive