Abstract
In the theory of stream ciphers, an important complexity measure to assess the (pseudo-)randomness of a stream generator is the linear complexity, essentially the complexity to approximate the sequence (seen as formal power series) by rational functions.
For multisequences with several, i.e. M, streams in parallel (e.g. for broadband applications), simultaneous approximation is considered.
This paper improves on previous results by Niederreiter and Wang, who have given an algorithm to calculate the distribution of linear complexities for multisequences, obtaining formulae for M = 2 and 3. Here, we give a closed formula numerically verified for M up to 8 and for M = 16, and conjectured to be valid for all M ∈ ℕ.
We model the development of the linear complexity of multisequences by a stochastic infinite state machine, the Battery–Discharge–Model, and we obtain the asymptotic probability for the linear complexity deviation d(n) : = L(n) − ⌈n·M/(M + 1)⌉ for M sequences in parallel as
The precise formula is given in the text.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Canales Chacón, M. del P., Vielhaber, M.: Structural and Computational Complexity of Isometries and their Shift Commutators. Electronic Colloq. on Computational Complexity, ECCC TR04–057 (2004)
Dai, Z., Feng, X., Yang, J.: Multi-continued Fraction Algorithm and Generalized B-M Algorithm over F 2. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 339–354. Springer, Heidelberg (2005)
Niederreiter, H., Wang, L.–P.: The Asymptotic Behavior of the Joint Linear Complexity Profile of Multisequences. Monatshefte für Mathematik 150, 141–155 (2007)
Rosenblatt, M.: Random Processes. Springer (1974)
Shoup, V.: The Number Theory Library NTL, http://shoup.net/ntl
Sloane, N.J.A.: Online Encyclopedia of Integer Sequences, http://oeis.org
Vielhaber, M., Canales Chacón, M. del P.: The Battery–Discharge–Model: A Class of Stochastic Finite Automata to Simulate Multidimensional Continued Fraction Expansion, published at: arXiv.org/abs/0705.4134
Vielhaber, M., Canales Chacón, M. del P.: Towards a General Theory of Simultaneous Diophantine Approximation of Formal Power Series: Linear Complexity of Multisequences, arXiv.org/abs/cs.IT/0607030
Vielhaber, M.: A Unified View on Sequence Complexity Measures as Isometries. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 143–153. Springer, Heidelberg (2005)
Wang, L.–P., Niederreiter, H.: Enumeration results on the joint linear complexity of multisequences. Finite Fields Appl. 12, 613–637 (2006)
Rosen, K.H. (ed.): Handbook of Discrete and Combinatorial Mathematics. CRC, Boca Raton (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Vielhaber, M., del Pilar Canales Chacón, M. (2012). The Linear Complexity Deviation of Multisequences: Formulae for Finite Lengths and Asymptotic Distributions. In: Helleseth, T., Jedwab, J. (eds) Sequences and Their Applications – SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol 7280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30615-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-30615-0_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30614-3
Online ISBN: 978-3-642-30615-0
eBook Packages: Computer ScienceComputer Science (R0)