The Linear Complexity Deviation of Multisequences: Formulae for Finite Lengths and Asymptotic Distributions

  • Michael Vielhaber
  • Mónica del Pilar Canales Chacón
Conference paper

DOI: 10.1007/978-3-642-30615-0_16

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)
Cite this paper as:
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

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 deviationd(n) : = L(n) − ⌈n·M/(M + 1)⌉ for M sequences in parallel as
$${prob}(d(n)=d)=\Theta\left(q^{-|d|(M+1)}\right),\forall M\in \mathbb N, \forall d\in\mathbb Z, \forall q=p^k.$$
The precise formula is given in the text.

Keywords

Linear complexity multisequence Battery Discharge Model 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael Vielhaber
    • 1
    • 2
  • Mónica del Pilar Canales Chacón
    • 1
    • 2
  1. 1.Hochschule Bremerhaven, FB2BremerhavenGermany
  2. 2.Instituto de MatemáticasUniversidad Austral de ChileValdiviaChile

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