On the fractal nature of the set of all binary sequences with almost perfect linear complexity profile

  • Harald Niederreiter
  • Michael Vielhaber
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT)


Stream ciphers usually employ some sort of pseudo-randomly generated bit strings to be added to the plaintext. The cryptographic properties of such binary sequences can be stated in terms of the so-called linear complexity profile. This paper shows that the set of all sequences with an almost perfect linear complexity profile maps onto a fractal subset of [0, 1].

The space \( \mathbb{F}_2^\infty \) of all infinite binary sequences can be mapped onto [0, 1] by \( \iota :\left( {{a_i}} \right)_{i = 1}^\infty \mapsto \sum\nolimits_{i = 1}^\infty {{a_i}} {2^{ - 1}} \). Any such sequence admits a linear complexity profile (l.c.p.) , stating for each n that the initial string (al, …, a n ) can be produced by an LFSR of length L n (but not L n − 1). Usually Ln/2, and so m(n):= 2 • L n n should vary around zero.

Let A d be the set of those sequences from \( \mathbb{F}_2^\infty \) whose l.c.p. is almost perfect in the sense of |m(n)| < dn (Niederreiter, 1988a). The subset of [0,1] obtained as ι(A d ) is fractal and its Hausdorff dimension is bounded from above by
$$ {D_H}\left( {\iota \left( {{A_d}} \right)} \right) \leqslant \frac{{1 + {{\log }_2}\varphi d}}{2} $$
where φ d is the positive real root of \( {x^d} = \sum\nolimits_{i = 0}^{d - 1} {{x^i}} \), e.g.φ1 = 1, φ2 = 1.618… (Fibonacci’s golden ratio). Thus, although all the A d have Haar measure zero in FT, a sharper distinction can be made by looking at their Hausdorff dimension. As a by-product the paper gives explicit formulae for the number of sequences of length n in A d , for all n and d.


Linear complexity Hausdorff dimension 


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Copyright information

© IFIP International Federation for Information Processing 1995

Authors and Affiliations

  • Harald Niederreiter
    • 1
  • Michael Vielhaber
    • 1
  1. 1.Institute for Information ProcessingAustrian Academy of SciencesViennaAustria

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