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

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

## Abstract

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.

## Keywords

Linear complexity Hausdorff dimension

## References

1. Dai, Z.-D. (1989) Continued fractions and the Berlekamp–Massey algorithm, E.I.S.S. Report # 89/7, Europäisches Institut für Systemsicherheit, Karlsruhe. Lidl, R. and Niederreiter, H. (1994) Introduction to Finite Fields and Their Applications, revised ed., Cambridge University Press, Cambridge. Niederreiter, H. (1988a) Sequences with almost perfect linear complexity profile, in Advances in Cryptology–EUROCRYPT ‘87 (eds. D. Chaum, W.L. Price), LNCS 304, 37–51, Springer, Berlin.Google Scholar
2. Niederreiter, H. (1988b) The probabilistic theory of linear complexity, in Advances in Cryptology–EUROCRYPT ‘88 (ed. C.G. Günther), LNCS 330, 191–209, Springer, Berlin.Google Scholar
3. Peitgen, H.O., Jürgens, H. and Saupe, D. (1992) Chaos and Fractals — New Frontiers of Science, Springer, New York, Berlin.
4. Rueppel, R.A. (1986) Analysis and Design of Stream Ciphers, Springer, Berlin.

© 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