Showing papers by "Harald Niederreiter published in 2003"
08 Dec 2003
TL;DR: A survey of recent work on the linear complexity, thelinear complexity profile, and the k-error linear complexity of sequences and on the joint linear complex of multisequences and a new enumeration theorem on multiseaquences are presented.
Abstract: We present a survey of recent work on the linear complexity, the linear complexity profile, and the k-error linear complexity of sequences and on the joint linear complexity of multisequences. We also establish a new enumeration theorem on multisequences and state several open problems. The material is of relevance for the assessment of keystreams in stream ciphers.
99 citations
TL;DR: It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα and discrepancy are all small.
91 citations
TL;DR: A well-known relationship between the linear complexity of an N-periodic sequence and the (generalized) discrete Fourier transform of N- tuples is extended to the case of multisequences and the expected value of the joint linear complexity is computed.
43 citations
TL;DR: In this paper, the authors consider dynamical systems generated by iterations of rational functions over finite fields and residue class rings and present a survey of recent developments and outline several open problem.
Abstract: We consider dynamical systems generated by iterations of rational functions over finite fields and residue class rings We present a survey of recent developments and outline several open problem
35 citations
TL;DR: New error bounds for quasi-Monte Carlo integration for node sets with a special kind of uniformity property are established and proved for arbitrary probability spaces with structure of a metric space.
34 citations
TL;DR: The existence of periodic sequences over a finite field which simultaneously achieve the maximum value (for the given period length) of the linear complexity and of the k-error linear complexity for small values of k is established.
Abstract: We establish the existence of periodic sequences over a finite field which simultaneously achieve the maximum value (for the given period length) of the linear complexity and of the k-error linear complexity for small values of k. This disproves a conjecture of Ding, Xiao, and Shan (1991). The result is of relevance for the theory of stream ciphers.
32 citations
01 Jan 2003
TL;DR: A survey of recent developments in dynamical systems generated by iterations of rational functions over finite fields and residue class rings is presented and several open problem are outlined.
Abstract: We consider dynamical systems generated by iterations of rational functions over finite fields and residue class rings. We present a survey of recent developments and outline several open problem.
32 citations
TL;DR: In this paper, it was shown that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomials such that for all these infinitely many moduli and all dimensions s the quantity R(s) and the star discrepancy are small.
Abstract: Extensible (polynomial) lattice rules have been introduced recently and they are convenient tools for quasi-Monte Carlo integration. It is shown in this paper that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomial lattice rules such that for all these infinitely many moduli and all dimensions s the quantity R(s) and the star discrepancy are small. The case of Korobov-type polynomial lattice rules is also considered.
26 citations
TL;DR: The following issues in quasi-Monte Carlo methods are discussed: error bounds and error reduction, optimization of net constructions, and randomization and derandomization.
17 citations
TL;DR: Conditions under which there are many periodic sequences over an arbitrary finite field 𝔽q with period N, maximal linear complexity N, and k-error linear complexity close to N are established.
Abstract: The linear complexity and the k-error linear complexity are important concepts for the theory of stream ciphers in cryptology. Keystreams that are suitable for stream ciphers must have large values of these complexity measures. We study periodic sequences over an arbitrary finite field 𝔽q and establish conditions under which there are many periodic sequences over 𝔽q with period N, maximal linear complexity N, and k-error linear complexity close to N. The existence of many such sequences thwarts attacks against the keystreams by exhaustive search.
17 citations
TL;DR: Some recent achievements of analytic number theory are used to extend the class of period lengths N and the number of admissible errors k for which this conjecture fails for rather large values of k and discuss the relevance of this result for stream ciphers.
Abstract: C. Ding, W. Shan and G. Xiao conjectured a certain kind of trade-off between the linear complexity and the k-error linear complexity of periodic sequences over a finite field. This conjecture has recently been disproved by the first author, by showing that for infinitely many period lengths N and some values of k both complexities may take very large values (contradicting the above conjecture). Here we use some recent achievements of analytic number theory to extend the class of period lengths N and the number of admissible errors k for which this conjecture fails for rather large values of k. We also discuss the relevance of this result for stream ciphers.
TL;DR: In this article, the distribution of points in an orbit of PGL(2,q) acting on an element of GF(q^n) was shown to be uniformly distributed if n is small with respect to q.