Showing papers by "Harald Niederreiter published in 1994"
01 Jul 1994
TL;DR: Finite fields play a fundamental role in some of the most fascinating applications of modern algebra to the real world as mentioned in this paper, such as the construction of projective planes with a finite number of points and lines.
Abstract: Finite fields play a fundamental role in some of the most fascinating applications of modern algebra to the real world. These applications occur in the general area of data communication, a vital concern in our information society. Technological breakthroughs like space and satellite communications and mundane matters like guarding the privacy of information in data banks all depend in one way or another on the use of finite fields. Because of the importance of these applications to communication and information theory, we will present them in greater detail in the following chapters. Chapter 8 discusses applications of finite fields to coding theory, the science of reliable transmission of messages, and Chapter 9 deals with applications to cryptology, the art of enciphering and deciphering secret messages. This chapter is devoted to applications of finite fields within mathematics. These applications are indeed numerous, so we can only offer a selection of possible topics. Section 1 contains some results on the use of finite fields in affine and projective geometry and illustrates in particular their role in the construction of projective planes with a finite number of points and lines. Section 2 on combinatorics demonstrates the variety of applications of finite fields to this subject and points out their usefulness in problems of design of statistical experiments. In Section 3 we give the definition of a linear modular system and show how finite fields are involved in this theory.
162 citations
TL;DR: This note points out programs to implement Niederreiter's low-discrepancy sequences.
Abstract: This note points out programs to implement Niederreiter's low-discrepancy sequences.
60 citations
TL;DR: These results demonstrate that this explicit inversive congruential method recently introduced by Eichenauer-Herrmann is eminently suitable for the generation of parallel streams of pseudorandom numbers with desirable properties.
38 citations
TL;DR: In this paper, the authors apply two-dimensional lattice rules to continuous functions over the unit square which do not have a continuous periodic extension, and show that certain functions are integrated exactly whenever the lattice contains a (possibly rotated) square unit cell.
29 citations
01 Jan 1994
TL;DR: Based on theoretical results discussed by Bratley et al. as discussed by the authors, as well as on empirical comparisons, they believe that Niederreiter's sequences supersede earlier methods due to Faure and to Sobol' -implemented by Fox and Fox [1986] and BRatley and Fox[ 1988], respectively.
Abstract: Bratley et al. [1992] describe an algorithm to generate Niederreiter’s low-discrepancy sequences. Among other things, these sequences are useful for numerical integration in certain fixed dimensions. For further information and background, see Niederreiter [1992]. Based on theoretical results discussed by Bratley et al. [1992], as well as on empirical comparisons, we believe that Niederreiter’s sequences supersede earlier methods due to Faure and to Sobol’—implemented by Fox [1986] and Bratley and Fox [ 1988], respectively.
28 citations
TL;DR: This method can be viewed as an analog of the inversive congruential method for pseudorandom number generation and study, in particular, the periodicity properties and the behavior under the serial test for sequences of Pseudorandom vectors generated by theinversive method.
Abstract: Pseudorandom vectors are of importance for parallelized simulation methods. In this article we carry out a detailed analysis of the inversive method for the generation of uniform pseudorandom vectors. This method can be viewed as an analog of the inversive congruential method for pseudorandom number generation. We study, in particular, the periodicity properties and the behavior under the serial test for sequences of pseudorandom vectors generated by the inversive method.
27 citations
TL;DR: Digital inversive pseudorandom numbers satisfy statistical independence properties that are close to those of truly random numbers in the sense of asymptotic discrepancy.
Abstract: A new algorithm, the digital inversive method, for generating uniform pseudorandom numbers is introduced. This algorithm starts from an inversive recursion in a large finite field and derives pseudorandom numbers from it by the digital method. If the underlying finite field has q elements, then the sequences of digital inversive pseudorandom numbers with maximum possible period length q can be characterized. Sequences of multiprecision pseudorandom numbers with very large period lengths are easily obtained by this new method. Digital inversive pseudorandom numbers satisfy statistical independence properties that are close to those of truly random numbers in the sense of asymptotic discrepancy. If q is a power of 2, then the digital inversive method can be implemented in a very fast manner.
25 citations
TL;DR: The deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is developed in a general framework and it is shown that it is feasible for arbitrary finite fields, in the sense that the linearization can be achieved in polynomial time.
Abstract: The deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem. The main ingredients are differential equations in rational function fields and normal bases of field extensions. For finite fields of characteristic 2, it is known that this algorithm has several advantages over the classical Berlekamp algorithm. We develop the algorithm in a general framework, and we show that it is feasible for arbitrary finite fields, in the sense that the linearization can be achieved in polynomial time.
21 citations
02 Jan 1994
TL;DR: This paper improves and generalizes earlier results on the linear complexity of a termwise product of two shift-register sequences and provides information on the minimal polynomial of such a product.
Abstract: In the theory of stream ciphers the termwise product of shift-register sequences plays a crucial role. In this paper we improve and generalize earlier results on the linear complexity of a termwise product of two shift-register sequences and we also provide information on the minimal polynomial of such a product.
17 citations
TL;DR: In this article, the authors consider a finite field of order q, where q is an arbitrary prime power, and F q is a prime power in the finite field F q.
Abstract: 1. Introduction. Let F q be the finite field of order q, where q is an arbitrary prime power, and let F
16 citations
TL;DR: Further statistical independence properties of a general class of nonlinear congruential pseudorandom number generators are established and the results obtained are essentially best possible in an asymptotic sense and show that the generated Pseudorandom numbers model truly random numbers very closely in terms of asymPTotic discrepancy.
Abstract: Recently, several nonlinear congruential methods for generating uniform pseudorandom numbers have been proposed and analysed. In the present note, further statistical independence properties of a general class of nonlinear congruential pseudorandom number generators are established. The results that are obtained are essentially best possible in an asymptotic sense and show that the generated pseudorandom numbers model truly random numbers very closely in terms of asymptotic discrepancy.
09 May 1994
TL;DR: This work presents for the first time a lower bound for the linear complexity of the product of two shift-register sequences in the general case and provides information on the minimal polynomial of such a product.
Abstract: The determination of the linear complexity of the product of two shift-register sequences is a basic problem in the theory of stream ciphers. We present for the first time a lower bound for the linear complexity of the product of two shift-register sequences in the general case. Moreover, we provide information on the minimal polynomial of such a product.