On the Average Distribution of Inversive Pseudorandom Numbers
TL;DR: The inversive congruential method is an attractive alternative to the classical linear congruent method for pseudorandom number generation as mentioned in this paper, and it has been shown that, on average, much stronger results than those known for ''individual'' sequences can be obtained.
About: This article is published in Finite Fields and Their Applications.The article was published on 2002-10-01 and is currently open access. It has received 31 citations till now. The article focuses on the topics: Inversive congruential generator & Pseudorandomness.
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29 Nov 2010
TL;DR: This book presents survey articles on some of the new developments in the theory of algebraic function fields over finite fields, which have not yet been presented in other books or survey articles.
Abstract: The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.
86 citations
01 Jan 2002
TL;DR: A survey of recent developments in the theory of nonlinear generators for uniform pseudorandom numbers can be found in this article, where the emphasis is on discrepancy-based tests for inversive generators where most of the progress has taken place.
Abstract: We present a survey of recent developments in the theory of nonlinear generators for uniform pseudorandom numbers. The emphasis is on discrepancybased tests for inversive generators where most of the progress has taken place.
69 citations
TL;DR: Lower bounds are obtained on the linear and nonlinear complexity profile of a general nonlinear pseudorandom number generator, of the inversive generator, and of a new nonlinear generator called quadratic exponential generator.
Abstract: We obtain lower bounds on the linear and nonlinear complexity profile of a general nonlinear pseudorandom number generator, of the inversive generator, and of a new nonlinear generator called quadratic exponential generator. The results are interesting for applications to cryptography and Monte Carlo methods.
39 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: This paper uses estimates of the degree growth of iterations of multivariate polynomials to bound exponential sums along the orbits of these dynamical systems and shows that they admit much stronger estimates than in the general case and can be of use for pseudorandom number generation.
Abstract: In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.
34 citations
Cites background from "On the Average Distribution of Inve..."
...It is possible that some of the arguments of [39] may be applied to this problem....
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...We remark that, in the case of the so-called inversive generator , rather strong estimates are also available [38, 39], but this generator involves a modular inversion at each step, which is a computationally expensive operation....
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References
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Book•
01 Jan 1992TL;DR: This chapter discusses Monte Carlo methods and Quasi-Monte Carlo methods for optimization, which are used for numerical integration, and their applications in random numbers and pseudorandom numbers.
Abstract: Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.
3,815 citations
"On the Average Distribution of Inve..." refers background in this paper
...We refer to [5, 6, 7, 17, 18, 19, 20, 22, 23] for more details and references to original papers....
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Book•
20 Mar 1997819 citations
"On the Average Distribution of Inve..." refers background in this paper
...21 of [3]) for the discrepancy of a sequence of points of the s-dimensional unit cube, which we present in the following form....
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Book•
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01 Jan 1996
TL;DR: In this article, the authors present a generalization of Fields, Galois Theory, and Algebraic Number Theory to algebraic number theory, which is called Galois theory of inseparable field extensions.
Abstract: Preface Section 1A Linear Algebra Van der Waerden conjecture and applications (GP Egorychev) Random matrices (VL Girko) Matrix equations Factorization of matrix polynomials (AN Malyshev) Matrix functions (L Rodman) Section 1B Linear (In)dependence Matroids (JPS Kung) Section 1D Fields, Galois Theory, and Algebraic Number Theory Higher derivation Galois theory of inseparable field extensions (JK Deveney, JN Mordeson) Theory of local fields Local class field theory Higher local class field theory (IB Fesenko) Infinite Galois theory (M Jarden) Finite fields and their applications (R Lidl, H Niederreiter) Global class field theory (W Narkiewicz) Finite fields and error correcting codes (H van Tilborg) Section 1F Generalizations of Fields and Related Objects Semi-rings and semi-fields (U Hebisch, HJ Weinert) Near-rings and near-fields (GF Pilz) Section 2A Category Theory Topos theory (S MacLane, I Moerdijk) Categorical structures (RH Street) Section 2B Homological Algebra Cohomology Cohomological Methods in Algebra Homotopical Algebra The cohomology of groups (JF Carlson) Relative homological algebra Cohomology of categories, posets, and coalgebras (AI Generalov) Homotopy and homotopical algebra (JF Jardine) Derived categories and their uses (B Keller) Section 3A Commutative Rings and Algebras Ideals and modules (J-P Lafon) Section 3B Associative Rings and Algebras Polynomial and power series rings Free algebras, firs and semifirs (PM Cohn) Simple, prime, and semi-prime rings (VK Kharchenko) Algebraic microlocalization and modules with regular singularities over filtered rings (ARP van den Essen) Frobenius rings (K Yamagata) Subject Index
295 citations
TL;DR: In this paper, Woźniakowski et al. presented a program committee for the International Journal of Distributed Sensor Networks (GanIzerS) with the following members: Piotr krzyżanowski, Marek kwas, leszek Plaskota, and Grzegorz Wasilkowski.
Abstract: loCal orGanIzerS • Piotr krzyżanowski • Marek kwas • leszek Plaskota • Henryk Woźniakowski (chair) PROGRAM COMMITTEE • William Chen (australia) • ronald Cools (Belgium) • Josef dick (australia) • Henri Faure (France) • alan Genz (USa) • Paul Glasserman (USa) • Stefan Heinrich (Germany) • Fred J. Hickernell (USa) • Stephen Joe (new zealand) • aneta karaivanova (Bulgaria) • alexander keller (Germany) • Frances kuo (australia) • Gerhard larcher (austria) • Pierre l’ecuyer (Canada) • Christiane lemieux (Canada) • Makoto Matsumoto (Japan) • Peter Mathé (Germany) • thomas Müller-Gronbach (Germany) • Harald niederreiter (austria) • erich novak (Germany) • art B. owen (USa) • Friedrich Pillichshammer (austria) • leszek Plaskota (Poland) • klaus ritter (Germany) • Wolfgang Ch. Schmid (austria) • nikolai Simonov (russia) • Ian H. Sloan (australia) • Ilya M. Sobol’ (russia) • Jerome Spanier (USa) • Shu tezuka (Japan) • Xiaoqun Wang (China) • Grzegorz Wasilkowski (USa) • Henryk Woźniakowski (chair) (Poland/USa)
238 citations
TL;DR: A theorem on the period length of sequences produced by this type of generators is proved and it is shown that good results are obtained if a non-linear congruential generator of about the same period length is applied.
Abstract: A non-linear congruential pseudo random number generator is introduced. This generator does not have the lattice structure in the distribution of tuples of consecutive pseudo random numbers which appears in the case of linear congruential generators. A theorem on the period length of sequences produced by this type of generators is proved. This theorem justifies an algorithm to determine the period length. Finally a simulation problem is described where a linear congruential generator produces completely useless results whereas good results are obtained if a non-linear congruential generator of about the same period length is applied.
143 citations
"On the Average Distribution of Inve..." refers methods in this paper
...These pseudorandom numbers were introduced by Eichenauer and Lehn [4]....
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