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E-raamat: Applied Algebra and Number Theory

Edited by (Nanyang Technological University, Singapore), Edited by (Johannes Kepler Universität Linz), Edited by , Edited by (Johannes Kepler Universität Linz)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Dec-2014
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316121641
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Applied Algebra and Number TheorySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Dec-2014
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316121641
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Harald Niederreiter's pioneering research in the field of applied algebra and number theory has led to important and substantial breakthroughs in many areas. This collection of survey articles has been authored by close colleagues and leading experts to mark the occasion of his 70th birthday. The book provides a modern overview of different research areas, covering uniform distribution and quasi-Monte Carlo methods as well as finite fields and their applications, in particular, cryptography and pseudorandom number generation. Many results are published here for the first time. The book serves as a useful starting point for graduate students new to these areas or as a refresher for researchers wanting to follow recent trends.

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This book contains survey articles on modern topics related to the work of Harald Niederreiter, written by close colleagues and leading experts.
Preface xi
1 Some highlights of Harald Niederreiter's work
1(21)
Gerhard Larcher
Friedrich Pillichshammer
Arne Winterhof
Chaoping Xing
1.1 A short biography
1(3)
1.2 Uniform distribution theory and number theory
4(3)
1.3 Algebraic curves, function fields and applications
7(3)
1.4 Polynomials over finite fields and applications
10(3)
1.5 Quasi-Monte Carlo methods
13(9)
References
18(4)
2 Partially bent functions and their properties
22(17)
Ayca Cesmelioglu
Wilfried Meidl
Alev Topuzoglu
2.1 Introduction
22(2)
2.2 Basic properties
24(4)
2.3 Examples and constructions
28(1)
2.4 Partially bent functions and difference sets
29(6)
2.5 Partially bent functions and Hermitian matrices
35(1)
2.6 Relative difference sets revisited: a construction of bent functions
36(3)
References
38(1)
3 Applications of geometric discrepancy in numerical analysis and statistics
39(19)
Josef Dick
3.1 Introduction
39(1)
3.2 Numerical integration in the unit cube
40(4)
3.3 Numerical integration over the unit sphere
44(3)
3.4 Inverse transformation and test sets
47(1)
3.5 Acceptance-rejection sampler
48(3)
3.6 Markov chain Monte Carlo and completely uniformly distributed sequences
51(2)
3.7 Uniformly ergodic Markov chains and push-back discrepancy
53(5)
References
54(4)
4 Discrepancy bounds for low-dimensional point sets
58(33)
Henri Faure
Peter Kritzer
4.1 Introduction
58(8)
4.2 Upper discrepancy bounds for low-dimensional sequences
66(9)
4.3 Upper discrepancy bounds for low-dimensional nets
75(6)
4.4 Lower discrepancy bounds for low-dimensional point sets
81(6)
4.5 Conclusion
87(4)
References
88(3)
5 On the linear complexity and lattice test of nonlinear pseudorandom number generators
91(11)
Domingo Gomez-Perez
Jaime Gutierrez
5.1 Introduction
91(2)
5.2 Lattice test and quasi-linear complexity
93(1)
5.3 Quasi-linear and linear complexity
94(3)
5.4 Applications of our results
97(2)
5.5 An open problem
99(3)
References
99(3)
6 A heuristic formula estimating the keystream length for the general combination generator with respect to a correlation attack
102(7)
Rainer Gottfert
6.1 The combination generator
102(1)
6.2 The model
102(1)
6.3 Preliminaries
103(1)
6.4 The correlation attack
103(2)
6.5 The formula
105(4)
References
108(1)
7 Point sets of minimal energy
109(17)
Peter J. Grabner
7.1 Introduction
109(2)
7.2 Generalized energy and uniform distribution on the sphere
111(5)
7.3 Hyper-singular energies and uniform distribution
116(3)
7.4 Discrepancy estimates
119(3)
7.5 Some remarks on lattices
122(4)
References
123(3)
8 The cross-correlation measure for families of binary sequences
126(18)
Katalin Gyarmati
Christian Mauduit
Andras Sarkozy
8.1 Introduction
126(3)
8.2 The definition of the cross-correlation measure
129(4)
8.3 The size of the cross-correlation measure
133(2)
8.4 A family with small cross-correlation constructed using the Legendre symbol
135(4)
8.5 Another construction
139(5)
References
141(3)
9 On an important family of inequalities of Niederreiter involving exponential sums
144(20)
Peter Hellekalek
9.1 Introduction
144(4)
9.2 Concepts
148(9)
9.3 A hybrid Erdos--Turan--Koksma inequality
157(7)
References
161(3)
10 Controlling the shape of generating matrices in global function field constructions of digital sequences
164(26)
Roswitha Hofer
Isabel Pirsic
10.1 Introduction
164(5)
10.2 Global function fields
169(1)
10.3 Constructions revisited
170(6)
10.4 Designing morphological properties of the generating matrices
176(6)
10.5 Computational results
182(3)
10.6 Summary and outlook
185(5)
References
187(3)
11 Periodic structure of the exponential pseudorandom number generator
190(14)
Jonas Kaszian
Pieter Moree
Igor E. Shparlinski
11.1 Introduction
190(4)
11.2 Preparations
194(1)
11.3 Main results
195(4)
11.4 Numerical results on cycles in the exponential map
199(2)
11.5 Comments
201(3)
References
202(2)
12 Construction of a rank-1 lattice sequence based on primitive polynomials
204(12)
Alexander Keller
Nikolaus Binder
Carsten Wachter
12.1 Introduction
204(1)
12.2 Integro-approximation by rank-1 lattice sequences
205(1)
12.3 Construction
206(5)
12.4 Applications
211(3)
12.5 Conclusion
214(2)
References
214(2)
13 A quasi-Monte Carlo method for the coagulation equation
216(19)
Christian Lecot
Ali Tarhini
13.1 Introduction
216(3)
13.2 The quasi-Monte Carlo algorithm
219(3)
13.3 Convergence analysis
222(7)
13.4 Numerical results
229(1)
13.5 Conclusion
229(6)
References
231(4)
14 Asymptotic formulas for partitions with bounded multiplicity
235(20)
Pierre Liardet
Alain Thomas
14.1 Introduction
235(4)
14.2 Asymptotic expansion of MU,q
239(7)
14.3 Proof of Theorem 14.2
246(9)
References
253(2)
15 A trigonometric approach for Chebyshev polynomials over finite fields
255(25)
Juliano B. Lima
Daniel Panario
Ricardo M. Campello de Souza
15.1 Introduction
255(2)
15.2 Trigonometry in finite fields
257(8)
15.3 Chebyshev polynomials over finite fields
265(5)
15.4 Periodicity and symmetry properties of Chebyshev polynomials over finite fields
270(3)
15.5 Permutation properties of Chebyshev polynomials over finite fields
273(5)
15.6 Conclusions
278(2)
References
278(2)
16 Index bounds for value sets of polynomials over finite fields
280(17)
Gary L. Mullen
Daqing Wan
Qiang Wang
16.1 Introduction
280(3)
16.2 Value sets of univariate polynomials
283(2)
16.3 Permutation polynomial vectors
285(12)
References
294(3)
17 Rational points of the curve yqn -y = γxqh+1 -α over Fqm
297(10)
Ferruh Ozbudak
Zulfukar Saygi
17.1 Introduction
297(4)
17.2 Preliminaries
301(1)
17.3 Proof of the main theorem
302(5)
References
306(1)
18 On the linear complexity of multisequences, bijections between Zahlen and Number tuples, and partitions
307
Michael Vielhaber
18.1 Introduction and notation
307(2)
18.2 Single sequences
309(8)
18.3 Multilinear complexity
317(10)
18.4 Partitions, bijections, conjectures
327(4)
18.5 Open questions and further research
331(1)
18.6 Conclusion
332
References
332
Gerhard Larcher is Full Professor for Financial Mathematics and Head of the Institute for Financial Mathematics at the Johannes Kepler University Linz. Friedrich Pillichshammer is Associate Professor in the Institute for Financial Mathematics at the Johannes Kepler University Linz. Arne Winterhof is Senior Fellow at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) at the Austrian Academy of Sciences, Linz. Chaoping Xing is Full Professor in the Department of Physical and Mathematical Sciences at Nanyang Technological University, Singapore.
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