114 Result(s)
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Chapter
Ian Sloan and Lattice Rules
Lattice rules are a powerful and popular form of quasi-Monte Carlo rules that are based on integration lattices. The study of the theory and application of lattice rules is intimately connected with the name I...
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Article
Expansion complexity and linear complexity of sequences over finite fields
The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequ...
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Article
Multisequences with high joint nonlinear complexity
We introduce the new concept of joint nonlinear complexity for multisequences over finite fields and we analyze the joint nonlinear complexity of two families of explicit inversive multisequences. We also esta...
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Chapter and Conference Paper
Vandermonde Nets and Vandermonde Sequences
A new of digital nets Vandermonde nets was recently introduced by the authors. We generalize the construction of Vandermonde nets with a view to obtain digital nets that serve as stepping stones for new co...
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Article
A survey of some applications of finite fields
We present a survey of applications of finite fields to cryptography, digital nets, and pseudorandom number generation.
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Chapter and Conference Paper
Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve
This work star and errors of two quasi-random points constructions using a generator one-dimensional sequence and the Hilbert space-filling curve. This recursive fractal is proven to maximize localit...
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Book
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Reference Work Entry In depth
Finite Fields
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Chapter
A Review of Number Theory and Algebra
Elementary number theory may be regarded as a prerequisite for this book, but since we, the authors, want to be nice to you, the readers, we provide a brief review of this theory for those who already have som...
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Chapter
Coding Theory
Life is a comedy of errors, at least in the opinion of William Shakespeare, but you can make a concentrated effort to reduce the number of errors that you commit and thus increase the quality of your life. The...
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Chapter
Pseudorandom Numbers
We pointed out in Sect. 4.1.2 that the Monte Carlo method for numerical integration uses random samples, but we did not say anything about how to produce random samples. Maybe we were wise to keep quiet, becau...
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Chapter
Cryptography
Cryptology in the modern sense is the theory of data security and data integrity. Cryptology as a practical craft can be traced back several thousand years as it was already used in one form or other in the a...
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Chapter
Quasi-Monte Carlo Methods
There are many scientific as well as real-world applications where we run into the problem of computing a definite integral. In calculus courses you are taught that a definite integral $$\int _{a}^{b}f(u)du$...
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Chapter
Further Applications
Check-digit systems and error-correcting codes (see Chap. 3 for the latter) are birds of a feather, but it must be conceded that error-correcting codes are the more colorful birds. Just like error-correcting c...
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Article
Halton-type sequences from global function fields
For any prime power q and any dimension s ⩾ 1, a new construction of (t, s)-sequences in base q using global function fields is presented. The construction yields an analog of Halton sequences for global function...
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Article
Improved results on the probabilistic theory of the joint linear complexity of multisequences
We improve previous results on the asymptotic behavior and the expected value of the joint linear complexity of random multisequences over finite fields. These results are of interest for word-based stream cip...
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Chapter
Low-Discrepancy Simulation
This article presents a survey of low-discrepancy sequences and their applications to quasi-Monte Carlo methods for multidimensional numerical integration. Quasi-Monte Carlo methods are deterministic versions ...
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Article
Further discrepancy bounds and an Erdös–Turán–Koksma inequality for hybrid sequences
We consider hybrid sequences, that is, sequences in a multidimensional unit cube that are composed from lower-dimensional sequences of two different types. We establish nontrivial deterministic discrepancy bou...
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Article
Construction Algorithms for Good Extensible Lattice Rules
Extensible (polynomial) lattice rules have the property that the number N of points in the node set may be increased while retaining the existing points. It was shown by Hickernell and Niederreiter in a nonconstr...
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Article
Joint linear complexity of multisequences consisting of linear recurring sequences
The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences...