Low-Discrepancy Sequences and Global Function Fields with Many Rational Places

Niederreiter, Harald; Xing, Chaoping

doi:10.1006/ffta.1996.0016

Cited by 163 publications

(125 citation statements)

References 21 publications

(43 reference statements)

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“…sequences [55], Faure sequences [8], Niederreiter sequences [35,36] and NiederreiterXing sequences [37]. Although a bound of the form (1.14) indicates an ultimate order of convergence theoretically higher than the classical Monte Carlo rate of 1/ √ N, that bound is unsatisfactory when the dimensionality is high because, for fixed s, (ln N) s /N keeps growing with increasing N until N is exponentially large in s. In contrast to that somewhat negative observation is a remarkable result proved by Heinrich et al [12], which states that there exists a sequence of QMC point sets for which the star discrepancy is of order √ s/N with an unknown constant, or alternatively √ s ln s ln N/N with an explicit constant.…”

Section: The Classical Setting and What Goes Wrongmentioning

confidence: 99%

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Kuo¹,

Schwab²,

Sloan³

2012

ANZIAMJ

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This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called "product and order dependent" (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

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Section: The Classical Setting and What Goes Wrongmentioning

confidence: 99%

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Kuo¹,

Schwab²,

Sloan³

2012

ANZIAMJ

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“…Of course |J F [2]| is dealt with asymptotically with the torsion limit J 2 (q, A) which we have introduced in this paper. Stirling's Formula gives an asymptotical bound for the binomial coefficients n t when t is some fixed fraction of n. Finally the quotients A r /h can be bounded by means of algebraic geometric techniques which have been used before in the code theoretic literature, for instance [25], [28], [39], [40]. We state now an upper bound of this type.…”

Section: If the N-code C * I Is T-disconnected And If C * J Has (N − mentioning

confidence: 99%

The Torsion-Limit for Algebraic Function Fields and Its Application to Arithmetic Secret Sharing

Cascudo

Cramer

Xing

2011

Lecture Notes in Computer Science

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Abstract. An (n, t, d, n−t)-arithmetic secret sharing scheme (with uniformity) for F k q over Fq is an Fq-linear secret sharing scheme where the secret is selected from F k q and each of the n shares is an element of Fq. Moreover, there is t-privacy (in addition, any t shares are uniformly random in F t q ) and, if one considers the d-fold "component-wise" product of any d sharings, then the d-fold component-wise product of the d respective secrets is (n − t)-wise uniquely determined by it. Such schemes are a fundamental primitive in information-theoretically secure multi-party computation. Perhaps counter-intuitively, secure multi-party computation is a very powerful primitive for communication-efficient two-party cryptography, as shown recently in a series of surprising results from 2007 on. Moreover, the existence of asymptotically good arithmetic secret sharing schemes plays a crucial role in their communication-efficiency: for each d ≥ 2, if A(q) > 2d, where A(q) is Ihara's constant, then there exists an infinite family of such schemes over Fq such that n is unbounded, k = Ω(n) and t = Ω(n), as follows from a result at CRYPTO'06. Our main contribution is a novel paradigm for constructing asymptotically good arithmetic secret sharing schemes from towers of algebraic function fields. It is based on a new limit that, for a tower with a given Ihara limit and given positive integer , gives information on the cardinality of the -torsion sub-groups of the associated degree-zero divisor class groups and that we believe is of independent interest. As an application of the bounds we obtain, we relax the condition A(q) > 2d from the CRYPTO'06 result substantially in terms of our torsion-limit. As a consequence, this result now holds over nearly all finite fields Fq. For example, if d = 2, it is sufficient that q = 8, 9 or q ≥ 16.

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“…The construction of global function fields with many rational places, or equivalently of algebraic curves over F, with many F g -rational points, is of great theoretical interest. Moreover, it is also important for applications in the theory of algebraic-geometry codes (see [15], [16]) and in the recent constructions of low-discrepancy sequences introduced by the authors (see [5], [7], [10], [17]). …”

Section: N Q (G) = Maxn(k)mentioning

confidence: 99%

“…For the construction of sdimensional low-discrepancy sequences in a given base q by means of rational places (see e.g. [5]) we need a global function field K/W q with N(K) > s + 1. In order to cover the standard range 1 < s < 50 of applications of low-discrepancy sequences in an efficient manner, we need to find, for each dimension s in this range, a global function field K/F 9 of relatively small genus with N(K) > s + 1.…”

Section: N Q (G) = Maxn(k)mentioning

confidence: 99%

Global Function Fields With Many Rational Places Over the Quinary Field

Niederreiter¹,

Xing²

1997

Demonstratio Mathematica

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Low-Discrepancy Sequences and Global Function Fields with Many Rational Places

Abstract: We construct digital (t, s)-sequences in a prime-power base q for which the quality parameter t has the least possible order of magnitude. The construction uses algebraic function fields over the finite field of order q which contain many places of degree 1 relative to the genus.

Cited by 163 publications

References 21 publications

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

The Torsion-Limit for Algebraic Function Fields and Its Application to Arithmetic Secret Sharing

Global Function Fields With Many Rational Places Over the Quinary Field

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