Efficient lattice assessment for LCG and GLP parameter searches

Entacher, Karl; Schell, Thomas; Uhl, Andreas

doi:10.1090/s0025-5718-01-01415-6

Cited by 8 publications

(4 citation statements)

References 35 publications

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“…Our experiments use an approximation of this type. The high quality of the LLL-approximation and the speedup with respect to the "original" spectral test have recently been shown [6].…”

Section: The Spectral Testmentioning

confidence: 98%

Normalization of the Spectral Test in High Dimensions

Entacher

Laimer

Röck

et al. 2004

Monte Carlo Methods and Applications

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Abstracts -The spectral test provides a reliable measure for lattice assessment and can be computed very efficiently. It has extensively been applied to find good lattices for several MC and QMC applications. In order to enable comparisons across dimensions, a normalized spectral test is widely used. We empirically demonstrate significant shortcomings of this normalization in high dimensions, discuss the empirical distribution of the normalized spectral test values, and propose a new normalization strategy. The new normalization is shown to give more reliable results, especially concerning the comparability of the values accross dimensions.

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“…Our experiments use an approximation of this type. The high quality of the LLL-approximation and the speedup with respect to the "original" spectral test have recently been shown [6].…”

Section: The Spectral Testmentioning

confidence: 98%

Normalization of the Spectral Test in High Dimensions

Entacher

Laimer

Röck

et al. 2004

Monte Carlo Methods and Applications

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“…Following a suggestion in a paper by Entacher, Schell, and Uhl 21 and in the associated code, we compute the figures of merit from

f_{2}

f_{8}

using the implementation of the ubiquitous Lenstra–Lenstra–Lovász basis‐reduction algorithm 22 provided by Shoup's NTL library 23 . For

m = 2^{64}

and

m =^{128}

we recorded in an output file all tested multipliers whose minimum spectral score is at least 0.70 (we used a lower threshold for

m = 2^{32}

).…”

Section: Search Strategymentioning

confidence: 99%

Computationally easy, spectrally good multipliers for congruential pseudorandom number generators

Steele

Vigna

2021

Softw Pract Exp

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Congruential pseudorandom number generators rely on good multipliers, that is, integers that have good performance with respect to the spectral test. We provide lists of multipliers with a good lattice structure up to dimension eight and up to lag eight for generators with typical power-of-two moduli, analyzing in detail multipliers close to the square root of the modulus, whose product can be computed quickly. K E Y W O R D Scongruential generators, pseudorandom number generators 1 * We remark that these denominations, by now used for half a century, are completely wrong from a mathematical viewpoint. The map x  → ax is indeed a linear map, but the map x  → ax + c is an affine map: 2 what we call an "MCG" or "MLCG" should called an "LCG" (this in fact happens in some books) and what we call an "LCG" should be called an "ACG". The mistake originated probably in the interest of Lehmer in (truly) linear maps with prime moduli. 1 Constants were added later to obtain large-period generators with nonprime moduli, but the "linear" name stuck (albeit some authors are using the term "mixed" instead of "linear"). At this point it is unlikely that the now-traditional names will be corrected.

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“…Entacher et al [7] consider M as a measure of the minimum test value of the multiplier a itself and additionally the subsequence generators with multipliers a k for a set of different k values. Kao and Tang [15] derived the upper bounds of spectral test for multiple recursive random number generators and conducted several searches.…”

Section: Assessing the Lattice Structurementioning

confidence: 99%

Distribution of Lattice Points

Sezgin

2006

Computing

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We discuss the lattice structure of congruential random number generators and examine figures of merit. Distribution properties of lattice measures in various dimensions are demonstrated by using large numerical data. Systematic search methods are introduced to diagnose multiplier areas exhibiting good, bad and worst lattice structures. We present two formulae to express multipliers producing worst and bad laice points. The conventional criterion of normalised lattice rule is also questioned and it is shown that this measure used with a fixed threshold is not suitable for an effective discrimination of lattice structures. Usage of percentiles represents different dimensions in a fair fashion and provides consistency for different figures of merits.

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Efficient lattice assessment for LCG and GLP parameter searches

Cited by 8 publications

References 35 publications

Normalization of the Spectral Test in High Dimensions

Normalization of the Spectral Test in High Dimensions

Computationally easy, spectrally good multipliers for congruential pseudorandom number generators

Distribution of Lattice Points

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