Recent Advances in the Theory of Nonlinear Pseudorandom Number Generators

Niederreiter, Harald; Shparlinski, Igor E.

doi:10.1007/978-3-642-56046-0_6

Cited by 61 publications

(41 citation statements)

References 32 publications

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“…This generator has proved to be extremely useful for quasi-Monte Carlo type applications, and in particular it exhibits very attractive uniformity of distribution and nonlinearity properties; see [30], [31], [32], [33] for surveys or recent results. It is certainly natural to study its cryptographic properties as well.…”

Section: Introductionmentioning

confidence: 99%

Predicting nonlinear pseudorandom number generators

et al. 2004

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Abstract. Let p be a prime and let a and b be elements of the finite field Fp of p elements. The inversive congruential generator (ICG) is a sequence (un) of pseudorandom numbers defined by the relation u n+1 ≡ au −1 n +b mod p. We show that if sufficiently many of the most significant bits of several consecutive values un of the ICG are given, one can recover the initial value u 0 (even in the case where the coefficients a and b are not known). We also obtain similar results for the quadratic congruential generator (QCG),. This suggests that for cryptographic applications ICG and QCG should be used with great care. Our results are somewhat similar to those known for the linear congruential generator (LCG), x n+1 ≡ axn + b mod p, but they apply only to much longer bit strings. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for LCG.

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Section: Introductionmentioning

confidence: 99%

Predicting nonlinear pseudorandom number generators

et al. 2004

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“…See Eichenauer-Herrmann (1993), Eichenauer-Herrmann et al (1998), Niederreiter and Shparlinski (2002), and the references therein. Maximal-period instances are easy to find and these generators typically perform well in empirical tests.…”

Section: Nonlinear Generatorsmentioning

confidence: 99%

History of uniform random number generation

L’Ecuyer¹

2017

2017 Winter Simulation Conference (WSC)

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Random number generators were invented before there were symbols for writing numbers, and long before mechanical and electronic computers. All major civilizations through the ages found the urge to make random selections, for various reasons. Today, random number generators, particularly on computers, are an important (although often hidden) ingredient in human activity. In this article, we give a historical account on the design, implementation, and testing of uniform random number generators used for simulation.

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“…There are at least two ways of getting rid of this regular linear structure: (1) use a nonlinear transition function f or (2) keep the transition function linear but use a nonlinear output function g. Several types of nonlinear RNGs have been proposed over the years; see, e.g., Blum et al (1986);Eichenauer-Herrmann (1995); Eichenauer-Herrmann et al (1997); Hellekalek and Wegenkittl (2003); Knuth (1998); L'Ecuyer (1994); Shparlinski (2002), andTezuka (1995). Their nonlinear mappings are defined in various ways by multiplicative inversion in a finite field, quadratic and cubic functions in the finite ring of integers modulo m, and other more complicated transformations.…”

Section: Nonlinear Rngsmentioning

confidence: 99%

“…Many of them have output sequences that tend to behave much like i.i.d. U (0, 1) sequences even over their entire period length, in contrast with "good" linear RNGs, whose point sets Ψ t are much more regular than typical random points (Eichenauer-Herrmann et al, 1997;L'Ecuyer and Hellekalek, 1998;L'Ecuyer and Granger-Piché, 2003;Niederreiter and Shparlinski, 2002). On the other hand, their statistical properties have been analyzed only empirically or via asymptotic theoretical results.…”

Section: Nonlinear Rngsmentioning

confidence: 99%