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Topic: Linear congruential generator

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	Linear congruential generator - Wikipedia, the free encyclopedia
		Linear congruential generators (LCGs) represent one of the oldest and best-known pseudorandom number generator algorithms.
		The period of a general LCG is at most M, and in most cases less than that.
		A further problem of LCGs is that the lower-order bits of the generated sequence have a far shorter period than the sequence as a whole if M is set to a power of 2.
	en.wikipedia.org /wiki/Linear_congruential_generator (546 words)

	prng (Site not responding. Last check: 2007-11-06)
		A pseudorandom number generator (PRNG) is an algorithm which when run generates a sequence of numbers, the elements of which are approximately independent.
		The outputs of pseudorandom number generators are not random—they only approximate some of the properties of random numbers.
		Common classes of algorithms are linear congruential generators, lagged Fibonacci generators, linear feedback shift registers and generalised feedback shift registers.
	www.yourencyclopedia.net /PRNG.html (783 words)

	Linear congruential generator: LCG
		This generator corresponds to the URN12 generator in [31, 64].
		Generator 1 and 6 are used in [58] and [57] to study transformation methods for non-uniform variates.
		This LCG is also used as an example to show that LCGs are not recommended to generate random variates by the ratio of uniforms method [58].
	random.mat.sbg.ac.at /~charly/server/node3.html (2119 words)

	Random Number Generators
		Generators using a modulus that is a power of two are the fastest, but suffer from a hierarchy of patterns [
		The prime modulus generators provided in the RNG library have the useful feature that the value zero does not occur as a valid random number state.
		where X is replaced with a token that indicates the type of linear congruential recurrence relation being used for the random number generator.
	nuclear.llnl.gov /CNP/rng/rngman/node3.html (460 words)

	Linear congruential generator (Site not responding. Last check: 2007-11-06)
		Linear congruential generators (LCGs) represent one of the oldest and pseudorandom number generator algorithms.
		These are fastest-evaluated of all random number generators; a Mersenne twister implementation uses it to generate data.
		Neither this nor any other LCG should used for applications where high-quality randomness is For example it is not suitable for Monte Carlo simulation because of the serial correlation (among things).
	www.freeglossary.com /Linear_congruential_generator (482 words)

	Linear Congruential Generator: LCG
		Linear congruential generators are very sensitive with respect to changing the parameters.
		Similar as RANDU this LCG produces a fine lattice structure in dimension 2 whereas in dimension 3 strong correlations are exhibited, which are predicted by the spectral test above.
		Unfortunately, his tests on this generator were not published; our own tests and use of the generator confirmed that it is an exceptionally good pseudorandom number generator.
	crypto.mat.sbg.ac.at /results/karl/server/node3.html (640 words)

	[No title]
		For multiplicative congruential generators with modulus 2[sup]k[/sup], k [less or = to] 52, and period 2[sup]k-2[/sup], we show that the cost per random number for these two distributions is 3 and 3.125 multiply-adds on RS/6000(R) processors.
		Using the Algorithm and Architecture approach, we describe a new concept called "generalized unrolling." Finally, we present a multiplicative congruential generator for which the modulus is not a power of 2.
		Finally, we present a multiplicative congruential generator for the interval (0, 1) of the form s[sub]i+1[/sub] = as[sub]i[/sub] mod (2[sup]k[/sup] - 1), s[sub]i+1[/sub] x[sub]i+1[/sub] = -----------------, (2) 2[sup]k[/sup] - 1 (discussed in [3]), for which the modulus is not a power of 2.
	www.research.ibm.com /journal/rd/461/agarwal.txt (3623 words)

	Additive Congruential Random Number Generators (Site not responding. Last check: 2007-11-06)
		This example is the generator used for the BSD random() random number generator for 32 bit machines.
		Also the period of the lowest bit is quite long, compared to the linear congruential generator in which the lowest bit cycles 1,0,1,0,...
		Generating polynomial: x^31 + x^3 + 1 (primitive polynomial) Initialize circular queue of 31 elements using ANSI C linear congruential generator.
	home1.gte.net /deleyd/random/additive.html (493 words)

	[No title]
		linear congruential generators A Linear Congruential Generator (LCG) produces a sequence of pseudorandom integers according to the canonical linear recurrence x (n) = (a (x (n - 1) + c) mod p, n (1.
		For a prime p the parameter a must be a generator g of the multiplicative group Z*p of the integers modulo p.
		A multiplicative LCG for a prime p produces a pseudorandom sequence of integers according to the canonical linear recurrence x (n) = (gn (S) mod p, n (1, where every value from 1 to p - 1 inclusive occurs once for every value of n between 1 and p - 1 inclusive.
	www.homeworlds.com /papers/SECLCG.doc (2528 words)

	Citations: How to predict congruential generators - Krawczyk (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		The linear congruential generator is an example of a classical method for generating random looking bits that pass a variety of standard statistical tests.
		The generators mostly used in simulation (linear congruential, multiple recursive, GFSR, are known not to be PT perfect.
		....congruential generator is still predictable given some of the X i.
	citeseer.ist.psu.edu /context/51889/0 (843 words)

	Pseudo Random Number Generator.
		A linear congruential generator (LCG) is based on the formulae developed by Lehmer (1948), which is written {rnd(i+1)=(rnd(i)*b+a) mod max}.
		LCG's are at the basis of almost all random number generators currently in use, ranging from random numbers used for screensavers to random numbers used in Fortran, MsWindows and SPSS.
		This generator works on the principle that the sum of a very large number (vln) of random numbers in the range 0 to 1 drawn from random distributions is normally distributed with mean vln/2 and variance vln/12.
	home.clara.net /sisa/randhlp.htm (1987 words)

	Citations: Linear congruential generators do not produce random sequences - Frieze, Kannan, Lagarias (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		....are unknown the sequence of numbers produced by a linear congruential generator is still predictable given some of the X i.
		Truncated LCG were suggested by Knuth [12] as a possible way to make a linear congruential generator secure.
		as have more general congruential generators [4, 13] However, as indicated above, this predictability does not directly mean a cryptographic algorithm using the generator is breakable, since it is possible none of the bits of the random numbers used by the algorithm are ever made public.
	citeseer.ist.psu.edu /context/51883/0 (645 words)

	Linear Congruential Generators (Site not responding. Last check: 2007-11-06)
		This generator (with m a power of 2 and c=0) is the de facto standard included with FORTRAN and C compilers.
		Since PRNs are generated with this algorithm, some cautions to the reader are in order: (i) The PRN should not be split apart to make several random numbers since the higher order bits are much more random than the lower order bits.
		Generally LCGs are best parallelized by parameterizing the iteration process, either through the multiplier or the additive constant.
	tell11.utsi.edu /sprng/paper/node9.html (325 words)

	Linear Congruential Generators (Site not responding. Last check: 2007-11-06)
		The quality of the generator is strongly dependent upon the choice of these constants (a significant part of Knuth's chapter on random numbers is dedicated to this topic).
		A scatter plot for this for 2000 pairs from this generator reveals linear bands emerging from the plot.
		This generator often known as the minimal standard random number generator, it is often (but not always) the generator that used for the built in random number function in compilers and other software packages.
	www.taygeta.com /rwalks/node1.html (288 words)

	linear congruential generator (Site not responding. Last check: 2007-11-06)
		Definition: A class of algorithms that are pseudo-random number generators.
		Karl Entacher's thorough review and comparison of many linear congruential generators.
		Bob Bockholt, "linear congruential generator", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/linearCongruentGen.html (176 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Note that the default generator (i.e., the one that has been in Dataplot for many years) is based on Fibonacci sequence as defined by Marsagalia.
		Note that this is equivalent to the generator UNI of Jim Blue, David Kahaner, and George Marsagalia that is in the NIST CMLIB library.
		AS183 is based on the fractional part of the sum of 3 multiplicative congruential generators.
	www.itl.nist.gov /div898/software/dataplot/refman1/auxillar/randnumb (435 words)

	Random Library
		A `split' generator is a long-period generator for which we are able to split the period into arbitrary sub-periods, which we can access quickly.
		Generators may now be started with a single seed, or with a vector of seeds whose length is generator dependent.
		The generators have been subjected to a battery of statistical tests, and the results are described in the documentation.
	www.swarm.org /swarmdocs/refbook/swarm.random.sgml.reference.html (2522 words)

	Random Number Generation
		Unfortunately, generating random numbers is a task that looks a lot easier than it really is, primarily because it is fundamentally impossible to produce truly random numbers on any deterministic device.
		To evaluate a random number generator, several different tests should be used and the statistical significance of the results established.
		Generating random numbers according to a given nonuniform distribution can be a tricky business.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE142.HTM (1580 words)

	random.f: Linear congruential random number generator (Site not responding. Last check: 2007-11-06)
		It is a C modification of the RANDOM subroutine C in V.A. Dyck, J.D. Lawson and J.A. Smith, C "FORTRAN 77 : an introduction to C structured problem solving", Prentice-Hall, C New Jersey (1984).
		This modified generator is used C since it is easy to keep track of the C current value of the seed (iseed).
		endif C C Generate a new random deviate using a linear congruential C generator.
	euripides.gws.uky.edu /~trevor/Public/random_f.html (294 words)

	Generating Random Numbers
		A good random number generator produces sequences of bits which are indistinguishable from random coin flips.
		Generating random numbers is one of the most subtle and interesting problems in computer science, because seemingly reasonable solutions can have disastrous consequences.
		This generator produces a sequence of coin flips which has some of the properties of a truly random sequence.
	www.cs.sunysb.edu /~skiena/jaialai/excerpts/node7.html (1135 words)

	3.1 Approach (Site not responding. Last check: 2007-11-06)
		We begin by discussing the linear congruential generator - the one most commonly used for generating random integers.
		is generated, modulo arithmetic using the modulus m is performed, to yield the new "random" integer
		This is a characteristic of linear, congruential generators which minimizes storage requirements, but at the same time, imposes restrictions on the period.
	csep1.phy.ornl.gov /rn/node10.html (150 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		In practice, the neighbors of each encoding symbol are chosen by using a linear congruential generator: neighbor i is the input symbol at position a · i + b mod n, where a and b are randomly chosen, and n is the number of input symbols.
		In practice, and in simulations of LT, the behavior of LT codes has been observed to be extremely close in its statistical behavior to the analysis that assumes all neighbors are chosen independently, even though in the simulations and in practice they are chosen according to the linear congruential generator.
		We show that even using a linear congruential generator, only k+o(k) encoding symbols are required to decode with high probability.
	www.ocf.berkeley.edu /~lip/projects/LT-codes.html (184 words)

	6502.org: Source: Linear Congruential Pseudo-Random Number Generator Routines
		While no method of generating random numbers is perfect, the linear congruential method is widely considered to be a reasonable method.
		The strengths and weakness of the common random number generators are discussed, and numerous empirical and theoretical tests for randomness are described.
		The term "linear congruential" sounds scary, but there are only two steps involved: multplication and addition.
	www.6502.org /source/integers/random/random.html (2107 words)

	Citations: Linear congruential generators over elliptic curves - Hallgren (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		It is know that the linear congruential generator produces cryptographically weak sequences, see [6,12,13,16] thus partially motivated by this fact an elliptic curve congruential generator has been introduced in [10] and then in [8]....
		On the other hand, one of the advantages of the linear congruential generator (1) has been a variety of results about the distribution of its elements [14,18,19] Here we use some....
		Mahassni and Shparlinski [9] obtained a general bound for the 0 1 distribution of the one stage LFSR on elliptic curves over nite elds.
	citeseer.ist.psu.edu /context/1212541/228056 (537 words)

	Fast pseudorandom-number generators with modulus 2k or (2k)-1 using fused multiply-add
		These generators have implementations that are nearly the same as the three-multiply–add generator given by Equations (6), (7), and (8).
		Thus, the new generator is 53 times faster than the generic generator for both data in cache and data not in cache.
		Generators of type (77) and (79), for the interval (0, 1), are available in the RANDOM_NUMBER intrinsic function of IBM XL Fortran [4] and XL High Performance Fortran [5].
	www.research.ibm.com /journal/rd/461/agarwal.html (7296 words)

	A collection of classical pseudorandom number generators with linear structures - advanced version
		of selected linear pseudorandom number generators that were implemented in commercial software, used in applications, and some of which have extensively been tested.
		The quality of these generators is examined using scatter plots and the spectral test.
		In addition, the spectral test is applied to study the applicability of linear congruential generators on parallel architectures.
	random.mat.sbg.ac.at /results/karl/server/server.html (107 words)

	Niven Lectures (Site not responding. Last check: 2007-11-06)
		In honor of their generous support, the Department of Mathematics has established a permanent endowment fund, "The Ivan and Betty Niven Distinguished Lectures Fund", the income from which will fund a series of annual lectures on a broad array of topics in Mathematics.
		The first, due to D.H. Lehmer, is the linear congruential generator, where the (n+1)-st iterate x(n+1) is ax(n)+b (mod m) (where a,b,m and an initial seed x(0) are given).
		Among other results we have that for any nontrivial choice of parameters a,b,x(0), the linear congruential generator has period m/exp((1+o(1))loglog m logloglog m) for almost all m, while the power generator has period m/exp((1+o(1))(loglog m)^2 logloglog m) for almost all m.
	www.math.ubc.ca /~boyd/niven.html (324 words)

	SPRNG: Scalable Parallel Pseudo-Random Number Generator Library
		Each generator may contain several variants, which can be selected by setting the generator parameter during initialization.
		For example, the multiplier is a parameter for the 48 bit linear congruential generator.
		Tests to verify the quality of each generator are based on the default parameter.
	sprng.cs.fsu.edu /Version1.0/parameters.html (289 words)

	Mahalanobis
		Two fundamental types of uniform generators are in common use and are the basis for many variants: the linear congruential generators (LCGs) and the feedback shift-register (FSR) generators.[2]
		This problem was pointed out by Marsaglia (1968) who wrote: "[A]ll multiplicative (b=0) congruential random number generators have a defect--a defect that makes them unsuitable for many Monte Carlo problems and that cannot be removed by adjusting the starting value, multiplier, or modulus.
		of uniform variates produced by the generator are viewed as points in the unit cube of n dimensions, then all the points will be found to lie in a relatively small number of parallel hyperplanes.
	mahalanobis.twoday.net /stories/582175 (834 words)

	Swarm 2.2 Reference Guide: Package swarm.random
		The common functionality of simple and split generators.
		Archiving routines for internal generator and distribution state.
		This module consists of a set of random number generation classes and a set of distribution classes for transforming random number sequences into various simulated probability distributions.
	www.swarm.org /swarmdocs/refbook-java/swarm/random/package-summary.html (75 words)

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