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Random NumbersRandom Numbersandand

Monte Carlo MethodsMonte Carlo Methods

Charles Erwin and Jay Hatcher

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Random NumbersRandom Numbers

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Introduction Introduction

What are “Random Numbers?”

Where do Random Numbers Come from?

How to make Random numbers?

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Random NumbersRandom Numbers

Not just looking for a random number.– Looking for a sequence of numbers that

doesn’t repeat.– Do not want a sequence that is deterministic!

Seed: Number that begins the sequence.– Generally the only number that is known

initially.

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Why use Random Numbers?Why use Random Numbers?

Statistical SamplingCryptography/CryptanalysisGamingAnalysis of AlgorithmsSimulation Input

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Where to find Random NumbersWhere to find Random Numbers

Tables of Random Numbers– Classic Example: RAND corp’s

A Million Random Digits with 100,000 Normal Deviates

– Generated with Electronic simulation of roulette wheel attached to a computer.

– Early example of hardware Random Number generator.

Random Number Generator

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Ideas for Random Sequences?Ideas for Random Sequences?

von Neumann’s “Squaring” Method– Take a number, square it, and then use the middle

digits as the random number.

– What about a seed of 0000? Linear Congruential Generator

– Xn = (aXn-1 + c) mod m

– Xo = seed, modulus m, muliplier a, and increment c

– Repeats due to the modular arithmetic that forces wrapping of values into the desired range.

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Linear Congruential GeneratorLinear Congruential Generator Problems (cont)

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Random Numbers from HardwareRandom Numbers from HardwareLinux Example: /dev/random

– Generated by many different factors observed by the operating systems

– Avoids tracking things such as network traffic that can be manipulated by outsiders.

Other inputs for random information:– Readings from a Geiger counter– Detected noise from a radio receiver– Thermal or quantum-mechanical noise, amplified to provide a

random voltage source.

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Random Numbers from HardwareRandom Numbers from Hardware(cont)(cont)

Internal CPU Random Number Generator– Modern Intel chips feature an internal RNG

– Samples thermal noise by amplifying the voltage measured across undriven resistors.

– Analysis determines it cryptographically sound.

Unconventional Source: Lava Lamps– Random bits are extracted from images of the erupting blobs

inside six Lava Lite lamps. No longer available.

– LavaRnd provided random data using similar methods.

– http://www.lavarnd.org/what/index.html

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Random Numbers from HardwareRandom Numbers from Hardware(cont)(cont)

Commercial Random Number Generators– Operate from USB or Serial connections

– Standard mode is to deliver one byte of data at a time.

– Current models pass DIEHARD battery of tests

– Very fast generation.

– Prices range anywhere from ~$100 to more than $1,000 per unit.

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Software Random GeneratorsSoftware Random Generators

Random Number Algorithm an oxymoron.– von Neumann: “Anyone who uses software to

produce random numbers is in a state of sin”– Deterministic Random number generators are

called “Pseudorandom Number generators”– Clever algorithms can create sequences that

are relatively random.

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Modern Generator Example 1Modern Generator Example 1

Mersenne Twister by Makoto Matsumoto and Takuji Nishimura – Period size: 2^19937-1– Freely Available in C source code– Fast (used to be much faster than the C rand()

functionality but rand() has since been improved)

– Not cryptographically secure

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Modern Generator Example 2Modern Generator Example 2

Blum Blum Shub (BBS) by Lenore Blum, Manuel Blum and Michael Shub

– Xn+1 = (Xn)2 mod M– M=pq is the product of two large primes p and q

– Proven to be cryptographically secure– Very slow

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Summary of AlgorithmsSummary of Algorithms

Different generators are more appropriate for specific circumstances.– For Cryptanalysis or Cryptography, finding an

algorithm which is cryptographically secure is essential.

– For Simulations such as Monte-Carlo, a fast algorithm is preferred.

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Summary (cont.)Summary (cont.)

If your application is security based, your choices are limited to cryptographically secure random number generators.

Otherwise, use the normal algorithm analysis questions of correctness and complexity.

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Quick list of ReferencesQuick list of References Linear Congruential Generator:

http://eternallyconfuzzled.com/tuts/random.htmlhttp://www.taygeta.com/rwalks/node1.html

Randomness as a Resource: http://www.americanscientist.org/template/AssetDetail/assetid/20829/page/3

Hardware Random Number Generators: http://www.robertnz.net/hwrng.htm Mersenne Twister: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt

.html Blum Blum Shub:

http://en.wikipedia.org/wiki/Blum-Blum-Shub_pseudorandom_number_generator

DIEHARD Random Tests: http://www.cs.hku.hk/~diehard/cdrom/ Security Requirements for Random Numbers:

http://www.ietf.org/rfc/rfc4086.txt Intel’s internal RNG: http://www.cryptography.

com/resources/whitepapers/IntelRNG.pdf

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Monte Carlo MethodsMonte Carlo Methods

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Introduction Introduction

“Monte Carlo” was coined by Metropolis during the Manhattan Project of WWII

Named after a resort town in Monaco famous for its casinos

Monte Carlo methods use statistical simulation to approximate a solution to a problem

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Integration MethodsIntegration Methods

Direct Sampling

Random Walk and Markov Chains

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Integration Methods (cont)Integration Methods (cont)

Direct Sampling– Importance Sampling

– Stratified Sampling

– Adaptive Monte Carlo– VEGAS algorithm

– Recursive Stratified Sampling

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Integration Methods (cont)Integration Methods (cont)

Markov Chain Monte Carlo (MCMC)– Random Walks

– Metropolis-Hastings Algorithm

– Gibbs Sampling

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Optimization MethodsOptimization Methods

Genetic Algorithms

Simulated Annealing

Parallel Tempering

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Optimization Methods (cont)Optimization Methods (cont)

Genetic Algorithms– Explore solution space using subsequent

generations

– Generations that are better are combined and/or mutated to try to produce an even better solution

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Optimization Methods (cont)Optimization Methods (cont)

Simulated Annealing– Inspired by annealing in metallurgy

– Nearby solutions are compared to the global temperature T

– T is gradually decreased

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Optimization Methods (cont)Optimization Methods (cont)

Parallel Tempering– Run two or more simulations concurrently at

different temperatures

– Pay attention to overlap between the two systems and use the overlap to pick more likely solutions

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Other MethodsOther Methods

Direct Simulation– Stochastic simulation of rarefied gas flows– DSMC Demo

Dynamic Monte Carlo– models the dynamic behaviors of molecules

Quantum Monte Carlo– Applies MC to the Schrödinger wave equation

in quantum mechanics

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Example: Approximating Example: Approximating ππ

π is 4 times the number of dartsin the shaded part of the quadrant divided by the total darts in the quadrant

Buffon’s Needle

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ApplicationsApplications

Radiation transport Operations researchNuclear criticality Design of nuclear reactors Design of nuclear weapons Statistical physicsPhase transitions Wetting and growth of thin filmsReinforcement Learning Genetic ProgrammingAtomic wave functions and Intranuclear cascade reactions eigenvalues Thermodynamic propertiesLong chain coiling polymers Reaction kineticsPartial differential equations Large sets of linear equationsNumerical integration Uncertainty analysisDevelopment of statistical tests Cell population studies Combinatorial problem Search and optimizationSignal detection WarGames

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References and Additional InfoReferences and Additional Info Introduction to Monte Carlo Methods

– http://csep1.phy.ornl.gov/mc/mc.html Adaptive and Recursive Monte Carlo Methods

– http://www.library.cornell.edu/nr/bookcpdf/c7-8.pdf The WWW Virtual Library: Monte Carlo Methods

– http://random.mat.sbg.ac.at/links/monte.html The Basics of Monte Carlo Simulations

– http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html Advanced Monte Carlo Methods

– http://www.cs.pitt.edu/~milos/courses/cs3750/lectures/class10.pdf Monte Carlo Methods (Reinforcement Learning)

– http://www.cs.ualberta.ca/~sutton/book/5/node1.html Buffon’s Needle (applet)

– http://www.angelfire.com/wa/hurben/buff.html Direct Simulation MC

– http://www.simba.us/misc/dsmc/dsmca.html