Pseudorandom Number Generators

1
Pseudorandom Number Generators
2
Random Number - Definition

• A random selection of a number from a set or
  range of numbers is one in which each number in
  the range is equally likely to be selected.

3
Applications of Random Numbers

• Cryptography, games, and many statistical models
  rely on random numbers.
• Example from cryptography keys for encryption
  of data.
• Example from games the behavior of a
  computer-controlled character.
• Example from statistics - the Monte Carlo method.
  

4
Random Numbers

• True random numbers can only be generated by
  observations of random physical events, like dice
  throws or radioactive decay.
• Generation of random numbers by observation of
  physical events can be slow and impractical.

5
Pseudorandom Numbers

• Instead, sequences of numbers that approximate
  randomness are generated using algorithms.
• These numbers are inherently nonrandom because
  they are generated by deterministic mathematical
  processes.
• Anyone who considers arithmetical methods of
  producing random digits is, of course, in a state
  of sin. John von Neumann
• Hence, these numbers are known as pseudorandom
  numbers.
• The algorithms used to generate them are called
  pseudorandom number generators.

6
Pseudorandom Number Generators

• Different PRNGs approximate different properties
  of random numbers, and desirable properties vary
  with application.
• Therefore, different PRNGs are suitable for
  different applications.
• For example, a generator that produces
  unpredictable but not uniformly distributed
  number sequences may be useful in cryptography
  but not in the Monte Carlo method.

7
Middle-Square Method - History

• The middle-square method was first suggested by
  John von Neumann in 1946 for use in models of
  neutron collisions in nuclear reactions.
• The method was flawed, but it was simple and fast
  enough to be implemented using an ENIAC computer.
  

John von Neumann
8
Middle-Square Method

• Begin with an n-digit seed number x0.
• Square it to obtain a 2n-digit number, adding a
  leading zero if necessary.
• Take the middle n digits as the next random
  number.
• Repeat.
• Numbers generated can be scaled to any interval
  by multiplication and/or addition.

9
Middle-Square Method - Example

• Lets generate four-digit numbers starting with
  the seed 2041.
• Square the seed and a leading zero to obtain
  04165681.
• Take the middle four digits, 1656 as the next
  random number.
• Repeat to get the following sequence
• 2041,1656, 7423, 1009, 180, 324, 1049, 1004, 80,
  64, 40,16, 2, 0, 0, 0, 0, 0

10
Middle-Square Method - Flaw

• This sequence illustrates a serious flaw in the
  middle-square method it tends to degenerate to
  zero. (It degenerates after a number with n/2 or
  less digits is generated.)

11
Middle-Square Method - Example

• Lets try to generate numbers starting with 7600.
• 76002 57,760,000, so the next number is also
  7600. If this is repeated, the same number will
  be obtained indefinitely.
• This example illustrates the importance of
  choosing good seed values (and good parameters
  in general) for pseudorandom number generators.

12
Linear Congruence Method

• Due to its tendency to quickly degenerate to zero
  and/or repeat, the middle-square method is not a
  very practical algorithm.
• The linear congruence method provides more
  reliable results.
• Derrick H. Lehmer developed this method in 1951.
  Since then, it has become one of the most
  commonly used PRNGs.

13
Linear Congruence Method

• The method uses the following formula
• Xn1 (a Xn b) mod c
• given seed value X0 and integer values of a, b,
  and c.
• (y mod z means the remainder of the division
  of y by z.)

14
Linear Congruence Method Example

• Let a 1, b 7, c 10, and X0 7.
• X1 (1 7 7) mod (10) 4
• Repeat to get the following sequence
• 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 7, 4, 1, 5, 2, 9
• Note that the sequence cycles after every ten
  terms.
• Pseudorandom numbers always cycle eventually.

15
Linear Congruence Method Choosing Parameters

• Xn1 (a Xn b) mod c.
• The period (number of terms in a cycle) depends
  on the choice of parameters .
• a, b, c and X0 can be chosen such that the
  generator has a full period of c.
• Large values of c ensure long cycles.

16
Linear Congruence Method - Flaws

• The cycles of linear congruential generators may
  be too short for some applications.
• Issues arise from the easily detectable
  statistical interdependence of the members of
  sequences generated with this method. For
  example, it makes the method unsuitable for
  cryptography.
• The correlation of members of the sequences
  results in the uneven distribution of points
  generated in greater than 2 dimensions.
• Ordered triples of numbers generated by the
  algorithm lie on a finite number of planes.

17
Linear Congruence Method- RANDU

• The linear congruential generator RANDU is
  perhaps the most infamous example of a poorly
  chosen set of parameters for a PRNG.
• The generator was used widely throughout
  scientific community until the fact that ordered
  triples generated by it fell into only fifteen
  planes was taken into account.
• Many results produced using RANDU are now
  doubted.

3000 triples generated by RANDU.
18
Recent PRNGs Mersenne Twister

• The Mersenne Twister is now often used in place
  of the linear congruential generator.
• The Mersenne Twister was developed by
  mathematicians Makoto Matsumoto and Takuji
  Nishimura in 1997.
• The generator runs faster than all but least
  statistically sound PRNGs.
• It is distributed uniformly in 623 dimensions.
• The generator passes numerous tests for
  randomness.
• The Mersenne Twister gets its name from its huge
  period of 219937-1. This number is a Mersenne
  prime.
• It would probably take longer to cycle than the
  entire future existence of humanity (and,
  perhaps, the universe.)

19
Mersenne Twister

• Observing enough numbers generated by the
  Mersenne Twister allows all future numbers to be
  predicted.
• The Mersenne Twister is, therefore, not suitable
  in cryptography.
• This illustrates the fact that no single PRNG is
  the best choice for all applications.

20
Summary

• PRNGs are algorithms that produce sequences of
  numbers that simulate randomness.
• PRNGs are useful in game design, cryptography,
  and statistical modeling.
• Different PRNGs are suitable for different
  applications.
• It is important to choose a good set of
  parameters for a PRNG.
• The middle-square method uses the middle digits
  of the square of the nth term to generate the
  (n1)th term.
• The linear congruence method is defined by the
  recursive formula Xn1 (a Xn b) mod c

21
Sources

• Carter, Skip. Linear Congruential Generators. 9
  Jan 1996. Taygeta Scientific Incorporated. 15
  Jul 2006 lthttp//www.taygeta.com/rwalks/node1.html
  gt.
• "Hardware random number generator." Wikipedia,
  The Free Encyclopedia. 15 Jul 2006, 0450 UTC.
  Wikimedia Foundation, Inc. 17 Jul
  2006 lthttp//en.wikipedia.org/w/index.php?title
  Hardware_random_number_generator
  oldid63907837gt.
• Hutchinson, Mark. An Examination of Visual
  Basics Random Number Generation. 15 Seconds.
  14 Jul 2006 lthttp//www.15seconds.com/Issue/051110
  .htmgt.
• "Mersenne twister." Wikipedia, The Free
  Encyclopedia. 12 Jul 2006, 1846 UTC.
  Wikimedia Foundation, Inc. 17 Jul
  2006 lthttp//en.wikipedia.org/w/index.php?title
  Mersenne_twisteroldid63455933gt.
• "Middle-square method." Wikipedia, The Free
  Encyclopedia. 5 May 2006, 0506 UTC. Wikimedia
  Foundation, Inc. 17 Jul 2006 lthttp//en.wikipe
  dia.org/w/index.php?titleMiddle -square_metho
  doldid51635932gt.
• Pseudorandom number generator." Wikipedia, The
  Free Encyclopedia. 11 Jul 2006, 0722 UTC.
  Wikimedia Foundation, Inc. 17 Jul 2006
  lthttp//en.wikipedia.org/w/index.php?titlePseudor
  andom_number_generatoroldid63187601gt.
• "RANDU." Wikipedia, The Free Encyclopedia. 11 May
  2006, 1106 UTC. Wikimedia Foundation, Inc. 17
  Jul 2006 lthttp//en.wikipedia.org/w/index.ph
  p?titleRANDUoldid52640788gt.