History of Random Number Generators

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History of Random Number Generators
Bob De Vivo Probability and Statistics Summer 2005
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Early RNGs

• Dice
• Forerunners were marked disks or bones, used for
  betting games
• Earliest six-sided dice date from around 2750
  B.C.
• Found in both northern Iraq and India
• Marked with pips, possibly because they pre-date
  numbering systems
• Playing Cards
• Used in China for gambling games as early as the
  7th century
• Introduced to Europe in the late 1300s
• Coins
• Popular in ancient Rome, from whence they spread
  across Europe
• 1900, English statistician Karl Pearson tossed
  a coin 24,000 times, resulting in 12,012 heads (f
  0.5005)
• Spinning Wheels
• Ancient Greeks spun a shield balanced on the
  point of a spear. The shield was marked into
  section, and players would bet on where the
  shield would stop.
• Evolved into fairground wheel of fortune, then
  into roulette

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Types of Modern RNGs

• True random number generators
• Unpredictable (in theory), but slow and possibly
  biased
• Usually based on physical processes, which may be
  microscopic or macroscopic
• Examples
• Cards, coins, dice, roulette wheels
• Radioactive decay, thermal noise, photoelectric
  effect
• Keyboard stroke timing, lava lamps, fishtank
  bubbles
• Macroscopic processes are subject to Newtonian
  laws of motion and are therefore deterministic
  they appear unpredictable, however, because we
  cannot determine the initial conditions with
  sufficient accuracy
• RNGs based on quantum properties are completely
  unpredictable
• Pseudo random number generators
• Usually based on a mathematical algorithm
• Fast
• Suitable for computers
• Predictable, if algorithm and initial state are
  known
• Must repeat eventually, since algorithm has
  finite state, but period may be long enough to
  avoid repeating on any conceivable computer
  within a time span longer than the age of the
  universe.

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John von Neumann

• Born 1903 in Budapest
• With Stanislaw Ulam, developed a formal
  foundation for the Monte Carlo method
• In 1951, proposed one of the first pseudo-RNG
  algorithms for use on an electronic computer
• Middle-Square Method
• Start with x0, n digits long
• X1 middle n or n1 digits of x02
• Example
• x0 157
• 1572 24649
• x1 464
• Disadvantage very sensitive to choice of x0

ENIAC, completed in 1945
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Modern Pseudo-RNGs

• Linear congruential generators
• Form xn a xn-1 b (mod M)
• Most common type of PRNG in use today
• Period depends on M, but can be as long as 109
  when using 32-bit words
• Disadvantage serial correlation, which means
  successive numbers are not equidistributed
  (equally likely to fall anywhere in the range)
• Not suitable for Monte Carlo simulations
• Mersenne Twister
• developed in 1997 by Makoto Matsumoto and Takuji
  Nishimura
• Very fast
• Negligible serial correlation
• Period is 219937 - 1 (a Mersenne prime)
• Suitable for Monte Carlo, but not cryptography
• Blum Blum Shub
• proposed in 1986 by Lenore Blum, Manuel Blum, and
  Michael Shub
• Not very fast, so poor choice for simulations
• Good for cryptography because of the difficulty
  of finding any non-random patterns through
  calculation (strong security proof)
• Form xn1 (xn)2 mod(M), where M is the product
  of two large primes

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Mathematicas RNG

• Uses Wolfram rule 30 cellular automaton generator

Table for rule 30
Evolution of cells, starting with a single black
cell
Central column is chaotic