History of Random Number Generators

1. History of Random Number Generators Bob De Vivo Probability and Statistics Summer 2005

2. Early RNG’s • 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 1300’s • 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

3. Types of Modern RNG’s • “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 • RNG’s 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.

4. 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 n+1 digits of x02 Example: x0 = 157 1572 = 24649 x1 = 464 Disadvantage: very sensitive to choice of x0 ENIAC, completed in 1945

6. Modern Pseudo-RNG’s • 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: xn+1 = (xn)2 mod(M),where M is the product of two large primes

7. Mathematica’s 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