History of Random Number Generators Bob De Vivo

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

Early RNG’s • Dice o o • Playing Cards o o • Used in China for gambling games as early as the 7 th century Introduced to Europe in the late 1300’s Coins o o • 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 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 o o 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

Types of Modern RNG’s • “True” random number generators o o o • 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 o o o 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.

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 Ø Start with x 0, n digits long Ø X 1 = middle n or n+1 digits of x 02 Example: x 0 = 1572 = 24649 x 1 = 464 Disadvantage: very sensitive to choice of x 0 ENIAC, completed in 1945

Modern Pseudo-RNG’s • Linear congruential generators – – – • Mersenne Twister – – – • 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 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 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

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