Pseudorandom Number Generators

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.

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.

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.

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.

Pseudorandom Number Generators Different PRNG’s approximate different properties of random numbers, and desirable properties vary with application. Therefore, different PRNG’s 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.

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

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.

Middle-Square Method - Example Let’s 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…

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.)

Middle-Square Method - Example Let’s try to generate numbers starting with 7600. 7600^2= 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.

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 PRNG’s.

Linear Congruence Method The method uses the following formula: Xn+1 = (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.)

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.

Linear Congruence Method – Choosing Parameters Xn+1 = (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.

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.

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.

Recent PRNG’s – 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 PRNG’s. 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 2^19937-1. This number is a Mersenne prime. It would probably take longer to cycle than the entire future existence of humanity (and, perhaps, the universe.)

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.

Summary PRNG’s are algorithms that produce sequences of numbers that simulate randomness. PRNG’s are useful in game design, cryptography, and statistical modeling. Different PRNG’s 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 (n+1)th term. The linear congruence method is defined by the recursive formula Xn+1 = (a * Xn + b) mod c

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