congruentiegenerator

The congruentiegenerator is a class of pseudo-random number generators based on modular arithmetic. It generates sequences of integers by repeatedly applying a linear recurrence modulo a fixed modulus. In its simplest form, the generator uses the relation x_{n+1} = (a x_n + c) mod m, with m > 1, and parameters a, c chosen to influence the period and distribution of the output. The produced integers typically lie in the range [0, m-1] and can be scaled to other ranges as needed.

The concept rests on the idea of congruence in modular arithmetic. The sequence’s quality and period depend on the choice of parameters. A full-period generator visits every integer in [0, m-1] before repeating. The Hull-Dobell theorem gives necessary and sufficient conditions for a full period: c and m must be relatively prime; a-1 must be a multiple of every prime factor of m; and if m is divisible by 4, then a-1 must also be a multiple of 4. These conditions guide parameter selection for a robust congruentiegenerator.

Variants include the multiplicative congruentiegenerator, where c is zero, and the mixed (additive) version with c nonzero. Some generators employ more complex schemes or combine multiple congruentiegenerators to improve statistical properties, though modern practice often favors other RNG families for critical uses.

Applications typically include simulations and statistical sampling where performance is valued over cryptographic security. Limitations include potential correlations and dimensional defects, especially in higher-dimensional outputs, making congruentiegenerators less suitable for high-stakes randomness requirements.

See also: pseudo-random number generator, linear congruential generator, modular arithmetic.