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# Linear Congruential Generators

Linear Congruential Generators or LCGs represent one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is easy to understand, and they are easily implemented and fast. It is, however, well known that the properties of this class of generator are far from ideal. If higher quality random numbers are needed, and sufficient memory is available (~ 2 KBytes), then the Mersenne Twister algorithm is a preferred choice.

LCGs are defined by the recurrence relation:

Vj+1 = A · Vj + B (mod M)

Where Vn is the sequence of random values and A, B and M are generator-specific constants. See modular arithmetic for an explanation of the "mod M" notation.

The period of a general LCG is at most M, and very often less than that. In addition, they tend to exhibit severe defects. For instance, if an LCG is used to choose points in an n-dimensional space, triples of points will lie on, at most, M1/n hyperplanes. This is due to serial correlation between successive values of the sequence Vn.

A further problem of LCGs is that the lower-order bits of the generated sequence may cycle with a far shorter period than the sequence as a whole, particularly if M is set to a power of 2.

Today, with the advent of the Mersenne Twister, which both runs faster than and generates higher-quality deviates than almost any LCG, only LCGs with M = 232 (or 264) make sense at all (these are the fastest-evaluated of all random number generators). Numerical Recipes in C advocates a generator of this form with:

A = 1664525, B = 1013904223, M = 232

Neither this, nor any other LCG should be used for applications where high-quality randomness is critical. For example, it is not suitable for a Monte Carlo simulation[?] because of the serial correlation (among other things). Nevertheless, LCGs may be the only option in some cases. For instance, in an embedded system, the amount of memory available is often very severely limited. Similarly, in an environment such as a video game console taking a small number of high-order bits of an LCG may well suffice.

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