, Volume 55, Issue 3-4, pp 455-465

The sub-lattice structure of linear congruential random number generators

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Abstract

Sequences of pseudo-random numbers are discussed which are generated by the linear congruential method where the period is equal to the modulus m. Such sequences are divided into non-overlapping vectors with n components. In this way for each initial number exactly m/gcd(n, m) different vectors are obtained. It is shown that the periodic continuation (with period m) of these vectors forms a grid which is a sub-grid of the familiar grid generated by all m overlapping vectors. A sub-lattice structure also exists for certain multiplicative congruential generators which are often used in practice.