Generalized inversive congruential pseudorandom numbers

An approach to nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0,1) is the Inversive congruential generator with prime modulus. A generalization for arbitrary composite moduli denotes the number of positive integers less than m which are relatively prime to m.

Example

Let take m = 15 =

Theorem 1

Let , which implies the desired result.

Generalized Inversive Congruential Pseudorandom Numbers are well equidistributed in one dimension. A reliable theoretical approach for assessing their statistical independence properties is based on the discrepancy of s-tuples of pseudorandom numbers.

Discrepancy bounds of the GIC Generator

We use the notation according to the law of the iterated logarithm for discrepancies. In this sense, Generalized Inversive Congruential Pseudo-random Numbers model true random numbers very closely.
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