If we want to implement ICG or cICG, we shall need pairs *a*,*b* of parameters such that ICG(*p*,*a*,*b*) will have period *p*. As we have pointed out in Section 3, the polynomial will have to be an IMP polynomial.

We would have to apply Chou's algorithm every time we need a different ICG, even if the modulus *p* remains constant. This is the common situation with pseudorandom number generator s. For example, in the case of the LCG, we would have to carry out complex computations with the spectral test to determine new parameters. This is a task for specialists. Again, inversive methods are different. There is a new approach that allows us to implement many ``descendants'' from one single ICG with maximal period.

For every ``mother'' ICG(*p*,*a*,1) with period *p*, every ``son'' ICG() will have maximal period *p*, provided we choose in , see Eichenauer-Herrmann and Emmerich (1994). As we have seen in Section 3, all these ICG will have the same excellent theoretical properties. Hundreds of empirical tests provide strong evidence that this extraordinary fact is also true for the performance of ICG in empirical tests.

We present four tables of mother ICG for small prime moduli *p*. These parameters allow the implementation of compound ICG with three components on 32-bit architectures. The last two tables exhibit families of ICG, one mother and five sons each, where each son has a multiplier below Such multipliers are preferable on 32-bit processors for reasons of computational efficiency of the modular inversion involved.

Peter Hellekalek