For random number generators on parallel computers, it is vital that there are no correlations between the random number streams on different processors. For example, we don't want one processor repeating part of another processors sequence.
This could occur if we just use the naive method of running a LCG on each different processor and just giving randomly chosen seeds to each processor.
In many applications we also need to ensure that we get the same results for any number of processors. This is certainly the case for programs written in High Performance Fortran (HPF).