We consider the classical problem of generating random permutations with the uniform distribution. That is, we require that for an arbitrary permutation
of
n elements, with probability 1/n! the machine halts with the
ith output cell containing
(i), for 1
in. We study this problem on two models of parallel computations: the CREW PRAM and the EREW PRAM.
The main result of the paper is an algorithm for generating random permutations that runs in O(log log n) time and uses O(n
1+o(1)) processors on the CREW PRAM. This is the first o(log n)-time CREW PRAM algorithm for this problem.
On the EREW PRAM we present a simple algorithm that generates a random permutation in time O(log n) using n processors and O(n) space. This algorithm matches the running time and the number of processors used of the best previously known algorithms for the CREW PRAM, and performs better as far as the memory usage is considered.
The common and novel feature of both our algorithms is to design first a suitable random network generating a permutation and then to simulate this network on the PRAM model in a fast way.
Partially supported by KBN grant 8 S503 002 07, TEMPUS project JEP 8145, EU ESPRIT Long Term Research Project 20244 (ALCOM-IT), DFG-Sonderforschungsbereich 376 
Massive Parallelität
and DFG Leibniz Grant Me872/6-1.