# Random access random permutations

I have a large number of parallel processes and a large integer $$n$$, and want to randomly partition the integers $$[0,n)$$ among the processes with only $$O(1)$$ communication.

One nice way to do this would to generate a pseudorandom permutation $$\pi \in S_n$$ represented as a small function, so that only the random key/seed need be exchanged. Is there a good way to do this?

• I'm about to answer this myself. I was about to answer it on stackoverflow, but it was closed as "not a question": stackoverflow.com/questions/14077953/… – Geoffrey Irving Dec 30 '12 at 5:39
• Totally cool. I think it's an interesting question. – Geoff Oxberry Dec 30 '12 at 7:49
• One fun bit is that the inverse of the permutation gives which process owns a given index. – Geoffrey Irving Dec 30 '12 at 8:36

Pick $2^k$ slightly larger than $n$, generate a block cypher $f \in S_{2^k}$ operating on $k$ bit blocks, and construct a permutation on $[0,n)$ by walking along cycles of $f$ until we get back in the desired range. Specifically, given $x < n$ we set $$g(x) = f^p(x) = f(f(f(...x...)))$$ where $p$ is the least positive integer s.t. $f^p(x) < n$.
If $2^k = O(n)$, and the block cypher is good, the walk takes $O(1)$ expected time. Note that $p$ is necessarily finite, since eventually we will walk back around the cycle and find $f^p(x) = x$.