Almost k-Wise vs. k-Wise Independent Permutations, and Uniformity for General Group Actions

A family of permutations in Sn is k-wise independent if a uniform permutation chosen from the family maps any sequence of k distinct elements to any sequence of k distinct elements with equal probability. Efficient constructions of k-wise independent permutations are known for k = 2 and k = 3 based on multiply transitive permutation groups but are unknown for k≥ 4. In fact, it is known that there are no nontrivial subgroups of Sn for n≥ 25 which are 4-wise independent (“4-transitive”). Faced with this obstacle, research has turned towards constructing almost k-wise independent families, where small errors are allowed. Constructions of almost k-wise independent families of permutations, with optimal size up to polynomial factors, have been achieved by several authors. Motivated by this problem, we give several results relating almost k-wise and k-wise distributions over permutations. ∗An earlier version of this paper appeared in the Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM ’12), pages 350–361, 2012. †Supported in part by an ERC advanced grant and by NSF grant DMS-0835373. ‡Supported by NSF grant DMS-0835373. ACM Classification: G.3 AMS Classification: 68W20,68Q25