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

A family of permutations in S n is k-wise independent if a uniform permutation chosen from the family maps any distinct k elements to any distinct k elements equally likely. Efficient constructions of k-wise independent permutations are known for k = 2 and k = 3, but are unknown for k ≥ 4. In fact, it is known that there are no nontrivial subgroups of S n for n ≥ 25 which are 4-wise independent. Faced with this adversity, research has turned towards constructing almost k-wise independent families, where small errors are allowed. Optimal constructions of almost k-wise independent families of permutations were achieved by several authors.

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