Simple Permutations Mix Well

We study the random composition of a small family of O(n3 ) simple permutations on {0, 1}n . Specifically we ask what is the number of compositions needed to achieve a permutation that is close to k-wise independent. We improve on a result of Gowers [7] and show that up to a polylogarithmic factor, n3 k3 compositions of random permutations from this family suffice. Additionally, we introduce a new notion analogous to closeness to k-wise independence against adaptive adversaries and show the constructed permutation has the stronger property. This question is essentially about the rapid mixing of the random walk on a certain graph which we establish using a new approach to construct the so called canonical paths, which may be of independent interest. We also show that if we are willing to use a much larger family of simple permutations then we can guaranty closeness to k-wise independence with fewer compositions and fewer random bits.

• Type: Conference
• Year: 2004
• Where: ICALP
• Authors: Shlomo Hoory, Avner Magen, Steven Myers, Charles Rackoff
• Keywords: ICALP 2004, Permutations, Random, Simple Permutations, Theoretical Computer Science