Abstract
We study the random composition of a small family of O(n 3) 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 [8] and show that up to a polylogarithmic factor, n 3 k 3 compositions of random permutations from this family suffice. We further show that the result applies to the stronger notion of k-wise independence against adaptive adversaries. This question is essentially about the rapid mixing of the random walk on a certain graph, and we approach it using a new technique to construct canonical paths. 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.
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Hoory, S., Magen, A., Myers, S., Rackoff, C. (2004). Simple Permutations Mix Well. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_65
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DOI: https://doi.org/10.1007/978-3-540-27836-8_65
Publisher Name: Springer, Berlin, Heidelberg
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