Abstract
Constructions of k-wise almost independent permutations have been receiving a growing amount of attention in recent years. However, unlike the case of k-wise independent functions, the size of previously constructed families of such permutations is far from optimal. This paper gives a new method for reducing the size of families given by previous constructions. Our method relies on pseudorandom generators for space-bounded computations. In fact, all we need is a generator, that produces “pseudorandom walks” on undirected graphs with a consistent labelling. One such generator is implied by Reingold’s log-space algorithm for undirected connectivity [21,22]. We obtain families of k-wise almost independent permutations, with an optimal description length, up to a constant factor. More precisely, if the distance from uniform for any k tuple should be at most δ, then the size of the description of a permutation in the family is \(O(kn +\log \frac 1 {\delta})\).
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Kaplan, E., Naor, M., Reingold, O. (2005). Derandomized Constructions of k-Wise (Almost) Independent Permutations. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_30
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DOI: https://doi.org/10.1007/11538462_30
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