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Derandomized Constructions of k-Wise (Almost) Independent Permutations

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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 (Reingold/Reingold et al. in Proc. of the 37th/38th Annual Symposium on Theory of Computing, pp. 376–385/457–466, 2005/2006). 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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Author information

Correspondence to Omer Reingold.

Additional information

A preliminary version of this paper appeared in Random 2005 [19].

The research of M. Naor was supported in part by a grant from the Israel Science Foundation. The research of O. Reingold was supported by US–Israel Binational Science Foundation Grants 2002246 and 2006060.

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Kaplan, E., Naor, M. & Reingold, O. Derandomized Constructions of k-Wise (Almost) Independent Permutations. Algorithmica 55, 113–133 (2009). https://doi.org/10.1007/s00453-008-9267-y

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Keywords

  • Pseudo-randomness
  • Card shuffling
  • Block ciphers
  • Random walk
  • Connectivity