## 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 1/δ).

Original language | English |
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Pages (from-to) | 113-133 |

Number of pages | 21 |

Journal | Algorithmica |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2009 |

## Keywords

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