Derandomized Constructions of k-Wise (Almost) Independent Permutations

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 .