Derandomization of Families of Entropy-Preserving Functions

In probabilistic computations, random bits are viewed as a resource. An important goal is to consume as little of it as possible. We study functions, which decrease the number of random bits needed by an algorithms. Loss-less condensers take as input a low entropy source, and output a source, which is statistically close to a high entropy source. This closeness is referred to as the error of the condenser. We show a general error reduction for loss-less condensers, building on the error reduction technique of Raz, Reingold and Vadhan [42]. Families of k-wise almost independent permutations, permute their input in a way that looks random, to any party, which inspects only k values of the output. We give a new method for reducing the number of random bits needed to sample a permutation from such a family. Our method relies on a pseudorandom walk generator, implied by Reingold’s log-space algorithm for undirected connectivity [45, 47]. We obtain families of k-wise almost independent permutations, with an optimal number of random bits.