A family of functions F that map [0; n] 7! [0; n], is said to be h-wise independent if any h points in [0; n] have an image, for randomly selected f 2 F , that is uniformly distributed. This paper gives both probabilistic and explicit randomized constructions of n ffl -wise independent functions, ffl ! 1, that can be evaluated in constant time for the standard random access model of computation. Simple extensions give comparable behavior for larger domains. As a consequence, many probabilistic algorithms can for the first time be shown to achieve their expected asymptotic performance for a feasible model of computation. This paper also establishes a tight tradeoff in the number of random seeds that must be precomputed for a random function that runs in time T and is h-wise independent. Categories and Subject Descriptors: E.2 [Data Storage Representation]: Hash-table representation; F.1.2 [Modes of Computation]: Probabilistic Computation; F2.3 [Tradepffs among Computational Measures]...