Fast pseudorandomness for independence and load balancing

We provide new constructions of several fundamental pseudorandom objects. Loosely speaking, these constructions obtain exponential improvements in efficiency compared to previous constructions with comparable randomness complexity. Our measure of efficiency is the number of word operations, as captured by the well-established unit-cost word RAM model. Our main results are the following: 1. A family of (1/n)-almost logn-wise independent Boolean hash functions with O(logn) description length (or seed length) and O(loglogn) operations per evaluation. Prior constructions with similar seed lengths required Θ(logn) operations. 2. ε-biased sequences for ε = 1/poly(n) with seed length O(logn loglogn) and O((loglogn)²) operations (to evaluate an output bit or a block of up to logn consecutive bits). Prior constructions achieve O(logn) seed length, but require Θ(logn) operations. This construction implies pseudorandom generators with similar efficiency that fool classes such as low-degree polynomials and read-once CNFs. 3. Hash functions for placing n balls in n bins such that with all but probability 1/n the maximal load is O(logn/loglogn) (which is optimal), with seed-length O(logn loglogn) and O((loglogn)²) operations per evaluation. The previously known construction with similar seed length required Θ(logn loglogn) operations. Indeed, our construction is an efficient instantiation of that construction, due to Celis, Reingold, Segev and Wieder (FOCS 2011). These constructions are all simultaneously within loglogn factors of the optimal seed length, and within (loglogn)² factors of the optimal computational efficiency.