Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs

Let f (xl, .. . xk) be a Boolean function that k parties wish to collaboratively evaluate, where each xi is a bit-string of length n. The ith party knows each input argument except x,; and each party has unlimited computational power. They share a blackboard, viewed by all parties, where they can exchange messages. The objective is to minimize the number of bits written on the board. We prove lower bounds of the form Q(n c?), for the number of bits that need to be exchanged in order to compute some (explicitly given) polynomial time computable functions. Our bounds hold even if the parties only wish to have a 1% advantage at guessing the value off on random inputs. The lower bound proofs are based on discrepancy upper bounds for specilic functions over “cylinder intersection” sets. These results may be of independent interest. We give several applications of the lower bounds. The tirst application is a pseudorandom generator for Logspace. We explicitly construct (in polynomial time) pseudorandom sequences of length n from a random seed of length exp(c &) that no Logspace Turing machine will be able to distinguish from truly random sequences. As a corollary we give an explicit construction of a universal traversal sequence of length