Randomness versus fault-tolerance

We investigate the relations between two major properties of multiparty protocols: fault tolerance (or resilience) and randomness. Fault-tolerance is measured in terms of the maximum number of colluding faulty parties, t, that a protocol can withstand and still maintain the privacy of the inputs and the correctness of the outputs (of the honest parties). Randomness is measured in terms of the total number of random bits needed by the parties in order to execute the protocol. Previously, the upper bound on the amount of randomness required by general constructions for securely computing any nontrivial function f was polynomial both in n, the total number of parties, and the circuit-size C (f). This was the state of knowledge even for the special case t = 1 (i.e., when there is at most one faulty party). In this pa per we show that for any linear-size circuit, and for any number t < n/3 of faulty parties, O (poly(t).log n) randomness is sufficient. More generally, we show that, for any function f with circuit-size C(f), we need only O (poly(t).log n + poly(t).(C(f)/n)) randomness in order to withstand any coalition of size at most t. Furthermore, in our protocol only t + 1 parties flip coins and the rest of the parties are deterministic. Our results generalize to the case of adaptive adversaries as well.