A Randomness-Rounds Tradeoff in Private Computation

We study the role of randomness in multi-party private computations. In particular, we give several results that prove the existence of a randomness-rounds tradeoff in multi-party private computation of xor. We show that with a single random bit, Θ(n) rounds are necessary and sufficient to privately compute xor of n input bits. With d ≥ 2 random bits, Ω(log n/d) rounds are necessary, and O(log n/log d) are sufficient.More generally, we show that the private computation of a boolean function f, using d ≥ 2 random bits, requires Ω(log S(f)/d) rounds, where S(f) is the sensitivity of f. Using a single random bit, Ω(S(f)) rounds are necessary.