Randomness in private computations

We consider the amount of randomness used in private distributed computations. Specifically, we show how n players can compute the exclusive-or (xor) of n boolean inputs t-privately, using only O(t²log(n/t)) random bits (the best known upper bound is O(tn)). We accompany this result by a lower bound on the number of random bits required to carry out this task; we show that any protocol solving this problem requires at least t random bits (again, this significantly improves over the known lower bounds). For the upper bound, we show how, given m subsets of {1, ..., n}, to construct in (deterministic) polynomial time a probability distribution of n random variables such that (1) the parity of random-variables in each of these m subsets is 0 or 1 with equal probability; and (2) the support of the distribution is of size at most 2m. This construction generalizes previously considered types of sample spaces (such as k-wise independent spaces and Schulman's spaces [S92]). We believe that this construction is of independent interest and may have various applications.