Private Computations in Networks: Topology versus Randomness
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Private Computation - k-Connected versus 1-Connected Networks
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
Lower bounds on the amount of randomness in private computation
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On the randomness complexity of efficient sampling
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
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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(t2 log (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 (i.e., a probability distribution over {0,1}n) 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 [Sample spaces uniform on neighborhoods, in Proc. of the 24th Annual ACM Symposium on Theory of Computing, ACM, New York, 1992, pp. 17--25]). We believe that this construction is of independent interest and may have various applications.