Technical Report CS0859

Authors: E. Kushilevitz and Y. Mansour
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Abstract: We consider the amount of randomness used in {\em private} computations. Specifically, we show how to compute the exclusive-or (xor) of $n$ boolean inputs $t$-privately, using only $O(t^2 \log(n/t))$ 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 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.
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