# Technical Report CS0859

 TR#: CS0859 Class: CS Title: RANDOMNESS IN PRIVATE COMPUTATIONS. Authors: E. Kushilevitz and Y. Mansour PDF Not Available 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. Copyright The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information

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