NDLI: Randomness in Private Computations

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Randomness in Private Computations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kushilevitz, Eyal Mansour, Yishay
Copyright Year	1997
Abstract	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.
Starting Page	647
Ending Page	661
Page Count	15
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480196306130
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	10
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-01
Access Restriction	Subscribed
Subject Keyword	small probability spaces privacy randomness Network design and communication Combinatorics Cryptography xor function
Content Type	Text
Resource Type	Article
Subject	Mathematics

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