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Characterizing Linear Size Circuits in Terms of Privacy

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Abstract

In this paper we prove a perhaps unexpected relationship between the complexity class of the boolean functions that have linear size circuits andn-party private protocols. Specifically, letfbe a boolean function. We show thatfhas a linear size circuit if and only iffhas a 1-privaten-party protocol in which the total number of random bits used byallplayers is constant. From the point of view of complexity theory, our result gives a characterization of the class of linear size circuits in terms of another class of a very different nature. From the point of view of privacy, this result provides 1-private protocols that use a constant number of random bits, for many important functions for which no such protocol was previously known. On the other hand, our result suggests that proving, for anyNPfunction, that it has no 1-private constant-random protocol, might be difficult.

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An early version of this paper appeared in the “Proceedings of the 28thACM Symp. on the Theory of Computing (STOC), May, 1996,” pp. 541–550.

Part of this research was done while visiting ICSI Berkeley. Supported by MANLAM Fund 120-857 and by the Fund for Promotion of Research at the Technion.

Part of this research was done while with the Dept. of Computer Science, Tel-Aviv University, Tel-Aviv, Israel.

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E-mail: [email protected]

f2

E-mail: [email protected]

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E-mail: [email protected]