NDLI: A Zero-One Law for Boolean Privacy

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A Zero-One Law for Boolean Privacy

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kushilevitz, Eyal Chor, Benny
Copyright Year	1991
Abstract	A Boolean function $f:A_1 \times A_2 \times \cdots \times A_n \to \{ 0,1 \}$ is t-private if there exists a protocol for computing f so that no coalition of size $\leqq t$ can infer any additional information from the execution, other than the value of the function. It is shown that f is $\lceil n/2 \rceil $-private if and only if it can be represented as \[ f ( x_1 ,x_2 , \cdots ,x_n ) = f_1 ( x_1 ) \oplus f_2 ( x_2 ) \oplus \cdots \oplus f_n ( x_n ), \] where the $f_i $ are arbitrary Boolean functions. It follows that if f is $\lceil n/2 \rceil $-private, then it is also n-private. Combining this with a result of Ben-Or, Goldwasser, and Wigderson, and of Chaum, Crepeau, and Damgard, [Proc. 20th Symposium on Theory of Computing, 1988, pp. 110 and pp. 1119] an interesting zero-one law for private distributed computation of Boolean functions is derived: every Boolean function defined over a finite domain is either n-private, or it is $\lfloor ( n - 1 )/2 \rfloor $-private but not $\lceil n/2 \rceil $-private.A weaker notion of privacy is also investigated, where (a) coalitions are allowed to infer a limited amount of additional information, and (b) there is a probability of error in the final output of the protocol. It is shown that the same characterization of $\lceil n/2 \rceil $-private Boolean functions holds, even under these weaker requirements.In particular, this implies that for Boolean functions, the strong and the weak notions of privacy are equivalent.
Starting Page	36
Ending Page	47
Page Count	12
File Format	PDF
ISSN	08954801
DOI	10.1137/0404004
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	4
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-08
Access Restriction	Subscribed
Subject Keyword	Boolean functions Combinatorics Cryptography private distributed computations Information theory, general
Content Type	Text
Resource Type	Article
Subject	Mathematics

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