Sequences II pp 345-359 | Cite as

# Privacy of Dense Symmetric Functions

## Abstract

An *n* argument function, *f*, is called *t* — private if there exists a distributed protocol for computing *f*, so that no coalition of ≤ *t* processors can infer any additional information from the execution of the protocol. It is known that every function defined over a finite domain is \( \left[ {\frac{{n - 1}}{2}} \right] \)-private. The general question of *t* — privacy (for *t* ≥ \( \left[ {\frac{n}{2}} \right] \)) is still unresolved.

In this work we relate the question of \( \left[ {\frac{n}{2}} \right] \)-privacy for the class of *symmetric* functions of Boolean arguments *f*: {0,1}^{ n } → {0,1,..., *n*} to the structure of weights in *f* ^{-1}(*b*) (*b* ∈ {0,1,..., *n*}). We show that if *f* is \( \left[ {\frac{n}{2}} \right] \)-private, then every set of weights *f* ^{-1}(*b*) must be an *arithmetic progression*. For the class of *dense* symmetric functions (defined in the sequel), we refine this to the following necessary and sufficient condition for \( \left[ {\frac{n}{2}} \right] \) — privacy of *f*: Every collection of such arithmetic progressions must yield distinct remainders, when computed modulo the greatest common divisor of their differences. This condition is used to show that for dense symmetric functions, \( \left[ {\frac{n}{2}} \right] \)-privacy implies *n*-privacy.

## Keywords

Boolean Function Symmetric Function Arithmetic Progression Boolean Variable Great Common Divisor## Preview

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## References

- [1]Benor M., S. Goldwasser, and A. Wigderson, “Completeness Theorems for Non-Cryptographic Fault-Tolerant Distributed Computation”
*Proc. of 20th STOC*, 1988, pp. 1–10.Google Scholar - [2]Beaver, D., “Perfect Privacy for Two Party Protocols”, Technical Report TR-11–89, Harvard University, 1989.Google Scholar
- [3]Benaloh Cohen, J.D., “Secret Sharing Homomorphisms: Keeping Shares of a Secret Secret”,
*Advances in Cryptography — Crypto86 (proceedings)*, A.M. Odlyzko (ed.), Springer-Verlag, Lecture Notes in Computer Science, Vol. 263, pp. 251–260, 1987.Google Scholar - [4]Chaum, D., C. Crepeau, and I. Damgard, “Multiparty Unconditionally Secure Protocols”
*Proc. of 20th STOC*, 1988, pp. 11–19.Google Scholar - [5]Chor, B., M. Geréb-Graus, and E. Kushilevitz, “Private Computations Over the Integers”,
*31th IEEE Conference on the Foundations of Computer Science*, October 1990, pp. 335–344.Google Scholar - [6]Chor, B., and E. Kushilevitz, “A Zero-One Law for Boolean Privacy”,
*SIAM J. Discrete Math.*, Vol 4, No 1, 1991, pp. 36–47. Early version in*Proc. of 21th STOC*, 1989, pp. 62–72.MathSciNetMATHCrossRefGoogle Scholar - [7]Kushilevitz, E., “Privacy and Communication Complexity”,
*Proc. of 30th FOCS*, 1989, pp. 416–421. To appear in*SIAM Jour. Disc. Math.*Google Scholar