Private Computation — k-Connected versus 1-Connected Networks
We study the role of connectivity of communication networks in private computations under information theoretic settings. It will be shown that some functions can be computed by private protocols even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be computed on non-2-connected networks.
Furthermore, a general technique for simulating private protocols on arbitrary networks will be presented. Using this technique every private protocol can be simulated on arbitrary k-connected networks using only a small number of additional random bits.
Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks.
KeywordsBoolean Function Parity Function Input String Random String Communication Step
- 1.N. Alon, J. H. Spencer, and P. Erdös. The Probabilistic Method. John Wiley and Sons, 1992.Google Scholar
- 2.J. Bar-Ilan and D. Beaver. Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction. In Proc. 8th Ann. Symp. on Principles of Distributed Comput. (PODC), pages 201–209. ACM, 1989.Google Scholar
- 3.M. Ben-Or, S. Goldwasser, and A. Wigderson. Completeness theorems for noncryptographic fault-tolerant distributed computation. In Proc. 20th Ann. Symp. on Theory of Comput. (STOC), pages 1–10. ACM, 1988.Google Scholar
- 5.R. Canetti and R. Ostrovsky. Secure computation with honest-looking parties: What if nobody is truly honest? In Proc. 31st Ann. Symp. on Theory of Comput. (STOC), pages 255–264. ACM, 1999.Google Scholar
- 6.D. Chaum, C. Crépeau, and I. Damgøard. Multiparty unconditionally secure protocols. In Proc. 20th Ann. Symp. on Theory of Comput. (STOC), pages 11–19. ACM, 1988.Google Scholar
- 13.M. Franklin and M. Yung. Secure hypergraphs: Privacy from partial broadcast. In Proc. 27th Ann. Symp. on Theory of Comput. (STOC), pages 36–44. ACM, 1995.Google Scholar
- 14.A. Gál and A. Rosén. A theorem on sensitivity and applications in private computation. In Proc. 31st Ann. Symp. on Theory of Comput. (STOC), pages 348–357. ACM, 1999.Google Scholar
- 15.O. Goldreich, S. Micali, and A. Wigderson. How to play any mental game or a completeness theorem for protocols with honest majority. In Proc. 19th Ann. Symp. on Theory of Comput. (STOC), pages 218–229. ACM, 1987.Google Scholar
- 21.A. C.-C. Yao. Protocols for secure computations. In Proc. 23rd Ann. Symp. on Foundations of Comput. Sci. (FOCS), pages 160–164. IEEE, 1982.Google Scholar
- 22.A. C.-C. Yao. How to generate and exchange secrets. In Proc. 27th Ann. Symp. on Foundations of Comput. Sci. (FOCS), pages 162–167. IEEE, 1986.Google Scholar