# Threshold computation and cryptographic security

## Abstract

Threshold machines [21] are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines [11] are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. Simon [21] proved that for unbounded-error polynomial-time machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored.

In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class, BPP_{path}, and, more generally, we study the structural properties of BPP_{path}. We prove that BPP_{path} contains both NP^{BPP} and P^{NP[log]}, and that BPP_{path} is contained in \(P^{\Sigma _2^p [\log ]}\), BPP^{NP}, and PP. We conclude that, unless the polynomial hierarchy collapses, bounded-error threshold computation is strictly more powerful than bounded-error probabilistic computation.

We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length [5]. We show, for both BPP and BPP_{path}, that if there is *any* database for which this formalization of security differs from the security given by oblivious [9] database access, then BPP≠PP. It follows that if any set lacking small circuits can be securely accepted, then BPP≠PP.

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## References

- 1.M. Abadi, J. Feigenbaum, and J. Kilian. On hiding information from an oracle.
*Journal of Computer and System Sciences*, 39:21–50, 1989.Google Scholar - 2.L. Babai. Trading group theory for randomness. In
*Proceedings of the 17th ACM Symposium on Theory of Computing*, pages 421–429, April 1985.Google Scholar - 3.T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question.
*SIAM Journal on Computing*, 4(4):431–442, 1975.CrossRefGoogle Scholar - 4.J. Balcázar, R. Book, and U. Schöning. The polynomial-time hierarchy and sparse oracles.
*Journal of the ACM*, 33(3):603–617, 1986.CrossRefGoogle Scholar - 5.D. Beaver and J. Feigenbaum. Hiding instances in multioracle queries. In
*Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science*, pages 37–48. Springer-Verlag*Lecture Notes in Computer Science #415*, 1990.Google Scholar - 6.R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In
*Proceedings of the 7th Structure in Complexity Theory Conference*, pages 14–19. IEEE Computer Society Press, June 1992.Google Scholar - 7.R. Book. Restricted relativizations of complexity classes. In J. Hartmanis, editor,
*Computational Complexity Theory*, Proceedings of Symposia in Applied Mathematics #38, pages 47–74. American Mathematical Society, 1989.Google Scholar - 8.R. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs?
*Information Processing Letters*, 25:127–132, 1987.CrossRefGoogle Scholar - 9.J. Feigenbaum, L. Fortnow, C. Lund, and D. Spielman. The power of adaptiveness and additional queries in random-self-reductions. In
*Proceedings of the 7th Structure in Complexity Theory Conference*, pages 338–346. IEEE Computer Society Press, June 1992.Google Scholar - 10.S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. In
*Proceedings of the 6th Structure in Complexity Theory Conference*, pages 30–42. IEEE Computer Society Press, June/July 1991.Google Scholar - 11.J. Gill. Computational complexity of probabilistic Turing machines.
*SIAM Journal on Computing*, 6(4):675–695, 1977.CrossRefGoogle Scholar - 12.Y. Han, L. Hemachandra, and T. Thierauf. Threshold computation and cryptographic security. Technical Report TR-461, University of Rochester, Department of Computer Science, Rochester, NY, 1993.Google Scholar
- 13.J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP.
*Theoretical Computer Science*, 58:129–142, 1988.CrossRefGoogle Scholar - 14.R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In
*Proceedings of the 12th ACM Symposium on Theory of Computing*, pages 302–309, April 1980.Google Scholar - 15.K. Ko. Some observations on the probabilistic algorithms and NP-hard problems.
*Information Processing Letters*, 14(1):39–43, 1982.Google Scholar - 16.J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP.
*Journal of Computer and System Sciences*, 44(2):272–286, 1992.CrossRefGoogle Scholar - 17.R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities.
*Theoretical Computer Science*, 1(2):103–124, 1975.CrossRefGoogle Scholar - 18.T. Long and A. Selman. Relativizing complexity classes with sparse oracles.
*Journal of the ACM*, 33(3):618–627, 1986.CrossRefGoogle Scholar - 19.A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In
*Proceedings of the 13th IEEE Symposium on Switching and Automata Theory*, pages 125–129, 1972.Google Scholar - 20.M. Ogiwara and L. Hemachandra. A complexity theory for feasible closure properties. In
*Proceedings of the 6th Structure in Complexity Theory Conference*, pages 16–29. IEEE Computer Society Press, June/July 1991.Google Scholar - 21.J. Simon.
*On Some Central Problems in Computational Complexity*. PhD thesis, Cornell Univeristy, Ithaca, N.Y., January 1975. Available as Cornell Department of Computer Science Technical Report TR75-224.Google Scholar - 22.M. Sipser. A complexity theoretic approach to randomness. In
*Proceedings of the 15th ACM Symposium on Theory of Computing*, pages 330–335, 1983.Google Scholar - 23.L. Stockmeyer. The polynomial-time hierarchy.
*Theoretical Computer Science*, 3:1–22, 1977.CrossRefGoogle Scholar - 24.N. Vereshchagin. On the power of PP. In
*Proceedings of the 7th Structure in Complexity Theory Conference*, pages 138–143. IEEE Computer Society Press, June 1992.Google Scholar - 25.S. Zachos. Robustness of probabilistic complexity classes under definitional perturbations.
*Information and Control*, 54:143–154, 1982.CrossRefGoogle Scholar - 26.S. Zachos. Probabilistic quantifiers and games.
*Journal of Computer and System Sciences*, 36:433–451, 1988.CrossRefGoogle Scholar - 27.S. Zachos and H. Heller. A decisive characterization of BPP.
*Information and Control*, 69:125–135, 1986.CrossRefGoogle Scholar