Algebraic methods for interactive proof systems

5 min read Original article ↗

Abstract

A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].

Formats available

You can view the full content in the following formats:

References

[1]

~AIELLO, W., GOLDWASSER, S., AND H~xSTAD, J. On the power of interaction. Combinatorica, ~10, 1 (1992), 3-25.

[2]

~BAUAI, L. Trading group theory for randomness. In Proceedings' of the 17th Annual ACM ~Symposium on the Theory of Computing (Providence, R.I., May 6 8). ACM, New York, 1985, ~pp. 421-429.

[3]

~BABAI, L., AND FORTNOW, L. Arithmetization: a new method in structural complexity theory. ~Computat. Complex. 1 (1991), 41-66.

[4]

~BABAI, L., FORTNOW, L. AND LUND, C. Non-deterministic exponential time has two-prover ~interactive protocols. Comp,tat. Complex. I (1991), 3-40.

[5]

~BABAI, L., AND MORAN, S. Arthur-Merlin games: A randomized proof system, and a ~hierarchy of complexity classes. J. Co,zput. Syst. Sci. 36 2 (1988), 254-276.

[6]

~BEAVER, D. AND FEIGENBAUM, J. Hiding instances in multioracle queries. In Proceedings of ~the 7th Symposium on the Theoretical Aspects of Computer Sctence. Lecture Notes in Computer ~Science, vol. 415. Springer Verlag, New York, 1990, pp. 37-48.

[7]

~BEN-OR, M., GOLDREICH, O., GOLDWASSER, S., H~STAD, J., KILIAN, J., MICAL1, S., AND ~ROGAWAY, P. Everything provable is provable in zero-knowledge. In Proceedings of Crypto ~88. Lecture notes in Computer Science, vol. 403. Springer-Verlag, New York, 1988, pp. 37-56.

[8]

~BEN-OR, M., GOLDWASSER, S., KILIAN, J., AND WIGDERSON, m. Multi-prover interactive ~proofs: How to remove intractability assumptions. In Proceedings' of the 20th Annual ACM ~Symposium on the Theory of Computing (Chicago, Ill, May 2-4). ACM, New York, 1988, pp. ~113-131.

[9]

~BLUM, M., AND KANNAN, S. Designing programs that check their work. In Proceedings of the ~21th Annual ACM Symposittm on the Theory of Computing (Seattle, Wash., May 15-17). ACM. ~New York, 1989. pp. 86-97.

[10]

~BLUM. m., LUBY, M., AND RUBINFELD, R. Self-testing correcting with applications to numeri- ~cal problems. In Proceedtngs of t/re 22nd Annual A CM Symposium on the Theory of Comt)ttting ~(Baltimore, Md., May 14-16). ACM, New York, 1990, pp. 73-83.

[11]

~BOPPANA, R., H&STAD, J., AND ZACHOS, $. Does co-NP have short interactive proofs? b~f ~Proc. Lett. 25, 2 (1987). 127-132.

[12]

~CAt, J., CONDON, A., AND LIPTON, R.J. PSPACE is provable by two provers in one round. In ~Proceedings of the 6th Annual Conference on Structure in Complexity Theory (Chicago, Ill., June ~30-July 3). IEEE, New York, 1991, pp. 110-115.

[13]

~CHOR, B., GOLDREICH, O., AND H,~STAD, J. The random oracle hypothesis is false. ~Manuscript, Technion, Haifa, Israel, 1990.

[14]

~COOK, S. A. The complexity of theorem-proving procedures. In Proceedzngs of the 3rd ~Annual ACM Sympostum on the Theory of Computing (Shaker Heights, Oh., May 3-5). ACM, ~New York, 1971, pp. 151 158.

[15]

~FELDMAN, P. The optimum prover lives in PSPACE. Manuscript. M.I.T., Cambridge, Mass., ~1986.

[16]

~FORTNOW, L., AND LUND, C. Interactive proof systems and alternating time-space complex- ~ity. In Proceedings' of the 8th Symposium on Theoretzcal Aspects of Computer Science Lecture ~Notes in Computer Science, vol. 480, Sprmger-Verlag, New York, 1991, pp. 263 274.

[17]

~FORTNOW, L., ROMPEL, J., AND SIPSER, M. On the power of multi-prover interactwe ~protocols. In Proceedings of the 3rd Con}erence on Structure zn Complexity Theory (Washington, ~D.C., June 14-17). IEEE, New York, 1988, pp. 156 161.

[18]

~FORTNOW, L., AND SIPSER, M. Are there interactive protocols for co-NP languages? inf. ~Proc. Lett. 28 (1988), 249-251.

[19]

~FURER, M., GOLDREICH, O., MANSOUR, Y., SIPSER, M., AND ZACHOS, S. On completeness ~and soundness in interactive proof systems. In S. Micali, ed. Randomness and Computation, ~(volume 5 of Advances m Compzmng Research). JAI Press, Greenwich, Conn. 1989, pp. ~429-442.

[20]

~GOLDREICH, O., MICALI, S., AND WiGDERSON, A. Proofs that yield nothing but their validity ~and a methodology of cryptographic protocol design. In Proceedings of the 27th IEEE ~Symposium on Foundations of Computer Science. IEEE, New York, 1986, pp. 174 187.

[21]

~GOLDWASSER, S., MICAL1, S., AND RACKOFF, C. The knowledge complexity of interactive ~proof-systems. SIAMJ. Comput. 18, 1 (1989), 186-208.

[22]

~GOLDWASSER, S. AND SIPSER, M. Private coins versus public coins in interactive proof ~systems. In S. Micali, ed. Randomness and Compzttatzon, (volume 5 of AdL,ances bz Computing ~Research). JAI Press, Greenwich, Conn. 1989, pp. 73-90.

[23]

~IMPAGLIAZZO, R., AND YUNG, M. Direct minimum-knowledge computation. In Proceeding3 ~of Crypto 87. Lecture Notes in Computer Science, vol. 293, Springer-Verlag, New York, 1987, ~pp. 40-51.

[24]

~KARP, R., AND LIPTON, R. Some connection between nonuniform and uniform complexity ~classes. In Proceedings of the 12th Annual ACM Synzposzum on the Theory of Computzng (Los ~Angeles, Calif., Apr. 28-30). ACM, New York, 1980, pp. 302 309.

[25]

~LIPTON, R. New directions in testing. In J. Feigenbaum and M. Merntt, eds. Dtstributed ~Computing and Cr37~tography (volume 2 of DIMACS Se~es in Dtscrete Mathematics and ~Theoretical Computer Science). American Mathematical Society, Providence, R.I., 1991, pp. ~191-202.

[26]

~NIVEN, I., AND ZUCKERMAN, H. S. An tntroductzon to the theory of numbers 4th ed., Wiley, ~New York, 1980, pp. 224-225.

[27]

~PRATt, V. Every prime has a succinct certificate. SIAMJ. Comput. 4 (1975), 214-220.

[28]

~SHAMIR, h. IP - PSPACE. J. ACM 39, 4 (Oct. 1992), 869-877.

[29]

~SHEN, A. IP PSPACE: Simplified proof. J ACM 39, 4 (Oct. 1992), 878-880.

[30]

~StMOr% J. On some central problems in computational complexity. PhD thesis, Cornell ~University, Computer Science, 1975. Tech Report TR 75-224.

[31]

~SOLOVAY, R., AND STRASSEN, V. h fast Monte-Carlo test for primality. SIAM Jr. Conzput. 6 ~(1977), 84-85. See also erratum 7 (1978), I18.

[32]

~TODA, S. On the computational power of PP and ~ P. In Proceedings of the 30th iEEE ~Symposntm on Foundatzons of Computer Science. IEEE, New York, 1989, pp. 514-519.

[33]

~VALIANT, L. The complexity of computing the permanent. Theoret. Conzput. Scz 8 (1979), ~189-201.