Abstract
Under the assumption that encryption functions exist, we show that all languages in NP possess zero-knowledge proofs. That is, it is possible to demonstrate that a CNF formula is satisfiable without revealing any other property of the formula. In particular, without yielding neither a satisfying assignment nor weaker properties such as whether there is a satisfying assignment in which x 1 = TRUE, or whether there is a satisfying assignment in which x 1 = x 3 etc.
The above result allows us to prove two fundamental theorems in the field of (two-party and multi-party) cryptographic protocols. These theorems yield automatic and efficient transformations that, given a protocol that is correct with respect to an extremely weak adversary, output a protocol correct in the most adversarial scenario. Thus, these theorems imply powerful methodologies for developing two-party and multi-party cryptographic protocols.
Work done while the first author was in the Laboratory for Computer Science, MIT, partially sup ported by an IBM Postdoctoral Fellowship, and NSF Grant DCR-8509905 The second author was supported by NSF Grant DCR-8413577 and an LBM Faculty Development Award Work done while the third author was in Mathematical Sciences Research Institute, Berkeley
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References
Aho, A.V., J.E. Hopcroft, and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley Publ. Co., 1974.
Alexi, W., B. Chor, O. Goldreich, and C.P. Schnorr, “RSA and Rabin Functions: Certain Parts Are As Hard As The Whole”, to appear in SIAM Jour. on Computing. Extended Abstract in Proc. 25th FOCS, 1984.
Alon, N., Z. Galil, and M. Yung, “A Fully Polynomial Simultaneous Broadcast in the Presence of Faults”, preprint, 1985.
Babai, L., “Trading Group Theory for Randomness”, Proc. 17th STOC, 1985, pp. 421–429.
Benaloh, (Cohen) J.D., “Secret Sharing Homomorphisms: Keeping Shares of a Secret Secret”, these proceedings.
Blum, M., “Coin Flipping by Phone”, IEEE Spring COMPCOM, pp. 133–137, February 1982.
Blum, M., and Micali, S., “How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits”, SIAM Jour. on Computing, Vol. 13, 1984, pp. 850–864.
Brassard, G., and C. Crepeau, “Zero-Knowledge Simulation of Boolean Circuits”, manuscript 1986.
Brassard, G., and C. Crepeau, “Non-Transitive Transfer of Confidence: A Perfect Zero-Knowledge Interactive Protocol for SAT and Beyond”, manuscript, 1986.
Broder, A.Z., and D. Dolev, “Flipping Coins in Many Pockets (Byzantine Agreement on Uniformly Random Values”, Proc. 25th FOCS, 1984, pp. 157–170.
Chaum, D., “Demonstrating that a Public Predicate can be Satisfied Without Revealing Any Information About How”, these proceedings.
Chaum, D., J.H. Evertse, J. van de Graaf, and R. Peralta, “Demonstrating Possession of a Discrete Logarithm without Revealing It”, these proceedings.
Chor, B., O. Goldreich, and S. Goldwasser, “The Bit Security of Modular Squaring given Partial Factorization of the Modulos”, Proc. of Crypto85, to appear (1986).
Chor, B., S. Goldwasser, S. Micali, and B. Awerbuch, “Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults”, Proc. 26th FOCS, 1985, pp. 383–395.
Cook, S.A., “The Complexity of Theorem Proving Procedures”, Proc. 3rd STOC, pp. 151–158, 1971.
Diffie, W., and M.E. Hellman, “New Directions in Cryptography”, IEEE Trans. on Inform. Theory, Vol. IT-22, No. 6, November 1976, pp. 644–654.
Even, S., O. Goldreich, and A. Lempel, “A Randomized Protocol for Signing Contracts”, CACM, Vol. 28, No. 6, 1985, pp. 637–647.
Feldman, P., “A Practical Scheme for Verifiable Secret Sharing”, manuscript, 1986.
Feldman, P., and S., Micali, in preparation, 1985.
Fischer, M., S. Micali, C. Rackoff, and D.K. Wittenberg, “An Oblivious Transfer Protocol Equivalent to Factoring”, in preparation. Preliminary versions were presented in EuroCrypt84 (1984), and in the NSF Workshop on Mathematical Theory of Security, Endicott House (1985).
Galil, Z., S. Haber, and M. Yung, “A Private Interactive Test of a Boolean Predicate and Minimum-Knowledge Public-Key Cryptosystems”, Proc. 26th FOCS, 1985, pp. 360–371.
Garey, M.R., and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979.
Goldreich, O., “A Zero-Knowledge Proof that a Two-Prime Moduli Is Not a Blum Integer”, unpublished manuscript, 1985.
Goldreich, O., S. Goldwasser, and S. Micali, “How to Construct Random Functions”, Proc. of 25th Symp. on Foundation of Computer Science, 1984, pp. 464–479. To appear in Jour. of ACM.
Goldreich, O., S. Micali, and A. Wigderson, “Proofs that Yield Nothing But their Validity”, in preparation. An extended abstract will appear in the proceedings of 27th FOCS, 1986.
Goldreich, O., S. Micali, and A. Wigderson, “How to Automatically Generate Correct and Private Fault-Tolerant Protocols”, in preparations.
Goldwasser, S., and S. Micali, “Probabilistic Encryption”, JCSS, Vol. 28, No. 2, 1984, pp. 270–299.
Goldwasser, S., S. Micali, and C. Rackoff, “Knowledge Complexity of Interactive Proofs”, Proc. 17th STOC, 1985, pp. 291–304.
Goldwasser, S., S. Micali, and R.L. Rivest, “A Paradoxical Signature Scheme”, Proc. 25th FOCS, 1984.
Goldwasser, S., and M. Sipser, “Arthur Merlin Games versus Interactive Proof Systems”, Proc. 18th STOC, pp. 59–68, 1986.
Karp, R.M., “Reducibility among Combinatorial Problems”, Complexity of Computer Computations, R.E. Miller and J.W. Thatcher (eds.), Plenum Press, pp. 85–103, 1972.
Levin, L.A., “Universal Search Problems”, Problemy Peredaci Informacii 9, pp. 115–116, 1973. Translated in problems of Information Transmission 9, pp. 265–266.
Rivest, R.L., Shamir, A., and Adleman, L., “A Method for Obtaining Digital Signatures and Public Key Cryptosystems”, Comm. of the ACM, Vol. 21, February 1978, pp. 120–126.
Rabin, M.O., “Digitalized Signatures as Intractable as Factorization”, MIT/LCS/TR-212, 1979.
Rabin, M.O., “How to Exchange Secrets by Oblivious Transfer”, unpublished manuscript, 1981.
Shamir, A., “How to Share a Secret”, CACM, Vol. 22, 1979, pp. 612–613.
Yao, A.C., “Theory and Applications of Trapdoor Functions”, Proc. of the 23rd IEEE Symp. on Foundation of Computer Science, 1982, pp. 80–91.
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Goldreich, O., Micali, S., Wigderson, A. (1987). How to Prove All NP Statements in Zero-Knowledge and a Methodology of Cryptographic Protocol Design (Extended Abstract). In: Odlyzko, A.M. (eds) Advances in Cryptology — CRYPTO’ 86. CRYPTO 1986. Lecture Notes in Computer Science, vol 263. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47721-7_11
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