Abstract
It is well-known that one-way permutations (and even one-to-one one-way functions) imply the existence of non-interactive commitments. Furthermore the construction is black-box (i.e., the underlying one-way function is used as an oracle to implement the commitment scheme, and an adversary attacking the commitment scheme is used as an oracle in the proof of security).
We rule out the possibility of black-box constructions of non-interactive commitments from general (possibly not one-to-one) one-way functions. As far as we know, this is the first result showing a natural cryptographic task that can be achieved in a black-box way from one-way permutations but not from one-way functions.
We next extend our black-box separation to constructions of non-interactive commitments from a stronger notion of one-way functions, which we refer to as hitting one-way functions. Perhaps surprisingly, Barak, Ong, and Vadhan (Siam JoC ’07) showed that there does exist a non-black-box construction of non-interactive commitments from hitting one-way functions. As far as we know, this is the first result to establish a “separation” between the power of black-box and non-black-box use of a primitive to implement a natural cryptographic task.
We finally show that unless the complexity class \(\mathsf {NP} \) has program checkers, the above separations extend also to non-interactive instance-based commitments, and 3-message public-coin honest-verifier zero-knowledge protocols with \(O(\log n)\)-bit verifier messages. The well-known classical zero-knowledge proof for \(\mathsf {NP} \) fall into this category.
Chapter PDF
References
Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Report, Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur-208016, India (August 2002)
Alon, N., Spencer, J.H.: The probabilistic method, 3rd edn. Wiley, New York (2008)
Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P.: A new general derandomization method. JACM: Journal of the ACM 45 (1998)
Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P., Trevisan, L.: Weak random sources, hitting sets, and BPP simulations. SICOMP: SIAM Journal on Computing 28 (1999)
Barak, B.: How to go beyond the black-box simulation barrier. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 106–115 (2001)
Barak, B., Mahmoody, M.: Lower bounds on signatures from symmetric primitives. In: FOCS: IEEE Symposium on Foundations of Computer Science (2007)
Barak, B., Ong, S.J., Vadhan, S.: Derandomization in Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 299–315. Springer, Heidelberg (2003)
Blum, Impagliazzo: Generic oracles and oracle classes. In: FOCS: IEEE Symposium on Foundations of Computer Science (1987)
Blum, M.: Coin flipping by telephone. In: CRYPTO, pp. 11–15 (1981)
Blum, M., Kannan, S.: Designing programs that check their work. J. ACM 42(1), 269–291 (1995)
Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo random bits, pp. 112–117 (1982)
Boneh, Papakonstantinou, Rackoff, Vahlis, Waters: On the impossibility of basing identity based encryption on trapdoor permutations. In: FOCS: IEEE Symposium on Foundations of Computer Science (2008)
Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. Journal of Computer and System Sciences 37(2), 156–189 (1988)
Brassard, G., Crépeau, C., Yung, M.: Constant-round perfect zero-knowledge computationally convincing protocols. Theoretical Computer Science 84(1), 23–52 (1991)
Choi, S.G., Dachman-Soled, D., Malkin, T., Wee, H.: Black-Box Construction of a Non-malleable Encryption Scheme from Any Semantically Secure One. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 427–444. Springer, Heidelberg (2008)
Choi, S.G., Dachman-Soled, D., Malkin, T., Wee, H.: Simple, Black-Box Constructions of Adaptively Secure Protocols. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 387–402. Springer, Heidelberg (2009)
Dachman-Soled, D., Lindell, Y., Mahmoody, M., Malkin, T.: On the Black-Box Complexity of Optimally-Fair Coin Tossing. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 450–467. Springer, Heidelberg (2011)
Damgård, I.B., Pedersen, T.P., Pfitzmann, B.: Statistical secrecy and multibit commitments. IEEE Transactions on Information Theory 44(3), 1143–1151 (1998)
Gennaro, R., Gertner, Y., Katz, J., Trevisan, L.: Bounds on the efficiency of generic cryptographic constructions. SIAM Journal on Computing 35(1), 217–246 (2005)
Gennaro, R., Trevisan, L.: Lower bounds on the efficiency of generic cryptographic constructions. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 305–313 (2000)
Gertner, Y., Kannan, S., Malkin, T., Reingold, O., Viswanathan, M.: The relationship between public key encryption and oblivious transfer. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (2000)
Gertner, Y., Malkin, T., Reingold, O.: On the impossibility of basing trapdoor functions on trapdoor predicates. In: FOCS, pp. 126–135 (2001)
Goldreich, O., Kahan, A.: How to construct constant-round zero-knowledge proof systems for NP. Journal of Cryptology 9(3), 167–190 (1996)
Goldreich, O., Krawczyk, H.: Sparse pseudorandom distributions. Random Structures & Algorithms 3(2), 163–174 (1992)
Goldreich, O., Krawczyk, H., Luby, M.: On the existence of pseudorandom generators. SIAM Journal on Computing 22(6), 1163–1175 (1993)
Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC), pp. 25–32 (1989)
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority, pp. 218–229 (1987)
Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM 38(1), 691–729 (1991); Preliminary version in FOCS 1986
Goldreich, O., Wigderson, A.: Improved Derandomization of BPP Using a Hitting Set Generator. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM-APPROX 1999. LNCS, vol. 1671, pp. 131–137. Springer, Heidelberg (1999)
Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM Journal on Computing 18(1), 186–208 (1989); Preliminary version in STOC 1985
Goldwasser, S., Micali, S., Rivest, R.L.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM Journal on Computing 17(2), 281–308 (1988); Preliminary version in FOCS 1984
Goyal, V.: Constant round non-malleable protocols using one way functions (2011)
Haitner, I.: Semi-honest to Malicious Oblivious Transfer—The Black-Box Way. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 412–426. Springer, Heidelberg (2008)
Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols - A tight lower bound on the round complexity of statistically-hiding commitments. In: Proceedings of the 47th Annual Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society (2007)
Haitner, I., Horvitz, O., Katz, J., Koo, C.-Y., Morselli, R., Shaltiel, R.: Reducing Complexity Assumptions for Statistically-Hiding Commitment. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 58–77. Springer, Heidelberg (2005), See also preliminary draft of full version www.wisdom.weizmann.ac.il/~iftachh/papers/SCfromRegularOWF.pdf
Haitner, I., Ishai, Y., Kushilevitz, E., Lindell, Y., Petrank, E.: Black-box constructions of protocols for secure computation. SIAM J. Comput. 40(2), 225–266 (2011)
Haitner, I., Nguyen, M.-H., Ong, S.J., Reingold, O., Vadhan, S.: Statistically-hiding commitments and statistical zero-knowledge arguments from any one-way function. SIAM Journal on Computing (November 2007)
Haitner, I., Omri, E.: Coin flipping with constant bias implies one-way functions (2011)
Haitner, I., Reingold, O.: Statistically-hiding commitment from any one-way function. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC). ACM Press (2007)
Haitner, I., Reingold, O., Vadhan, S.P., Wee, H.: Inaccessible entropy (2009)
Hartmanis, J., Hemachandra, L.A.: One-way functions, robustness, and the non-isomorphism of \({NP}\)-complete sets. Technical Report, 86–796, Department of Computer Science, Cornell University, (January 1987)
Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM Journal on Computing 28(4), 1364–1396 (1999); Preliminary versions in STOC 1989 and STOC 1990
Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS), pp. 230–235 (1989)
Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC), pp. 44–61. ACM Press (1989)
Kahn, J., Saks, M., Smyth, C.: A dual version of Reimer’s inequality and a proof of Rudich’s conjecture. In: 15th Annual IEEE Conference on Computational Complexity, pp. 98–103 (2000)
Katz, J., Schröder, D., Yerukhimovich, A.: Impossibility of Blind Signatures from One-Way Permutations. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 615–629. Springer, Heidelberg (2011)
Kim, J.H., Simon, D.R., Tetali, P.: Limits on the efficiency of one-way permutation-based hash functions. In: FOCS, pp. 535–542 (1999)
Lenstra, A.K., Lenstra Jr., H.W. (eds.): The development of the number field sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Berlin (1993)
Levin, L.A.: One-way functions and pseudorandom generators. Combinatorica 7, 357–363 (1987)
Lin, H., Trevisan, L., Wee, H.: On Hardness Amplification of One-Way Functions. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 34–49. Springer, Heidelberg (2005)
Matsuda, T., Matsuura, K.: On Black-Box Separations among Injective One-Way Functions. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 597–614. Springer, Heidelberg (2011)
Miller, G.L.: Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences 13(3), 300–317 (1976)
Naor, M.: On Cryptographic Assumptions and Challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003)
Naor, M.: Bit commitment using pseudorandomness. Journal of Cryptology 4(2), 151–158 (1991); Preliminary version In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 128–136. Springer, Heidelberg (1990)
Naor, M., Ostrovsky, R., Venkatesan, R., Yung, M.: Perfect zero-knowledge arguments for NP using any one-way permutation. CRYPTO 1992 11(2), 87–108 (1998); Preliminary version in Brickell, E.F. (ed.): CRYPTO 1992. LNCS, vol. 740. Springer, Heidelberg (1993)
Nguyen, M.-H., Ong, S.J., Vadhan, S.: Statistical zero-knowledge arguments for NP from any one-way function. In: Proceedings of the 47th Annual Symposium on Foundations of Computer Science (FOCS), pp. 3–14 (2006)
Ostrovsky, R., Wigderson, A.: One-way fuctions are essential for non-trivial zero-knowledge. In: ISTCS, pp. 3–17 (1993)
Pass, R., Wee, H.: Black-Box Constructions of Two-Party Protocols from One-Way Functions. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 403–418. Springer, Heidelberg (2009)
Rabin, M.O.: Probabilistic algorithm for testing primality. Journal of Number Theory 12(1), 128–138 (1980)
Reingold, O., Trevisan, L., Vadhan, S.P.: Notions of Reducibility between Cryptographic Primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004)
Rudich, S.: Limits on the Provable Consequences of One-Way Functions. PhD. thesis, U.C. Berkeley (1988)
Simon, D.R.: Findings Collisions on a One-Way Street: Can Secure Hash Functions Be Based on General Assumptions? In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 334–345. Springer, Heidelberg (1998)
Tardos, G.: Query complexity, or why is it difficult to seperate \(\text{ NP }^{A}\) cap co \(\text{ NP }^{A}\) from \(\text{ P }^{A}\) by random oracles A? Combinatorica 9(4), 385–392 (1989)
Vahlis, Y.: Two Is a Crowd? A Black-Box Separation of One-Wayness and Security under Correlated Inputs. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 165–182. Springer, Heidelberg (2010)
Wee, H.: Black-box, round-efficient secure computation via non-malleability amplification. In: FOCS, pp. 531–540. IEEE Computer Society (2010)
Yao, A.C.: Theory and applications of trapdoor functions, pp. 80–91 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 International Association for Cryptologic Research 2012
About this paper
Cite this paper
Mahmoody, M., Pass, R. (2012). The Curious Case of Non-Interactive Commitments – On the Power of Black-Box vs. Non-Black-Box Use of Primitives. In: Safavi-Naini, R., Canetti, R. (eds) Advances in Cryptology – CRYPTO 2012. CRYPTO 2012. Lecture Notes in Computer Science, vol 7417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32009-5_41
Download citation
DOI: https://doi.org/10.1007/978-3-642-32009-5_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32008-8
Online ISBN: 978-3-642-32009-5
eBook Packages: Computer ScienceComputer Science (R0)