Reducing Complexity Assumptions for Statistically-Hiding Commitment

Determining the minimal assumptions needed to construct various cryptographic building blocks has been a focal point of research in theoretical cryptography. Here, we revisit the following question: what are the minimal assumptions needed to construct statistically-hiding commitment schemes? Previously, it was known how to construct such schemes based on one-way permutations. We improve upon this by constructing statistically-hiding commitment schemes based on approximable-preimage-size one-way functions. These are one-way functions for which there is an efficient way to approximate the number of preimages of a given output. A special case (for which we show a somewhat simpler construction) is that of regular one-way functions where all outputs have the same number of preimages. We utilize two different approaches in constructing statistically-hiding commitment schemes. Our first approach proceeds by showing that the scheme of Naor et al. can be implemented using any one-way function having an output distribution which is “sufficiently similar” to uniform. We then construct one-way functions with this property from approximable-preimage-size one-way functions. Our second approach begins by constructing a commitment scheme which is statistically hiding against an honest-but-curious receiver. We then demonstrate a compiler which transforms any such commitment scheme into one which is statistically hiding even against a malicious receiver. This compiler and its analysis may be of independent interest.

[1]  David Chaum,et al.  Minimum Disclosure Proofs of Knowledge , 1988, J. Comput. Syst. Sci..

[2]  Yehuda Lindell,et al.  Parallel Coin-Tossing and Constant-Round Secure Two-Party Computation , 2001, Journal of Cryptology.

[3]  Rafail Ostrovsky,et al.  Fair Games against an All-Powerful Adversary , 1990, Advances In Computational Complexity Theory.

[4]  Alexander Russell,et al.  Necessary and sufficient conditions for collision-free hashing , 1992, Journal of Cryptology.

[5]  Oded Goldreich,et al.  How to construct constant-round zero-knowledge proof systems for NP , 1996, Journal of Cryptology.

[6]  Omer Reingold,et al.  A New Interactive Hashing Theorem , 2007, Computational Complexity Conference.

[7]  Moni Naor,et al.  Bit commitment using pseudorandomness , 1989, Journal of Cryptology.

[8]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[9]  Aravind Srinivasan,et al.  Chernoff-Hoeffding bounds for applications with limited independence , 1995, SODA '93.

[10]  Oded Goldreich,et al.  Foundations of Cryptography: Volume 2, Basic Applications , 2004 .

[11]  Adi Shamir,et al.  Multiple NonInteractive Zero Knowledge Proofs Under General Assumptions , 1999, SIAM J. Comput..

[12]  Ran Raz.  A Parallel Repetition Theorem , 1998, SIAM J. Comput..

[13]  Leonid A. Levin,et al.  Security preserving amplification of hardness , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[14]  Oded Goldreich,et al.  Foundations of Cryptography: Basic Tools , 2000 .

[15]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[16]  Oded Goldreich,et al.  On the power of two-point based sampling , 1989, J. Complex..

[17]  Silvio Micali,et al.  On the Cryptographic Applications of Random Functions , 1984, CRYPTO.

[18]  Silvio Micali,et al.  Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems , 1991, JACM.

[19]  Silvio Micali,et al.  How to sign given any trapdoor permutation , 1992, JACM.

[20]  Moni Naor,et al.  Does parallel repetition lower the error in computationally sound protocols? , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[21]  S. Micali,et al.  Noninteractive Zero-Knowledge , 1990, SIAM J. Comput..

[22]  Rafail Ostrovsky,et al.  Secure Commitment Against A Powerful Adversary , 1992, STACS.

[23]  Moti Yung,et al.  On the Design of Provably Secure Cryptographic Hash Functions , 1991, EUROCRYPT.

[24]  Ivan Damgård,et al.  On the existence of statistically hiding bit commitment schemes and fail-stop signatures , 1994, Journal of Cryptology.

[25]  Manuel Blum,et al.  Coin Flipping by Telephone. , 1981, CRYPTO 1981.

[26]  John Rompel,et al.  One-way functions are necessary and sufficient for secure signatures , 1990, STOC '90.

[27]  Leonid A. Levin,et al.  A Pseudorandom Generator from any One-way Function , 1999, SIAM J. Comput..

[28]  Russell Impagliazzo,et al.  One-way functions are essential for complexity based cryptography , 1989, 30th Annual Symposium on Foundations of Computer Science.

[29]  Silvio Micali,et al.  A Digital Signature Scheme Secure Against Adaptive Chosen-Message Attacks , 1988, SIAM J. Comput..

[30]  Joan Boyar,et al.  A discrete logarithm implementation of perfect zero-knowledge blobs , 1990, Journal of Cryptology.

[31]  Russell Impagliazzo,et al.  Limits on the provable consequences of one-way permutations , 1988, STOC '89.

[32]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[33]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[34]  Oded Goldreich.  Foundations of Cryptography: Index , 2001 .

[35]  Salil P. Vadhan,et al.  Statistical Zero-Knowledge Arguments for NP from Any One-Way Function , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[36]  Manuel Blum,et al.  Noninteractive Zero-Knowledge , 1991, SIAM J. Comput..

[37]  Hugo Krawczyk,et al.  On the Existence of Pseudorandom Generators , 1993, SIAM J. Comput..

[38]  Rafail Ostrovsky,et al.  Perfect Zero-Knowledge Arguments for NP Using Any One-Way Permutation , 1998, Journal of Cryptology.

[39]  Ivan Damgård,et al.  Collision Free Hash Functions and Public Key Signature Schemes , 1987, EUROCRYPT.

[40]  Moni Naor,et al.  Universal one-way hash functions and their cryptographic applications , 1989, STOC '89.

[41]  Oded Goldreich,et al.  Foundations of Cryptography: List of Figures , 2001 .

[42]  Silvio Micali,et al.  Practical and Provably-Secure Commitment Schemes from Collision-Free Hashing , 1996, CRYPTO.

[43]  Silvio Micali,et al.  How to construct random functions , 1986, JACM.