## Uniform Generation of NP-witnesses using an NP-oracle (1997)

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Venue: | Information and Computation |

Citations: | 28 - 1 self |

### BibTeX

@ARTICLE{Bellare97uniformgeneration,

author = {Mihir Bellare and Oded Goldreich and Erez Petrank},

title = {Uniform Generation of NP-witnesses using an NP-oracle},

journal = {Information and Computation},

year = {1997},

volume = {163},

pages = {2000}

}

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### Abstract

A Uniform Generation procedure for NP is an algorithm which given any input in a fixed NP-language, outputs a uniformly distributed NP-witness for membership of the input in the language. We present a Uniform Generation procedure for NP that runs in probabilistic polynomial-time with an NP-oracle. This improves upon results of Jerrum, Valiant and Vazirani, which either require a \Sigma P 2 oracle or obtain only almost uniform generation. Our procedure utilizes ideas originating in the works of Sipser, Stockmeyer, and Jerrum, Valiant and Vazirani. Dept. of Computer Science & Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. E-Mail: mihir@cs.ucsd.edu. URL: http://www-cse.ucsd.edu/users/mihir. Supported in part by NSF CAREER Award CCR-9624439 and a 1996 Packard Foundation Fellowship in Science and Engineering. y Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. E-Mail: oded@wis...

### Citations

1041 | Knowledge Complexity of Interactive Proof Systems
- Goldwasser, Rackoff
- 1989
(Show Context)
Citation Context ...at any language with a PZK proof has one in which the prover is a probabilistic, polynomial algorithm with a NP oracle. Time complexity versus knowledge complexity. Knowledge complexity, suggested by =-=[13]-=- and defined by [12], provides a way to measure the number of bits of knowledge that a prover reveals to a verifier about some string x, in the process of proving that x belongs to some underlying lan... |

403 | Non-deterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ... did not use uniform generation.) Shared randomness in two prover proofs. We know that the class of languages recognized by two prover statistical zero-knowledge interactive proof systems equals NEXP =-=[2, 7]-=-. Bellare, Feige and Kilian [3] showed that a certain model feature ---namely the fact that the two provers are allowed to share a random string before the protocol begins--- is crucial to this result... |

281 |
Random generation of combinatorial structures from a uniform distribution
- Jerrum, Valiant, et al.
- 1986
(Show Context)
Citation Context ...llowing problem: Uniform Generation of NP-witnesses Given: x 2 L Output: A string w uniformly distributed in R x def = f w j R(x; w) = 1 g. 1 This was first considered by Jerrum, Valiant and Vazirani =-=[15], who-=- showed that it could be accomplished in probabilistic polynomial time given access to a \Sigma P 2 oracle. The same paper also considered a weaker version of the problem, called "almost uniform ... |

213 |
NP is as easy as detecting unique solutions
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...ynomial time with NP oracle solution to almost approximate counting is implicit in Sipser [19] and Stockmeyer [20], or can be derived via the probabilistic bisection technique of Valiant and Vazirani =-=[21]-=-. Details of the former solution were worked out and presented by Bellare and Petrank [5] in the context of applications to zero-knowledge proof systems, based on more recent versions of Sipser and St... |

147 |
A complexity theoretic approach to randomness
- Sipser
- 1983
(Show Context)
Citation Context ...r. Jerrum et. al. [15] obtained their results by reducing uniform generation to the problem of approximate counting. The latter problem can be solved using the "hashing paradigm" introduced =-=by Sipser [19] and emplo-=-yed in previous works on this problem [20, 15] (see Section 4.1 for a more complete description of the history.). In contrast, we directly apply the "hashing paradigm" to the problem of unif... |

131 | Multiprover interactive proofs: How to remove the intractability assumptions
- Ben-Or, Goldwasser, et al.
- 1988
(Show Context)
Citation Context ... did not use uniform generation.) Shared randomness in two prover proofs. We know that the class of languages recognized by two prover statistical zero-knowledge interactive proof systems equals NEXP =-=[2, 7]-=-. Bellare, Feige and Kilian [3] showed that a certain model feature ---namely the fact that the two provers are allowed to share a random string before the protocol begins--- is crucial to this result... |

102 | Chernoff-Hoeffding Bounds for Applications with Limited Independence - Schmidt, Siegel, et al. - 1994 |

87 | The Complexity of Perfect Zero-Knowledge
- Fortnow
- 1989
(Show Context)
Citation Context ...presented by Bellare and Petrank [5] in the context of applications to zero-knowledge proof systems, based on more recent versions of Sipser and Stockmeyer's hashing techniques that were developed in =-=[14, 9, 1]-=-. Putting this together yields the probabilistic, polynomial time with NP oracle solution to almost uniform generation. As an inspection of our procedure shows, we do not use the Jerrum et. al.'s redu... |

65 | Statistical zero-knowledge languages can be recognized in two rounds
- Aiello, HËšastad
- 1991
(Show Context)
Citation Context ....) Our results pertain to uniform generation. For the sake of discussing the history in the area, however, we also define the weaker notion of almost uniform generation. 3 Definition 2.4 Let ffi: N ! =-=[0; 1]-=- and let G be a generator for NP. We say that G is a ffi(\Delta)-uniform generator for R if for every x 2 LR we have kG succ (x) \Gamma Unif(R x )ksffi (jxj) : If ffi (\Delta)s2 \Gamma(\Delta) we call... |

64 | Randomness-efficient oblivious sampling - Bellare, Rompel - 1994 |

50 | The probabilistic method yields deterministic parallel algorithms - Motwani, Naor, et al. - 1989 |

42 | The (True) Complexity of Statistical Zero-Knowledge
- Bellare, Micali, et al.
- 1990
(Show Context)
Citation Context ... SZK languages are known to be in \Sigma P 2 " \Pi P 2 [9, 1], it is not a priori clear that the prover, as a function is even restricted to probabilistic PSPACE. The question was first considere=-=d by [4]-=- who reduced the complexity of the prover to that of (almost) uniform generation in such a way that SZK versus an honest verifier was maintained. A more general reduction, provided by [5], maintained ... |

42 |
The complexity of approximate counting
- Stockmeyer
- 1983
(Show Context)
Citation Context ...ucing uniform generation to the problem of approximate counting. The latter problem can be solved using the "hashing paradigm" introduced by Sipser [19] and employed in previous works on thi=-=s problem [20, 15] (see Sect-=-ion 4.1 for a more complete description of the history.). In contrast, we directly apply the "hashing paradigm" to the problem of uniform generation, rather than utilizing the above reductio... |

27 | On the Knowledge Complexity of N P
- Petrank, Tardos
- 1996
(Show Context)
Citation Context ... a PZK proof has one in which the prover is a probabilistic, polynomial algorithm with a NP oracle. Time complexity versus knowledge complexity. Knowledge complexity, suggested by [13] and defined by =-=[12]-=-, provides a way to measure the number of bits of knowledge that a prover reveals to a verifier about some string x, in the process of proving that x belongs to some underlying language L. Let SKC[(\D... |

18 | Making Zero-Knowledge Provers Efficient
- Bellare, Petrank
- 1992
(Show Context)
Citation Context ...ve proof systems and knowledge complexity that has exploited uniform generation to develop efficient decision procedures for certain languages based on their interactive proof or knowledge complexity =-=[5, 11, 3, 10]-=-. The complexity of the procedure is the crucial issue in these works, and it depends largely on the complexity of uniform generation. We survey these applications in Section 4.2. In one case that we ... |

18 | On the Complexity of Interactive Proofs with Bounded Communication
- Goldreich, Hastad
- 1998
(Show Context)
Citation Context ...ve proof systems and knowledge complexity that has exploited uniform generation to develop efficient decision procedures for certain languages based on their interactive proof or knowledge complexity =-=[5, 11, 3, 10]-=-. The complexity of the procedure is the crucial issue in these works, and it depends largely on the complexity of uniform generation. We survey these applications in Section 4.2. In one case that we ... |

14 | Simulating (log c n)-wise Independence - Berger, Rompel |

10 |
Private Coins versus Public Coins
- Goldwasser, Sipser
- 1989
(Show Context)
Citation Context ...presented by Bellare and Petrank [5] in the context of applications to zero-knowledge proof systems, based on more recent versions of Sipser and Stockmeyer's hashing techniques that were developed in =-=[14, 9, 1]-=-. Putting this together yields the probabilistic, polynomial time with NP oracle solution to almost uniform generation. As an inspection of our procedure shows, we do not use the Jerrum et. al.'s redu... |

9 | On the knowledge complexity of NP
- Petrank, Tardos
- 1996
(Show Context)
Citation Context ... language L. Let SKC[(\Delta)] denotes the class of languages possessing interactive proofs of negligible error probability and statistical knowledge complexity (SKC) at most (\Delta). A body of work =-=[5, 11, 17] has -=-sought extensions to non-zero SKC of the results of [9, 1] which showed SKC[0] ` \Sigma P 2 " \Pi P 2 . The first results used almost uniform generation. Specifically, Bellare and Petrank [5] pro... |

4 |
Making ZeroKnowledge Provers E cient
- Bellare, Petrank
- 1992
(Show Context)
Citation Context ...tive proof systems and knowledge complexity that has exploited uniform generation to develop e cient decision procedures for certain languages based on their interactive proof or knowledge complexity =-=[5, 11, 3, 10]-=-. The complexity ofthe procedure is the crucial issue in these works, and it depends largely on the complexity of uniform generation. We survey these applications in Section 4.2. In one case that we n... |

4 | Cherno -hoe ding bounds for applications with limited independence - Schmidt, Siegel, et al. - 1995 |

3 | On the role of shared randomness in two prover proof systems
- Bellare, Feige, et al.
- 1995
(Show Context)
Citation Context ...ve proof systems and knowledge complexity that has exploited uniform generation to develop efficient decision procedures for certain languages based on their interactive proof or knowledge complexity =-=[5, 11, 3, 10]-=-. The complexity of the procedure is the crucial issue in these works, and it depends largely on the complexity of uniform generation. We survey these applications in Section 4.2. In one case that we ... |

2 |
Knowledge Complexity and Computational Complexity
- Goldreich, Ostrovsky, et al.
- 1994
(Show Context)
Citation Context |

1 | Randomness-e cient oblivious sampling - Bellare, Rompel - 1994 |

1 |
On the knowledge complexity ofNP
- Petrank, Tardos
- 1996
(Show Context)
Citation Context ...underlying language L. LetSKC[ ( )] denotes the class of languages possessing interactive proofs of negligible error probability and statistical knowledge complexity (SKC) at most ( ). A body of work =-=[5, 11, 17]-=- has sought extensions to non-zero SKC of the results of [9, 1] which showedSKC[0] AM\co,AM. The rst results used almost uniform generation. Speci cally, Bellare and Petrank [5] provided a decision pr... |