## On the complexity of interactive proofs with bounded communication (1998)

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Venue: | Information Processing Letters |

Citations: | 19 - 1 self |

### BibTeX

@ARTICLE{Goldreich98onthe,

author = {Oded Goldreich and Johan Hastad},

title = {On the complexity of interactive proofs with bounded communication},

journal = {Information Processing Letters},

year = {1998}

}

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### Abstract

We investigate the computational complexity of languages which haveinteractive proof systems of bounded message complexity. In particular, denoting the length of the input by n, we show that If L has an interactive proof in which the total communication is bounded by c(n) bits then L can be recognized by a probabilistic machine in time exponential in O(c(n)+log(n)). If L has a public-coin interactive proof in which the prover sends c(n) bits then L can be recognized by a probabilistic machine in time exponential in O(c(n) log(c(n)) + log(n)). If L has an interactive proof in which the prover sends c(n) bits then L can be recognized by a probabilistic machine with an NP-oracle in time exponential in O(c(n) log(c(n)) + log(n)). Work done while being on a sabbatical leave at LCS, MIT. 0 1

### Citations

1076 | The knowledge complexity of interactive proof-systems
- Goldwasser, Micali, et al.
- 1985
(Show Context)
Citation Context ...ons, it is only natural to associate the notion of efficient computation with probabilistic and interactive polynomial-time computations. This leads to the notion of an interactive proof system (cf., =-=[9]) in which-=- the verification procedure is interactive and randomized, rather than being non-interactive and deterministic. Intuitively, one may think of this interaction as consisting of "tricky" quest... |

387 | Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems
- Goldreich, Micali, et al.
- 1991
(Show Context)
Citation Context ...laxed notion of zero-knowledge (aka computational zero-knowledge). In fact, assuming the existence of commitment schemes, all languages in NP do have (computational) zero-knowledge interactive proofs =-=[8]-=-. 1 Interactive proofs of bounded message complexity: Our first observation indicates that Kilian's result (as stated above) is unlikely for interactive proof (rather than computationallysound) system... |

316 |
Minimum Disclosure Proofs of Knowledge
- Brassard, Chaum, et al.
- 1988
(Show Context)
Citation Context ...ifference between interactive proof systems as defined by Goldwasser, Micali and Rackoff [9], and computationally-sound proof systems (aka argument systems) as defined by Brassard, Chaum and Cr'epeau =-=[6]-=-. A significant difference between interactive proof systems and computationally-sound proof systems has been observed in the domain of zero-knowledge. On one hand, it is widely believed that not all ... |

314 | Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes - Babai, Moran - 1988 |

303 | Trading group theory for randomness
- Babai
- 1985
(Show Context)
Citation Context ...f system in which both the bidirectional message complexity and the randomness complexity are polylogarithmic. Furthermore, this proof system is in the public-coins (aka Arthur-Merlin) model of Babai =-=[1]-=-. 1 Perfect zero-knowledge is a strict variant of zero-knowledge. The above stated belief does not refer to the more relaxed notion of zero-knowledge (aka computational zero-knowledge). In fact, assum... |

283 | Random generation of combinatorial structures from a uniform distribution - Jerrum, Valiant, et al. - 1986 |

147 | A Note on Efficient Zero-Knowledge Proofs and Arguments
- Kilian
- 1992
(Show Context)
Citation Context ...fs of bounded message complexity: In 1992, Kilian demonstrated that computationally-sound proof systems may be able to recognize any language in NP while using only polylogarithmic message complexity =-=[11]-=-. Specifically, assuming the existence of hashing functions for which collisions cannot be found by subexponential-size circuits, Kilian showed that any language in NP has a computationally-sound proo... |

143 | A complexity theoretic approach to randomness - Sipser - 1983 |

42 | The complexity of approximate counting - Stockmeyer - 1983 |

26 | Uniform generation of NP-witnesses using an NP-oracle - Bellare, Goldreich, et al. |

24 |
The knowledge complexity ofinteractive proofs
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ...ations, it is only natural to associate the notion of e cient computation with probabilistic and interactive polynomialtime computations. This leads to the notion of an interactive proof system (cf., =-=[8]-=-) in which the veri cation procedure is interactive and randomized, rather than being non-interactive and deterministic. Intuitively, one may think of this interaction as consisting of \tricky" questi... |

18 | Making Zero-Knowledge Provers Efficient - Bellare, Petrank - 1992 |

4 |
Making ZeroKnowledge Provers E cient
- Bellare, Petrank
- 1992
(Show Context)
Citation Context ...ue (bottom-up) within this time bound. Using Lemma 7, Part (1) of Proposition 5 follows. Evaluating Ax in the general case: In this case we use the uniform generation procedure of Bellare and Petrank =-=[3]-=- (see Appendix B). Loosely speaking, this procedure allows to uniformly select an NP-witness for a given input in an NP-language. The procedure runs in probabilistic polynomial-time using an NP-oracle... |

3 |
The Complexity of Perfect Zero-Knowledge. Advances in Computing Research: a research annual
- Fortnow
- 1989
(Show Context)
Citation Context ...utationally-sound proof systems has been observed in the domain of zero-knowledge. On one hand, it is widely believed that not all languages in NP have perfect zero-knowledge interactive proofs (cf., =-=[7]-=-). 1 In particular, the negation of this conjecture implies the collapse of the polynomial-time hierarchy (cf., [5]). On the other hand, assuming that factoring is hard, all languages in NP have perfe... |

3 |
A note on e cient zero-knowledge proofs and arguments (extended abstract
- Kilian
- 1992
(Show Context)
Citation Context ...fs of bounded message complexity: In 1992, Kilian demonstrated that computationally-sound proof systems may be able to recognize any language in NP while using only polylogarithmic message complexity =-=[10]-=-. Speci cally, assuming the existence of hashing functions for which collisions cannot be found by subexponential-size circuits, Kilian showed that any language in NP has a computationally-sound proof... |