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A hierarchy theorem for interactive proofs of proximity
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Gur, Tom and Rothblum, Ron D. (2017) A hierarchy theorem for interactive proofs of proximity. In: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Published in: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), 67 39:1-39:43. ISBN 9783959770293. ISSN 1868-8969. doi:10.4230/LIPIcs.ITCS.2017.39
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Official URL: http://drops.dagstuhl.de/opus/volltexte/2017/8153
Abstract
The number of rounds, or round complexity, used in an interactive protocol is a fundamental resource. In this work we consider the significance of round complexity in the context of Interactive Proofs of Proximity (IPPs). Roughly speaking, IPPs are interactive proofs in which the verifier runs in sublinear time and is only required to reject inputs that are far from the language. Our main result is a round hierarchy theorem for IPPs, showing that the power of IPPs grows with the number of rounds. More specifically, we show that there exists a gap function g(r) = Theta(r^2) such that for every constant r \geq 1 there exists a language that (1) has a g(r)-round IPP with verification time t=t(n,r) but (2) does not have an r-round IPP with verification time t (or even verification time t'=\poly(t)). In fact, we prove a stronger result by exhibiting a single language L such that, for every constant r \geq 1, there is an O(r^2)-round IPP for L with t=n^{O(1/r)} verification time, whereas the verifier in any r-round IPP for L must run in time at least t^{100}. Moreover, we show an IPP for L with a poly-logarithmic number of rounds and only poly-logarithmic erification time, yielding a sub-exponential separation between the power of constant-round IPPs versus general (unbounded round) IPPs. From our hierarchy theorem we also derive implications to standard interactive proofs (in which the verifier can run in polynomial time). Specifically, we show that the round reduction technique of Babai and Moran (JCSS, 1988) is (almost) optimal among all blackbox transformations, and we show a connection to the algebrization framework of Aaronson and Wigderson (TOCT, 2009).
| Item Type: | Conference Item (Paper) | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | |||||||||||||||
| Divisions: | Faculty of Science > Computer Science | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Computational complexity | |||||||||||||||
| Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) | |||||||||||||||
| Journal or Publication Title: | 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) | |||||||||||||||
| Publisher: | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik | |||||||||||||||
| Place of Publication: | Dagstuhl, Germany | |||||||||||||||
| ISBN: | 9783959770293 | |||||||||||||||
| ISSN: | 1868-8969 | |||||||||||||||
| Official Date: | 2017 | |||||||||||||||
| Dates: |
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| Volume: | 67 | |||||||||||||||
| Page Range: | 39:1-39:43 | |||||||||||||||
| DOI: | 10.4230/LIPIcs.ITCS.2017.39 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Open Access | |||||||||||||||
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| Conference Paper Type: | Paper | |||||||||||||||
| Title of Event: | 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) | |||||||||||||||
| Type of Event: | Conference |
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