On interactive proofs with a laconic prover

Question

1. What are the contributions mentioned in the paper "On interactive proofs with a laconic prover" ?

2. How do the authors cut the number of rounds by a constant?

3. what is the common input of the protocol?

4. what is the veri er s output?

Accepted Answer

The authors continue the investigation of interactive proofs with bounded communication as initiated by Goldreich and H astad IPL Let L be a language that has an interactive proof in which the prover sends few say b bits to the veri er. The authors prove that the com plement L has a constant round interactive proof of complexity that depends only exponentially on b. This provides the rst evidence that for NP complete languages the authors can not expect interactive provers to be much more laconic than the standard NP proof. When the proof system is further restricted e g when b or when the authors have perfect completeness they get signi cantly better upper bounds on the complex ity of L

Accepted Answer

In order to cut the number of rounds by a constant one may apply the MA to AM switch concurrently to disjoint segments of the game

Accepted Answer

The common input is an integer n a string f gpoly n specifying a set S def S f gn and an integer N which is supposed to equal jSj Error parameters a deviation error and a probability errorProtocol

Accepted Answer

The veri er is of the public coin type and can be implemented in probabilistic polynomial time Clearly the veri er s output is either an n bit long string or the error symbol A cheating prover may easily cause the veri er to always output but this means that the veri er detects that the prover is cheating

Accepted Answer

Since there are at most i b rb possible labelings to the vertices along this path with probability rb t r for every vertex labeling of levels i the value of v in T is within r of itsvalue in a random i st hybrid tree obtained from T by sampling its level i nodes

Accepted Answer

Thus the probability that the output resides in the set S is bounded by jS jN where the rst termaccounts for the probability that t def jS h j is greater than jS j N t and the second term is an upper bound on tt which holds otherwise Usingand recalling that the original S was possibly augmented so that jS j N the probability that the output resides in S is upper bounded bymax jS j NN max jS j NjS j Nand the lemma followsB

Accepted Answer

In case S is a slice of an NP set the protocol can be augmented with NP witnesses and the veri er may avoid outputting elements not in S and output whenever such an element is presented to it As the authors shall show the essential feature of the above protocol is that the prover cannot restrict the output to a too small set see Item ii belowLemma A analysis of the sampling protocol Suppose that and that the veri er follows the prescribed programi

Accepted Answer

This means that with probability at least r for every vertex labeling of levels i the values of all level i nodes in the i st hybrid tree obtained from T is within r of their corresponding values in T Considering all possible T s and doing the same for all neighboring hybrid pairs it follows that as claimed above with probability at least the value of a random subtree under the best labeling approximates the value of the full tree under the best labeling up to an additive term of Hence the authors obtain AM b r AM tr b and B follows since t O r b bO If one cares then AM b r AM br O r rApplied MathematicsWeizmann Institute of Science Rehovot Israeloded wisdom weizmann ac ilhttp www wisdom weizmann ac il odedSalil Vadhan Division of Engineering and Applied SciencesHarvard University Cambridge MA USAsalil eecs harvard eduhttp eecs harvard edu salilInstitute for Advanced Study Princeton NJ USAavi ias eduInstitute for Computer Science Hebrew University Jerusalem Israel

Accepted Answer

One e cient construction of such hash functions is obtained by considering the set of all k degree univariate polynomials over GF n f gn and taking the bit pre x of the value of such polynomials on an evaluation pointConstruction A a sampling protocol

Accepted Answer

The authors start by recalling the main idea underlying the transformation of Babai and Moran BM An Arthur& Merlin proof system can be viewed as a game between an honest Arthur and Merlin that alternate in taking moves such that Arthur takes random moves and Merlin takes optimal ones with respect to a xed predicate that is evaluated on the full transcript of the game s execution

Accepted Answer

If N jSj and the prover follows the prescribed program then with probability at least the veri er outputs an element of Sii For every set S f gn and every N n no matter what the prover does the probability that the veri er output resides in S is at most jS j NProof

On interactive proofs with a laconic prover

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Most frequently asked questions

1. What are the contributions mentioned in the paper "On interactive proofs with a laconic prover" ?

2. How do the authors cut the number of rounds by a constant?

3. what is the common input of the protocol?

4. what is the veri er s output?

5. how many vertex labels can be assigned to a leaf?

6. what is the probability that the output resides in the set?

7. what is the simplest way to prove that the veri er is cheating?

8. what is the way to label a level i?

9. what is the protocol for sampling NP sets?

10. What is the main idea of the Arthur& Merlin proof system?

11. what is the probability that the veri er outputs an element of Sii?

Citations

Foundations of Cryptography: Acknowledgments

The knowledge complexity of interactive proof-systems

Bulletproofs: Short Proofs for Confidential Transactions and More

SNARKs for C : verifying program executions succinctly and in zero knowledge

Separating succinct non-interactive arguments from all falsifiable assumptions

References

The knowledge complexity of interactive proof systems

Introduction to the Theory of Computation

The knowledge complexity of interactive proof-systems

Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems

Probabilistic checking of proofs: a new characterization of NP

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