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How to Delegate Computations:The Power of No-Signaling Proofs
Ran Raz(Weizmann Institute & IAS)
Joint work with:Yael Tauman Kalai
Ron Rothblum
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Delegation of Computation
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Delegation of Computation:Alice has Alice needs to compute , where is publicly knownBob offers to compute for AliceAlice sends to BobBob sends to Alice
π
π (π)
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1-Round Delegation Scheme for :
or 1) Completeness: if is honest:
2) Soundness: if:
3) Running time of : 4) Running time of :
π½ π·π ,π
π= π (π ) ,π
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Previous Work [GKR+KR]:If is a logspace-uniform circuit
ofsize and depth :1-round delegation scheme s.t.:Running time of : Running time of : (under exponential hardness
assumptions)
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Our Result:If 1-round delegation scheme s.t.:Running time of : Running time of : (under exponential hardness
assumptions)
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Variants of Delegation Schemes:1-Round or InteractiveComputational or Statistical
soundness:
β’ 1-Round, Computational: This talk!
β’ 1-Round, Statistical: Impossible!β’ Interactive, Computational:
Solved! (with only 2-rounds) [Killian,Micali], (based on ) [BFL]
β’ Interactive, Statistical: [GKR 08]
β’ Many other works, under unfalsifiable assumptions, or with preprocessing.
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The Approach of Aiello et al.
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2-Prover Interactive Proofs [BGKW]:Provers claim that sends a query to and to no communication between
and answers by answers by decides accept/reject by
π ππ¨ π©π π
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MIP=NEXP (scaled down) [BFL+FL]:, 2-provers MIP s.t.:1) Completeness: if are honest:
2) Soundness: if:
3) Running time of : 4) Running time of : 5) Communication:
π ππ¨ π©π π
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[Aiello Bhatt Ostrovsky Sivarama 00]:MIP 1-Round Argument ?!?
MIP:
= FHE of (with different keys) = FHE of
π ππ¨ π©
π=π¨(π) π=π©(π )
π½ π·οΏ½ΜοΏ½ ,ποΏ½ΜοΏ½ , οΏ½ΜοΏ½
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βProofβ of Soundness: generates and
decrypts to get
If we are doneIf then given , the answer gives information on . Thus, if first
keyis known, one can get information
on (without knowing the second key)
= FHE of (with different keys) = FHE of
π½ π·οΏ½ΜοΏ½ ,ποΏ½ΜοΏ½ , οΏ½ΜοΏ½
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The βProofβ is Wrong [DLNNR 00]: can depend on both as a function,but not as a random variables
Example: , (where is a random bit)
Given , the random variables are independent
Given , the random variables are independent
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No-Signaling Strategies:
, (where is a shared random
string):
Given , the random variables are independent
Given , the random variables are independent
π ππ¨ π©π π
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No-Signaling Strategies for provers:queries: , answers , ( random string)
For every : Given ,,, are independent
Soundness Against No-Signaling:, if:
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We Show (using [ABOS 00]):MIP with no-signaling
soundness 1-Round Argument (we need soundness for almost-no-signaling
strategies)
Corollary:Interactive Proof 1-Round Argument(under exponential hardness
assumptions)
Gives a simpler proof for [KR 09]
Challenge: Show stronger MIPs withno-signaling soundness
= FHE of (with different keys) = FHE of
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No-Signaling Strategies
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Entangled Strategies:
share entangled quantum state gets , gets measures , measures answers , answers Soundness Against Entangled Strategies:, if:
[IV12, V13]:
π ππ¨ π©π π
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Entangled vs. No-Signaling:Entangled strategies are no-signaling
Signaling information travels faster than
light
Hence, no-signaling is likely to hold in
any future ultimate theory of physics
No-signaling soundness is likely to ensure
soundness in any future physical theory
π ππ¨ π©π π
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MIPs with No-Signaling Soundness:No-Sig cheating provers are powerful:
(by linear programing)
In particular, all known protocols for
are not sound for no-signaling
Example: Assume: checks Let . Let . ( is random)Then always accepts (and the strategy is no-signaling)
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Our Result:
If MIP s.t.:Running time of : Running time of : Number of provers: Communication: Completeness: Soundness: against no-sig
strategies (with negligible error)(gives soundness against entangled
provers)
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Delegating Computation to the Martians:
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Delegating Computation to the Martians:
Running time of provers: Running time of : Number of provers: Number of provers: Completeness: Soundness: against no-sig
strategies (with negligible error)
ππ . .ππ
π· π π· π
ππ ππ
π· πβ¦β¦
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Steps of the Proof
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Step I: Switch to PCP:
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PCP:
A proof for that can be(probabilistically) verified by
readinga small number of its bits.
proof is correct ) P(V accepts) = 1 ) P(V accepts) β€ Ξ΅
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No-Signaling PCPs:For every subset of locationss.t. , distribution If queries locations , theanswers are given by Guarantee: if , then agree on their intersection
Soundness Against No-Signaling:, if:
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Our Result: PCP s.t.:Running time of prover: Running time of : Number of queries:Completeness: Soundness: against no-sig
strategies with (with negligible error)
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Step I: Switch to PCP:PCP with no-sig soundness
implies MIPwith no-sig soundness. with provers
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Step II: Amplification Lemma: If there exists a cheating no-sig
PCPthat cheats our verifier with exp
smallprob, then there exists a cheating
no-sig PCP that cheats (a slightly differentverifier) with prob close to 1
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Step III: Let be a circuit for . Let bethe outputs of all gates of . Our PCP contains the low degree extension ofWe βreadβ each by a procedure, as inlocally decodeable codes. Denote thesevalues by . Note that only of can be defined simultaneously.For every gate in the circuit, with parent and children , we show that satisfy the gate w.h.p.
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Step IV: If , the circuit is of width . So all the variables in the same level of can bedefined simultaneously. We can work our
wayup and prove that the output is computed correctly w.h.p.
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Thank You!