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Interactive Proofs
Lecture 19 And Beyond
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So far
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Interactive Proofs Lecture 19 And Beyond 1 So far 2 So far IP - - PowerPoint PPT Presentation
Interactive Proofs Lecture 19 And Beyond 1 So far 2 So far IP - - PowerPoint PPT Presentation
Interactive Proofs Lecture 19 And Beyond 1 So far 2 So far IP = PSPACE = AM[poly] 2 So far IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] 2 So far IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly]
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So far
IP = PSPACE = AM[poly]
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes]
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n)
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n) Using a public-coin set lower-bound proof
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n) Using a public-coin set lower-bound proof AM[k] = AM for constant k ! 2
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n) Using a public-coin set lower-bound proof AM[k] = AM for constant k ! 2 Using MA ⊆ AM and alternate characterization in terms of pairs of complementary ATTMs
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n) Using a public-coin set lower-bound proof AM[k] = AM for constant k ! 2 Using MA ⊆ AM and alternate characterization in terms of pairs of complementary ATTMs Perfect completeness: One-sided-error-AM = AM
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So far
IP = PSPACE = AM[poly] PSPACE enough to calculate max Pr[yes] AM[poly] protocol for TQBF using arithmetization In fact IP[k] ⊆ AM[k+2] for all k(n) Using a public-coin set lower-bound proof AM[k] = AM for constant k ! 2 Using MA ⊆ AM and alternate characterization in terms of pairs of complementary ATTMs Perfect completeness: One-sided-error-AM = AM Similar to BPP ⊆ Σ2P (yields MAM protocol; MAM=AM)
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AM ⊆ Π2P
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AM ⊆ Π2P
Consider any L with an AM protocol
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness:
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness: x∈L ⇒ ∀ yArthur ∃ zMerlin R(x,yArthur,zMerlin) = 1
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness: x∈L ⇒ ∀ yArthur ∃ zMerlin R(x,yArthur,zMerlin) = 1 And by (any positive) soundness:
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness: x∈L ⇒ ∀ yArthur ∃ zMerlin R(x,yArthur,zMerlin) = 1 And by (any positive) soundness: x∉L ⇒ ∃ yArthur ∀ zMerlin R(x,yArthur,zMerlin) = 0
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness: x∈L ⇒ ∀ yArthur ∃ zMerlin R(x,yArthur,zMerlin) = 1 And by (any positive) soundness: x∉L ⇒ ∃ yArthur ∀ zMerlin R(x,yArthur,zMerlin) = 0 i.e., x∈L ⇔ ∀y ∃z R(x,y,z) = 1
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AM ⊆ Π2P
Consider any L with an AM protocol By perfect completeness: x∈L ⇒ ∀ yArthur ∃ zMerlin R(x,yArthur,zMerlin) = 1 And by (any positive) soundness: x∉L ⇒ ∃ yArthur ∀ zMerlin R(x,yArthur,zMerlin) = 0 i.e., x∈L ⇔ ∀y ∃z R(x,y,z) = 1 Similarly, MA ⊆ Σ2P
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AM and coNP
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP MAM protocol for L: Merlin sends y, and then they run an AM protocol for (x,y) ∈ L ’
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP MAM protocol for L: Merlin sends y, and then they run an AM protocol for (x,y) ∈ L ’ But MAM = AM
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP MAM protocol for L: Merlin sends y, and then they run an AM protocol for (x,y) ∈ L ’ But MAM = AM Corollary: If GI is NP-complete, PH collapses (recall GNI ∈ AM)
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP MAM protocol for L: Merlin sends y, and then they run an AM protocol for (x,y) ∈ L ’ But MAM = AM Corollary: If GI is NP-complete, PH collapses (recall GNI ∈ AM)
P BPP coNP NP
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AM and coNP
If coNP ⊆ AM, then PH collapses to level 2 Will show coNP ⊆ AM ⇒ Σ2P ⊆ AM ⊆ Π2P L ∈ Σ2P: { x| ∃y (x,y) ∈ L ’} where L ’ ∈ coNP MAM protocol for L: Merlin sends y, and then they run an AM protocol for (x,y) ∈ L ’ But MAM = AM Corollary: If GI is NP-complete, PH collapses (recall GNI ∈ AM)
P BPP coNP NP AM
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BPP RP
Zoo
P
PSPACE
EXP NP NEXP L Σ2P MA AM Π2P IP
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Program Checking
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Program Checking
Suppose a special computer (using nano-bio-quantum technology!) is being sold for solving Graph Non-Isomorphism (GNI) efficiently
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Program Checking
Suppose a special computer (using nano-bio-quantum technology!) is being sold for solving Graph Non-Isomorphism (GNI) efficiently How do we trust this?
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Program Checking
Suppose a special computer (using nano-bio-quantum technology!) is being sold for solving Graph Non-Isomorphism (GNI) efficiently How do we trust this? Vendor: Trust me, this always works
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Program Checking
Suppose a special computer (using nano-bio-quantum technology!) is being sold for solving Graph Non-Isomorphism (GNI) efficiently How do we trust this? Vendor: Trust me, this always works User: In fact I just care if it works correctly on the inputs I want to solve. Maybe for each input I have, your machine could prove correctness using an IP protocol?
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Program Checking
Suppose a special computer (using nano-bio-quantum technology!) is being sold for solving Graph Non-Isomorphism (GNI) efficiently How do we trust this? Vendor: Trust me, this always works User: In fact I just care if it works correctly on the inputs I want to solve. Maybe for each input I have, your machine could prove correctness using an IP protocol? Vendor: But I don’ t have a (nano-bio-quantum) implementation of the prover’ s program...
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Program Checking
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Program Checking
Program checker
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Program Checking
Program checker
User
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Program Checking
Program checker
User
P
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Program Checking
Program checker
User checker
P
x
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Program Checking
Program checker
User checker
P
x
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Program Checking
Program checker
User checker
P
x
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Program Checking
Program checker
User checker
P
x f(x) or P!f
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Program Checking
Program checker On each input, either ensures (w.h.p) that P’ s output is correct,
- r finds out that P"f, efficiently
User checker
P
x f(x) or P!f
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Program Checking
Program checker On each input, either ensures (w.h.p) that P’ s output is correct,
- r finds out that P"f, efficiently
Completeness: Vendor need not fear being falsely accused
User checker
P
x f(x) or P!f
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Program Checking
Program checker On each input, either ensures (w.h.p) that P’ s output is correct,
- r finds out that P"f, efficiently
Completeness: Vendor need not fear being falsely accused Soundness: User need not fear using a wrong value as f(x)
User checker
P
x f(x) or P!f
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Program Checking
Program checker On each input, either ensures (w.h.p) that P’ s output is correct,
- r finds out that P"f, efficiently
Completeness: Vendor need not fear being falsely accused Soundness: User need not fear using a wrong value as f(x) Will consider boolean f (i.e., a language L)
User checker
P
x f(x) or P!f
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Program Checking and IP
User f(x) or P!f
P
x
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Program Checking and IP
PC for L from IP protocols (for L and Lc)
User f(x) or P!f
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc)
User f(x) or P!f Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be
User f(x) or P!f Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC!
User f(x) or P!f Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC!
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC!
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC!
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC!
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC! Retains completeness and soundness
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC! Retains completeness and soundness e.g. For PSPACE-complete L (why?)
User f(x) or P!f
L
Verifier
P
x
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Prover
Program Checking and IP
PC for L from IP protocols (for L and Lc) PC must be efficient. Provers may not be If provers (for L and Lc) are efficient given L-oracle, can construct PC! Retains completeness and soundness e.g. For PSPACE-complete L (why?) How about Graph Isomorphism?
User f(x) or P!f
L
Verifier
P
x
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Program Checking for GI
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism Pick node v1 in G0. For each node u in G1 attach a marker (say a large clique) to u and v1 and ask if the new graphs G0’ and G1’ are isomorphic.
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism Pick node v1 in G0. For each node u in G1 attach a marker (say a large clique) to u and v1 and ask if the new graphs G0’ and G1’ are isomorphic. If P says no for all u in G1, report “P bad”
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism Pick node v1 in G0. For each node u in G1 attach a marker (say a large clique) to u and v1 and ask if the new graphs G0’ and G1’ are isomorphic. If P says no for all u in G1, report “P bad” Else remember v1u, and continue with v2; keep old markers and use new larger markers to get G0’’ and G1’’
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism Pick node v1 in G0. For each node u in G1 attach a marker (say a large clique) to u and v1 and ask if the new graphs G0’ and G1’ are isomorphic. If P says no for all u in G1, report “P bad” Else remember v1u, and continue with v2; keep old markers and use new larger markers to get G0’’ and G1’’ On finding isomorphism, verify and output G0 ≡ G1
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Program Checking for GI
If P(G0,G1) says G0 ≡ G1, try to extract the isomorphism Pick node v1 in G0. For each node u in G1 attach a marker (say a large clique) to u and v1 and ask if the new graphs G0’ and G1’ are isomorphic. If P says no for all u in G1, report “P bad” Else remember v1u, and continue with v2; keep old markers and use new larger markers to get G0’’ and G1’’ On finding isomorphism, verify and output G0 ≡ G1 Note: An IP protocol (i.e., NP proof) for GI, where prover is in PGI
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Program Checking for GI
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle)
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle) Let H = π(Gb) where π is a random permutation and b = 0 or 1 at random
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle) Let H = π(Gb) where π is a random permutation and b = 0 or 1 at random Run P(G0,H) with many such H
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle) Let H = π(Gb) where π is a random permutation and b = 0 or 1 at random Run P(G0,H) with many such H If P says G0 ≡ H exactly whenever b=0, output G0 ≢ G1
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle) Let H = π(Gb) where π is a random permutation and b = 0 or 1 at random Run P(G0,H) with many such H If P says G0 ≡ H exactly whenever b=0, output G0 ≢ G1 Else output “Bad P”
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Program Checking for GI
If P(G0,G1) says G0 ≢ G1, test P similar to in IP protocol for GNI (coke from can/bottle) Let H = π(Gb) where π is a random permutation and b = 0 or 1 at random Run P(G0,H) with many such H If P says G0 ≡ H exactly whenever b=0, output G0 ≢ G1 Else output “Bad P” Note: Prover in the IP protocol for GNI is in PGI
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Multi-Prover Interactive Proofs
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately Provers can’ t talk to each other during the interrogation (but can agree on a strategy a priori)
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately Provers can’ t talk to each other during the interrogation (but can agree on a strategy a priori) Verifier cross-checks answers from the provers
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately Provers can’ t talk to each other during the interrogation (but can agree on a strategy a priori) Verifier cross-checks answers from the provers 2 provers as good as k provers
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately Provers can’ t talk to each other during the interrogation (but can agree on a strategy a priori) Verifier cross-checks answers from the provers 2 provers as good as k provers MIP = NEXP
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Multi-Prover Interactive Proofs
Interrogate multiple provers separately Provers can’ t talk to each other during the interrogation (but can agree on a strategy a priori) Verifier cross-checks answers from the provers 2 provers as good as k provers MIP = NEXP Parallel repetition theorem highly non-trivial!
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Probabilistically Checkable Proofs (PCPs)
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof Verifier reads some positions (probabilistically chosen) from the proof and decides to accept or reject
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof Verifier reads some positions (probabilistically chosen) from the proof and decides to accept or reject PCP[r,q]: length of proof 2r, number of queries q
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof Verifier reads some positions (probabilistically chosen) from the proof and decides to accept or reject PCP[r,q]: length of proof 2r, number of queries q Intuitively, in MIP, the provers cannot change their strategy (because one does not know what the other sees), so must stick to a prior agreed up on strategy
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof Verifier reads some positions (probabilistically chosen) from the proof and decides to accept or reject PCP[r,q]: length of proof 2r, number of queries q Intuitively, in MIP, the provers cannot change their strategy (because one does not know what the other sees), so must stick to a prior agreed up on strategy Which will be the written proof
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Probabilistically Checkable Proofs (PCPs)
Prover submits a (very long) written proof Verifier reads some positions (probabilistically chosen) from the proof and decides to accept or reject PCP[r,q]: length of proof 2r, number of queries q Intuitively, in MIP, the provers cannot change their strategy (because one does not know what the other sees), so must stick to a prior agreed up on strategy Which will be the written proof PCP[poly,poly] = MIP = NEXP
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PCP Theorem
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PCP Theorem
NP = PCP[log,const]
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PCP Theorem
NP = PCP[log,const] PCP is only poly long (just like usual NP certificate)
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PCP Theorem
NP = PCP[log,const] PCP is only poly long (just like usual NP certificate) But verifier reads only constantly many bits!
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PCP Theorem
NP = PCP[log,const] PCP is only poly long (just like usual NP certificate) But verifier reads only constantly many bits! Extensively useful in proving “hardness of approximation” results for optimization problems
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PCP Theorem
NP = PCP[log,const] PCP is only poly long (just like usual NP certificate) But verifier reads only constantly many bits! Extensively useful in proving “hardness of approximation” results for optimization problems Also useful in certain cryptographic protocols
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Zero-Knowledge Proofs
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Zero-Knowledge Proofs
Interactive Proof for membership in L
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L
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SLIDE 100
Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L
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SLIDE 101
Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L
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SLIDE 102
Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated”
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SLIDE 103
Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated”
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SLIDE 104
Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated”
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SLIDE 105
Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated” x i n L
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SLIDE 106
Ah, got it!
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Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated” x i n L
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SLIDE 107
Ah, got it!
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Ah, got it!
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Zero-Knowledge Proofs
Interactive Proof for membership in L Complete and Sound ZK Property: Verifier “learns nothing” except that x is in L Verifier’ s view could have been “simulated” For every adversarial strategy, there exists a simulation strategy x i n L
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Summary
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Summary
Interactive Protocols
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Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness
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Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness Zoo: MA and AM, between 1st and 2nd levels of PH
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SLIDE 112
Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness Zoo: MA and AM, between 1st and 2nd levels of PH Other related concepts
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SLIDE 113
Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness Zoo: MA and AM, between 1st and 2nd levels of PH Other related concepts MIP, PCP, ZK proofs
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Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness Zoo: MA and AM, between 1st and 2nd levels of PH Other related concepts MIP, PCP, ZK proofs Understanding power of interaction/non-determinism and randomness
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SLIDE 115
Summary
Interactive Protocols Public coins, ATTMs, collapse of AM[k], arithmetization, set lower-bound, perfect completeness Zoo: MA and AM, between 1st and 2nd levels of PH Other related concepts MIP, PCP, ZK proofs Understanding power of interaction/non-determinism and randomness Useful in “hardness of approximation”, in cryptography, ...
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