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CMU 15-251
Interactive proofs
Teachers:
Anil Ada
Ariel Procaccia (this time)
Our protagonists
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NP, revisited
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𝑥 ∈ 𝐴
NP, revisited
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• ¬
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o
o
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Coke vs. Pepsi
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IP for Coke vs. Pepsi
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Graph nonisomorphism
• 𝐺0 = (𝑉0, 𝐸0) 𝐺1 =𝑉1, 𝐸1
𝜋: 𝑉0 → 𝑉1𝑢, 𝑣 ∈ 𝐸0 ⇔ 𝜋 𝑢 , 𝜋 𝑣 ∈ 𝐸1
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1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
Graph nonisomorphism
• 𝐺0 𝐺1𝐺1 𝐺2 𝐺0
𝐺2•
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•
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IP for Graph nonisomorphism
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𝑏 ∈ {0,1}𝜋
𝜋(𝐺𝑏)
𝑏′
𝑏 = 𝑏′
IP for Graph nonisomorphism
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𝐺0:
𝐺1:
𝐺0: 𝜋(𝐺0):
0
1 2
3 4
1 2
3 4
IP for Graph nonisomorphism
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1. 1 1/𝑛!
2. 1 1/2
3. 1/2 1/𝑛!
4. 1/2 1/2
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𝑏 ∈ {0,1} 𝜋𝜋(𝐺𝑏)
𝑏′𝑏 = 𝑏′
Interactive proofs
• 𝐿
𝑃𝑉
o :
∀𝑥 ∈ 𝐿, Pr 𝑉 ↔ 𝑃 𝑥 = 1
o :
∀𝑥 ∉ 𝐿, ∀𝑃′, Pr 𝑉 ↔ 𝑃′ 𝑥 ≤ 1/2
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Interactive proofs
•
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½
Interactive proofs
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⊂
⊂
=
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Zero knowledge proofs
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•
•
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Why do we need ZKPs?
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Intuition for ZKPs
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ZKP for graph isomorphism
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𝑏 ∈ {0,1} 𝜋𝐻 = 𝜋(𝐺𝑏)
𝑏′
𝜋′
𝐻 = 𝜋′ 𝐺𝑏′
ZKP for graph isomorphism
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𝑏 ∈ {0,1} 𝜋𝐻 = 𝜋(𝐺𝑏)
𝑏′𝜋′
𝐻 = 𝜋′ 𝐺𝑏′
•
o
o
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ZKP for 3-coloring
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•
•
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ZKP for 3-coloring
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𝜋 𝑅, 𝐺, 𝐵
𝜋 𝛾 𝑣 𝑣 ∈ 𝑉
𝑢, 𝑣 ∈ 𝐸
𝑎 = 𝜋 𝛾 𝑢
𝑏 = 𝜋(𝛾 𝑣 )
𝑎 ≠ 𝑏
ZKP for 3-coloring
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𝑐 𝑑
𝑒
𝑏
𝑎
𝛾(𝐺)
𝑐 𝑑
𝑐 𝑑
• 𝐺0 𝐺1(𝐻, 𝑏′, 𝜋′)
𝐻 = 𝜋′ 𝐺𝑏′
3. 1 −1
3!
4. 1 −1
𝐸
ZKP for 3-coloring
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𝜋 𝑅, 𝐺, 𝐵
𝜋 𝛾 𝑣 𝑣 ∈ 𝑉
𝑢, 𝑣 ∈ 𝐸
𝑎 = 𝜋 𝛾 𝑢 𝑏 = 𝜋(𝛾 𝑣 )
𝑎 ≠ 𝑏
• 𝐺
1. 1 −1
2
2. 1 −1
𝑛!
ZKP for 3-coloring
•
•
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𝜋 𝑅, 𝐺, 𝐵
𝜋 𝛾 𝑣 𝑣 ∈ 𝑉
𝑢, 𝑣 ∈ 𝐸
𝑎 = 𝜋 𝛾 𝑢 𝑏 = 𝜋(𝛾 𝑣 )
𝑎 ≠ 𝑏
More generally
•
•
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What you need to know
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