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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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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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′

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o

o

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ZKP for 3-coloring

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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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