ZKP9 SNARKs based on Linear PCP (Pinocchio Groth16)

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ZKP学习笔记

ZK-Learning MOOC课程笔记

Lecture 9: SNARKs based on Linear PCP (Yupeng Zhang)

• SNARKs learned so far
  

• Earliest Implemented SNARKs

  • Pros
    • Shortest proof size (3 elements [Groth16])
    • Fast verifier (bilinear pairing)
  • Cons
    • FFT and group exponentiations on the prover
    • Circuit-specific trusted setup

• History of SNARKs
  

9.1 Quadratic Arithmetic Program (QAP)

• Recall: SNARKs for circuit-satisfiability
  

• Transcript/trace of Circuit

  • Interactive proof (lecture 4, slide 76): value of every gate
  • Plonk (lecture 5, slide 42): left input, right input, output of every gate
  • QAP: input + output of every multiplication gate

• QAP

  • Ignore the output of the addition gates
    

  • Labeling multiplication gates

  • Selector Polynomials

    • $l_i(x)$: is $c_i$ the left input of gate 𝑗, for 𝑗 = 1,2,3?

      • Examples:
        

        For $l_1(x)$:

        • 3 is the left input of gate 1? Yes! -> 1
        • 3 is the left input of gate 2? No! -> 0
        • 3 is the left input of gate 3? No! -> 0
          

        For $l_3(x)$:

        • 1 is the left input of gate 1? No! -> 0
        • 1 is the left input of gate 2? No! -> 0
        • 1 is the left input of gate 3?
          • Yes! -> 1
          • Because “1” is the input of the addition gate, and the addition gate is the left input of gate 3
            

    • Properties of the selector polynomials
      

    • More Selector Polynomials

      • $r_i(x)$: is $c_i$ the right input of gate 𝑗, for 𝑗 = 1,2,3?
        

      • $o_i(x)$: is $c_i$ the output of gate 𝑗, for 𝑗 = 1,2,3?
        

    • Master polynomial
      

    • Vanishing polynomial
      

• Circuit-SAT to QAP [GGPR13, PGHR13]
  

• The table is sparse.

9.2 From QAP to SNARK

• Probabilistically Checkable Proofs (PCP)
  

• IPCP [Kalai-Raz’08] and IOP [Ben-Sasson-Chiesa-Spooner’16]
  

• Polynomial IOP [Bünz-Fisch-Szepieniec’20]
  

• Linear PCP [Ishai-Kushilevitz-Ostrovsky’07]
  

• QAP and Linear PCP
  

  • We don’t use random checks.

• Key Generation

  • The $c_i$ and $q(x)$ are private
  • The selector polynomials and the vanishing polynomial are public.
  • The circuit can be pre-processed. (The preprocessing phase is circuit-dependent)
    

• Prove
  

• Verify
  

• Towards the real protocol

  • Q1: How to make sure ${\pi }_1$ is computed from $g^{l_i(\tau )}$

    • Solution: Knowledge of Exponent assumption (KoE) or Generic Group Model (GGM)

    • Recall: KoE
      

    • Recall: GGM
      

  • Q2: how to make sure the same $c$ is used in ${\pi }_1$,${\pi }_2$ and ${\pi }_3$?

    • Solution
      

  • Q3: What about public input and output?
    

    • $I_{mid}$: secret witness
    • $I_{io}$: public input and public output

• Putting everything together
  

• Properties of SNARK [PGHR13]
  

9.3 Other variants

• Rank-1-Constraint-System (R1CS)

  • QAP
    

  • R1CS：
    

    • Advantages
      • Can support generalized constraints or gates
      • more convenient to use in practice
    • Matrix View of R1CS
      

• Groth16
  

  • Combine the ${\pi }_3$, ${\pi }_4$, ${\pi }_5$ of [PGHR13] together
    • $\alpha$ and $\beta$ are secret keys in the trusted key generation, and $g^{\alpha }$ and $g^{\beta }$ are public parameters for the prover and the verifier
    • ${\pi }_3$: move the ${\Sigma }_{i=1}^m{c}_i\times o_i(x)$ to the right side of the equation -> ${\Sigma }_{i=1}^m{c}_i\times o_i(x)+V(x)q(x)$
  • Change the keygen accordingly
  • Proof size: 3 group elements, 144 bytes
  • Verifier time: 1 pairing equation

• Achieving Zero-Knowledge

  • The above is not zero-knowledge, because the adversary can infer some information by brute force attack.
  • Solution: add some random values (times the vanishing polynomial)
    • The [PGHR13] version:
      

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