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Day 47/100 of ZK
Today we focused on the prover side of Groth16: how the actual proof is constructed once the QAP is ready.
From Witness to Polynomial Evaluations
The prover starts with the witness vector w (public inputs, private inputs, and all intermediate values). Using this witness, the prover evaluates the three QAP polynomials at the secret point τ from the trusted setup:
A(τ) = sum of (witness coefficients × powers of τ from the proving key)
B(τ) = same process for the B polynomial
C(τ) = same process for the C polynomial
These evaluations are performed as multi-scalar multiplications against the precomputed powers of τ stored in the proving key. This is usually the most expensive part of proving.
Adding Blinding Factors
If the prover simply sent [A(τ)], [B(τ)], and [C(τ)], an attacker could potentially extract information or forge proofs by solving linear equations. To prevent this, the prover adds random blinding terms.
The final proof elements become:
π_A = [A(τ) + α·r + blinding terms]₁
π_B = [B(τ) + β·s + blinding terms]₂
π_C = [C(τ) + H(τ)·Z(τ) + more blinding terms]₁
Here r and s are fresh random field elements chosen by the prover. The extra terms cancel out correctly during verification for an honest prover but break the equations if someone tries to cheat.
Assembling the Three Proof Elements
After the blinded evaluations are computed, the prover packages them into the three group elements that form the complete Groth16 proof. These three elements are all that the verifier will ever see.
The entire process compresses the full witness and the entire QAP into a constant-size proof, independent of the number of constraints.
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Day 48/100 of ZK 🔐
Today we focused on the verification side of Groth16: the exact pairing checks the verifier performs and how public inputs are incorporated.
The Core Verification Equation
After receiving the three proof elements π_A, π_B, and π_C, the verifier uses the verification key to check a pairing equation of the following form:
e(π_A, π_B) = e([α]₁, [β]₂) · e(public_input_terms, [γ]₂) · e(π_C, [δ]₂)
In expanded form this checks that:
A(τ) · B(τ) = αβ + public_input_contribution + C(τ) + H(τ)·Z(τ) · δ-related terms
The left side comes from the product of the first two proof elements. The right side reconstructs the expected value using the verification key and the public inputs.
How Public Inputs Are Handled
Public inputs are not hidden. The verifier knows them and therefore can compute a linear combination of the corresponding verification-key elements. This combination is inserted into the pairing equation so that the public values are correctly accounted for without the prover having to hide them.
If the public inputs are wrong, or if the prover used inconsistent private values, the equation fails.
Why Blinding Terms Cancel
The random blinding factors the prover added on Day 47 are carefully constructed so that they cancel out on both sides of the pairing equation when the proof is honest. A malicious prover who tries to forge a proof without a valid witness cannot make the blinded terms cancel correctly, so the pairings will not match.
Result
If all pairing checks pass, the verifier is convinced that the original R1CS constraints were satisfied for the given public inputs, without learning anything about the private witness.