Zero-Knowledge Proofs in Payment Channels: Verifiable Settlement with zk-SNARKs

Zero-Knowledge Proofs in Payment Channels: Verifiable Settlement with zk-SNARKs

In traditional payment rails, auditing transactions requires complete exposure of sender identities, recipient wallets, and balance sums. In public distributed ledgers, this transparency creates severe privacy vulnerabilities.

How can a financial network prove with mathematical certainty that Account A had sufficient balance to send X units to Account B, without revealing the balances, the transaction amount, or the parties involved?

Enter zk-SNARKs (Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge). At Kone Pay, we study cryptographic circuits that enable private, compliant financial verification.

1. What Makes a Proof "Zero-Knowledge"?

A zero-knowledge proof satisfies three rigorous properties:

1. Completeness: If the statement is true and both prover and verifier are honest, the verifier will always be convinced.
2. Soundness: If the statement is false, no cheating prover can convince the verifier, except with negligible mathematical probability ($1 / 2^{128}$).
3. Zero-Knowledge: The verifier learns nothing beyond the validity of the statement. The verifier gains zero information about the prover's secret witness.

2. From Computation to Arithmetic Circuits

Computations in zk-SNARKs are translated into arithmetic circuits over a finite field $\mathbb{F}_p$.

Consider a basic transaction rule:

To express these constraints algebraically, they are represented as Rank-1 Constraint Systems (R1CS):

$(\vec{A} \cdot \vec{s}) \odot (\vec{B} \cdot \vec{s}) = (\vec{C} \cdot \vec{s})$

Where:

• $\vec{s}$ is the solution vector containing public inputs and private witness values ($1, \text{pub}_1, \dots, w_1, w_2$).
• $\vec{A}, \vec{B}, \vec{C}$ are coefficient matrices defining addition and multiplication gate wiring.

3. Quadratic Arithmetic Programs (QAP)

R1CS verifies gates one by one. To make proofs succinct (constant size, e.g., a few hundred bytes), R1CS is transformed into polynomials via Lagrange interpolation:

Where $T(x)$ is the target polynomial whose roots correspond to each constraint in the circuit. If $A(x)B(x) - C(x)$ is cleanly divisible by $T(x)$, all constraints in the financial transaction hold simultaneously!

4. Building a Confidential Transfer Circuit in Circom

Below is an illustrative Circom snippet validating a balance transfer without leaking values:

pragma circom 2.1.6;

include "bitify.circom";
include "comparators.circom";
include "poseidon.circom";

template PrivateTransfer() {
    // Private Witness Inputs
    signal input senderOldBalance;
    signal input amount;
    signal input senderPrivateKey;

    // Public Inputs
    signal input senderCommitment;
    signal input expectedNewCommitment;

    // 1. Check Sender has sufficient funds (no negative balances)
    component comp = GreaterEqThan(64);
    comp.in[0] <== senderOldBalance;
    comp.in[1] <== amount;
    comp.out === 1;

    // 2. Compute updated balance
    signal senderNewBalance;
    senderNewBalance <== senderOldBalance - amount;

    // 3. Verify public cryptographic commitments match via Poseidon Hash
    component hashOld = Poseidon(2);
    hashOld.inputs[0] <== senderOldBalance;
    hashOld.inputs[1] <== senderPrivateKey;
    hashOld.out === senderCommitment;

    component hashNew = Poseidon(2);
    hashNew.inputs[0] <== senderNewBalance;
    hashNew.inputs[1] <== senderPrivateKey;
    hashNew.out === expectedNewCommitment;
}

component main {public [senderCommitment, expectedNewCommitment]} = PrivateTransfer();

5. Settlement Performance and Verification

The beauty of zk-SNARKs lies in asymmetry:

• Proving Time: Proving generation requires significant polynomial evaluations and elliptic curve multi-scalar multiplications (MSMs), taking hundreds of milliseconds.
• Verification Time: Verification takes under 5 milliseconds with a pairing check ($e(A, B) = e(\alpha, \beta) \cdot e(C, \gamma)$), regardless of whether the circuit contained 100 or 100,000 constraints!

The Kone Pay Engineering Perspective

In Kone Pay's Cryptographic Systems Curriculum, engineers explore zero-knowledge rollups, balance proofs, and regulatory compliance circuits in our Fintech & Ledger Gateways Track.