Understanding Zero-Knowledge Proofs: Encoding Programs into zk-SNARKs

What are Zero-Knowledge Proofs?

Zero-knowledge proofs allow one party (the prover) to convince another party (the verifier) that they know a value or solution without revealing any information about the value itself. This seemingly paradoxical concept is essential in various applications, such as secure transactions, authentication, and privacy-preserving protocols.

The Concept of zk-SNARKs

zk-SNARKs are a specific type of zero-knowledge proof that provides succinct, non-interactive proofs. They enable efficient verification of computations without requiring interaction between the prover and verifier. zk-SNARKs are particularly useful in blockchain technologies, where they ensure the integrity and privacy of transactions without burdening the network with excessive data.

Encoding Programs into zk-SNARKs

To understand how programs are encoded into zk-SNARKs, let's break down the process into manageable steps. We'll use a simple example where we want to prove knowledge of two numbers a and b such that their sum equals a given number c.

1. Define the Computation: The first step is to define the computation we want to encode. In our case, the computation is the addition of two numbers:

c = a + b

2. Write the Program as a Circuit: We then express this computation as an arithmetic circuit. A circuit in this context consists of gates (like addition or multiplication) connected by wires that process input values and produce output values.

3. Translate the Circuit into Constraints: The circuit is converted into a set of constraints, specifically Rank-1 Constraint Systems (R1CS). Each constraint is a polynomial equation that ensures the computation is performed correctly. For our example, the constraint is:

a + b − c = 0

4. Generate the zk-SNARK Proof: Finally, we use a zk-SNARK library to create the proving and verification keys, generate the proof, and verify it.

Practical Implementation using snarkjs

To illustrate these steps, we’ll use the snarkjs library, a popular JavaScript tool for working with zk-SNARKs.

Step 1: Install the Necessary Tools

First, ensure you have node.js installed. Then, install snarkjs globally:

npm install -g snarkjs

Step 2: Define the Circuit

Create a file named circuit.circom with the following content:

pragma circom 2.0.0;

template Sum() {
    signal input a;
    signal input b;
    signal output c;

    c <== a + b;
}

component main = Sum();

This file defines a simple addition circuit with two inputs and one output.

Step 3: Compile the Circuit

Compile the circuit using the circom compiler:

circom circuit.circom --r1cs --wasm --sym --c

This command generates several files, including circuit.r1cs, circuit.wasm, and circuit.sym, which are essential for the next steps.

Step 4: Generate the Trusted Setup

Perform the trusted setup phase, which is crucial for generating the proving and verification keys:

snarkjs groth16 setup circuit.r1cs pot12_final.ptau circuit_0000.zkey

snarkjs zkey contribute circuit_0000.zkey circuit_final.zkey --name="First contribution"

snarkjs zkey export verificationkey circuit_final.zkey verification_key.json

Step 5: Generate the Proof

Create an input file input.json with the inputs for the circuit:

{
    "a": 3,
    "b": 4
}

Calculate the witness (intermediate computations):

snarkjs wtns calculate circuit.wasm input.json witness.wtns

Generate the proof:

snarkjs groth16 prove circuit_final.zkey witness.wtns proof.json public.json

Step 6: Verify the Proof

Finally, verify the proof:

snarkjs groth16 verify verification_key.json public.json proof.json

If the proof is valid, you will get an output indicating that the proof is correct.

Explanation of the Implementation

• Circuit Definition: The circuit.circom file defines a simple addition circuit where c=a+bc = a + bc=a+b.
• Compilation: The circom compiler translates the high-level circuit description into an R1CS format, which represents the constraints of the computation.
• Trusted Setup: The snarkjs groth16 setup and subsequent commands generate the proving and verification keys necessary for zk-SNARKs. This phase includes a ceremony where randomness is introduced to ensure the security of the keys.
• Proof Generation: Inputs are provided in a JSON file, and the witness (intermediate values) is calculated. The proof is then generated using the proving key.
• Verification: The generated proof is verified against the verification key to ensure that the computation was performed correctly without revealing the actual inputs.

Conclusion

Encoding programs into zk-SNARKs involves translating a computation into a series of constraints that can be efficiently verified without revealing the underlying data. This process ensures both the integrity and privacy of the computation, making zk-SNARKs a powerful tool for various applications, from secure transactions to privacy-preserving protocols.

By following the steps outlined in this article, you can begin experimenting with zk-SNARKs and explore their potential for enhancing security and privacy in your applications. The snarkjs library provides a practical and accessible way to work with zk-SNARKs, allowing you to implement and verify zero-knowledge proofs in your projects.

If you want to learn more about Zero Knowledge Proofs and how to implement a multi-party computation yourself using Rust, check out this article: Implementing Zero Knowledge Multi-Party Computation in Rust