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zkp-toolkit Gadget List

https://hackmd.io/@p0n1/ByEc6d928

This doc mainly lists the gadgets that the zkp-toolkit library has implemented in the first stage. In the implementation process, the code of bellman, libsnark, and ethsnarks is referenced. The code of each gadget is located in the gadgets folder. We briefly analyze the design and implementation of each gadget here.

Gadget list

Many other useful gadgets to come.

How to write zkp-toolkit gadget

In zkp-toolkit, there is a trait called ConstraintSynthesizer. To complete the verification in Groth16, it is necessary to inherit this trait. Under this trait, there is a function generate_constraints, which implements the construction of the circuit for with or without the substitution of the witness. It will be used twice, once in the setup phase, the witness is set to None, and once when the proof is generated and the witness value is substituted for specific calculations.

/// Computations are expressed in terms of rank-1 constraint systems (R1CS).
/// The `generate_constraints` method is called to generate constraints for
/// both CRS generation and for proving.
pub trait ConstraintSynthesizer<F: Field> {
    /// Drives generation of new constraints inside `CS`.
    fn generate_constraints<CS: ConstraintSystem<F>>(
        self,
        cs: &mut CS,
    ) -> Result<(), SynthesisError>;
}

After implementing the circuit in the generate_constraints function above, you may need to test this circuit. The general test steps are as follows:

Generate pvk from the generated random value and empty circuit.

let mut rng = &mut test_rng();
let n = 10u64; // |lhs - rhs| < 2^10

println!("Creating parameters...");
let params = {
  let c = RangeProof::<Fr> {
    lhs: None,
    rhs: None,
    n: n,
  };

  generate_random_parameters::<Bls12_381, _, _>(c, &mut rng).unwrap()
};

let pvk = prepare_verifying_key(&params.vk);

Assign withness to the circuit instance to generate a proof.

println!("Creating proofs...");

let c1 = RangeProof::<Fr> {
  lhs: Some(Fr::from(24u32)),
  rhs: Some(Fr::from(25u32)),
  n: n,
};

let proof = create_random_proof(c1, &params, &mut rng).unwrap();

Use vk, proof, and public input to verify that the proof.

println!("Proofs ok, start verify...");

assert!(verify_proof(&pvk, &proof, &[Fr::from(2u32).pow(&[n])]).unwrap());

The rangeproof test sample is as follows:

#[test]
fn test_rangeproof() {
    use ark_bls12_381::{Bls12_381, Fr};
    use ark_ff::Field;
    use ark_stdtest_rng;
    use zkp_groth16::{
        create_random_proof, generate_random_parameters, prepare_verifying_key, verify_proof,
    };

    let mut rng = &mut test_rng();
    let n = 10u64; // |lhs - rhs| < 2^10

    println!("Creating parameters...");
    let params = {
        let c = RangeProof::<Fr> {
            lhs: None,
            rhs: None,
            n: n,
        };

        generate_random_parameters::<Bls12_381, _, _>(c, &mut rng).unwrap()
    };

    let pvk = prepare_verifying_key(&params.vk);

    println!("Creating proofs...");

    let c1 = RangeProof::<Fr> {
        lhs: Some(Fr::from(24u32)),
        rhs: Some(Fr::from(25u32)),
        n: n,
    };

    let proof = create_random_proof(c1, &params, &mut rng).unwrap();
    println!("Proofs ok, start verify...");

    assert!(verify_proof(&pvk, &proof, &[Fr::from(2u32).pow(&[n])]).unwrap());
}

Details

rangeproof

Features

Compare the size of two variables (lhs, rhs)

Constraint

Calculate alpha_packed = 2 ^ n + B - A, to verify that the two are equal, add a constraint

1 * (2 ^ n + B-A) = alpha_packed
The alpha array represents the binary representation of alpha_packed, to verify that the two are equal, add a constraint

1 * sum (bits) = alpha_packed
To verify that each bit of the alpha array is binary

(1 - bits_i) * bits_i = 0

Calculate sum = ∑ bits_i (i = 0..n-1)

When sum = 0, it means that lhs = rhs, let output = 0, inv = 0

When sum != 0, it means lhs != rhs, let output = 1, inv = 1 / sum, output (that is, not_all_zeros) indicates whether two numbers are equal.
To ensure that the output is a binary number, verify

(1-output) * output = 0
Constrain the relationship between output and sum, there are two equality constraints: when the sum is not 0, the output must be 1

(1-output) * sum = 0
When sum = 0, the output must also be 0

inv * sum = output
less_or_eq is alpha[n], so the relationship between less_or_eq and less can be expressed as

less_or_eq * output = less
Verify whether the less relationship is established: less * 1 = 1

Code

rangeproof

isnonezero

Features

Prove if the variable value is non-zero.

Constraint

There are two constraints in the gadget in ethsnarks. X is the input value and Y is the output value. The two constraints are:

$$ X * (1-Y) = 0 $$
$$ X * (1 / X) = 0 $$

There is no result output variable Y in zkp-toolkit, so the constraints are:

$$ X * (1 / X) = 0 $$

Code

isnonzero

lookup_1bit

Features

The index of array c is represented by b of 1-bit size, the size of array c is 2, and finally, the result r is a specific element in the C array. This is the lookup table gadget as in ethsnarks.

Constraint

Input 1-bit b, variable array c, result value r:

$$ (c [0] + b * c [1] - (b * c [0])) * 1 = r $$

When b is 0, r = c [0]
When b is 1, r = c [1]

Code

lookup_1bit

lookup_2bit

Features

The index of array c is represented by b of 2-bit size, the size of array c is 4, and finally, the result r is a specific element in the C array.

Constraint

Input 2-bits b, variable array c, result value r:

$$ (c [1]-c [0] + (b [1] * (c [3]-c [2]-c [1] + c [0]))) * b [0] = -c [ 0] + r + (b [1] * (-c [2] + c [0])) $$

The above constraints refer to the implementations in ethsnarks. If the above constraint equation holds, it must be satisfied:

b: 00, r = c [0]

b: 01, r = c [1]

b: 10, r = c [2]

b: 11, r = c [3]

Code

lookup_2bit

lookup_3bit

Features

The index of array c is represented by b of 3-bit size, the size of array c is 8, and finally, the result r is a specific element in the C array.

Constraint

Input 3-bits b, variable array c, result value r:

$$ (c [0] + \ (b [0] *-c [0]) + \(b [0] * c [1]) + \(b [1] *-c [0]) + \(b [1] * c [2]) + \(b [0] * b [1] * (-c [1] + -c [2] + c [0] + c [3])) + \(b [2] * ( -c [0] + c [4])) + \(b [0] * b [2] * (c [0] -c [1] -c [4] + c [5])) + \(b [ 1] * b [2] * (c [0] -c [2] -c [4] + c [6])) + \(b [0] * b [1] * b [2] * (-c [0] + c [1] + c [2] -c [3] + c [4] -c [5] -c [6] + c [7])))) * 1 = r $$

The above constraints refer to the implementation of ethsnarks. If the above constraint equation holds, it must be satisfied:

b: 000, r = c [0]

b: 001, r = c [1]

b: 010, r = c [2]

b: 011, r = c [3]

b: 100, r = c [4]

b: 101, r = c [5]

b: 110, r = c [6]

b: 111, r = c [7]

Code

lookup_3bit.rs

merkletree

Features

Given a Merkle tree verification path (Merkle proof), and a leaf node and a root node, verify that the results computed for the leaf node on this verification path are the same as the root node.

Constraint

There are 3 types of constraints in merkletree:

merkletree calculates the constraints of each left_digests and right_digests on the path. Every bit on the hash needs to satisfy digests[i] * (1 - digests[i]) = 0 so the total number of constraints here is (2 * digest_size - 1) * tree_depth.
A hash constraint from the lowest level of merkletree to the root node tree_depth (where tree_depth is the actual tree height minus 1). Assuming each hash constraint x, the total constraint is tree_depth * x
Determine the left and right positions of the leaf hash node and the internal hash node by the bit of address_bit. The constraint of each hash bit is: is_right * (right.bits[i] - left.bits[i]) = (input.bits[i] - left.bits[i]) digest_size Constraint, looping tree_depth altogether. Number of constraints: digest_size * tree_depth.
Finally, the hash result calculated by the circuit is compared with the expected hash result. The circuit constraints are: root_digest[i] * 1 = computed_root[i] digests_size constraints.

Code

merkletree.rs

boolean

Features

This boolean logic gadget mainly refers to bellman/gadget/boolean.rs. There are various gadget operations on boolean variables, the Boolean is the encapsulation of AllocatedBit.

Constraint

The following bool operations are implemented:

AllocatedBit:
xor: Performs an XOR operation over the two operands
and: Performs an AND operation over the two operands
and_not Calculates: a AND (NOT b)
nor: Calculates (NOT a) AND (NOT b)
u64_into_boolean_vec_le: u64 to Vec<Boolean>
field_into_boolean_vec_le: Field to Vec<Boolean>
field_into_allocated_bits_le: Vec<AllocatedBit>

Code

boolean.rs

mimc

Features

mimc hash function

Constraint

xL, xR: = xR + (xL + Ci) ^ 3, xL

tmp = (xL + Ci) ^ 2

new_xL = xR + (xL + Ci) ^ 3

new_xL = xR + tmp * (xL + Ci)

new_xL-xR = tmp * (xL + Ci)

Constraint: MIMC_ROUNDS round new_xL = xR + (xL + Ci) ^ 3 constraints

Code

mimc.rs

Files

gadgets

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gadgets

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parent directory

README.md

zkp-toolkit Gadget List

Gadget list

How to write zkp-toolkit gadget

Details

rangeproof

Features

Constraint

Code

isnonezero

Features

Constraint

Code

lookup_1bit

Features

Constraint

Code

lookup_2bit

Features

Constraint

Code

lookup_3bit

Features

Constraint

Code

merkletree

Features

Constraint

Code

boolean

Features

Constraint

Code

mimc

Features

Constraint

Code