r1cs

This is a rust library for building R1CS gadgets over prime fields, which are useful in SNARKs and other argument systems.

An R1CS instance is defined by three matrices, A, B and C. These encode the following NP-complete decision problem: does there exist a witness vector w such that Aw ∘ Bw = Cw?

A gadget for some R1CS instance takes a set of inputs, which are a subset of the witness vector. If the given inputs are valid, it extends the input set into a complete witness vector which satisfies the R1CS instance.

Core types

Field is a trait representing prime fields. An Element<F> is an element of the prime field F.

A Wire is an element of the witness vector. An Expression<F> is a linear combination of wires.

A BooleanWire is a Wire which has been constrained in such a way that it can only equal 0 or 1. Similarly, a BooleanExpression<F> is an Expression<F> which has been so constrained.

A BinaryWire is a vector of BooleanWires. Similarly, a BinaryExpression<F> is a vector of BooleanExpression<F>s.

Basic example

Here's a simple gadget which computes the cube of a BN128 field element:

```rust // Create a gadget which takes a single input, x, and computes xxx. let mut builder = GadgetBuilder::::new(); let x = builder.wire(); let xexp = Expression::from(x); let xsquared = builder.product(&xexp, &xexp); let xcubed = builder.product(xsquared, x_exp); let gadget = builder.build();

// This structure maps wires to their (field element) values. Since // x is our input, we will assign it a value before executing the // gadget. Other wires will be computed by the gadget. let mut values = values!(x => 5u8.into());

// Execute the gadget and assert that all constraints were satisfied. let constraintssatisfied = gadget.execute(&mut values); assert!(constraintssatisfied);

// Check the result. asserteq!(Element::from(125u8), xcubed.evaluate(&values)); ```

This can also be done more succinctly with builder.exp(x_exp, 3), which performs exponentiation by squaring.

Custom fields

You can define a custom field by implementing the field::Field trait. As an example, here's the definition of Bn128 which was referenced above:

```rust pub struct Bn128 {}

impl Field for Bn128 { fn order() -> BigUint { BigUint::from_str( "21888242871839275222246405745257275088548364400416034343698204186575808495617" ).unwrap() } } ```

Boolean algebra

The example above involved native field arithmetic, but this library also supports boolean algebra. For example, here is a function which implements the boolean function Maj, as defined in the SHA-256 specification:

rust fn maj<F: Field>(builder: &mut GadgetBuilder<F>, x: BooleanExpression<F>, y: BooleanExpression<F>, z: BooleanExpression<F>) -> BooleanExpression<F> { let x_y = builder.and(&x, &y); let x_z = builder.and(&x, &z); let y_z = builder.and(&y, &z); let x_y_xor_x_z = builder.xor(x_y, x_z); builder.xor(x_y_xor_x_z, y_z) }

Binary operations

This library also supports bitwise operations, such as bitwise_and, and binary arithmetic operations, such as binary_sum.

Permutation networks

To verify that two lists are permutations of one another, you can use assert_permutation. This is implemented using AS-Waksman permutation networks, which permute n items using roughly n log_2(n) - n switches. assert_permutation uses two constraints per switch: one "is boolean" check and one constraint for routing.

Permutation networks make it easy to implement sorting gadgets, which we provide in the form of sort_ascending and sort_descending.

Non-determinism

Suppose we wish to compute the multiplicative inverse of a field element x. While this is possible to do in a deterministic arithmetic circuit, it is prohibitively expensive. What we can do instead is have the user compute x_inv = 1 / x, provide the result as a witness element, and add a constraint in the R1CS instance to verify that x * x_inv = 1.

GadgetBuilder supports such non-deterministic computations via its generator method, which can be used like so:

```rust fn inverse(builder: &mut GadgetBuilder, x: Expression) -> Expression { // Create a new witness element for xinv. let xinv = builder.wire();

// Add the constraint x * x_inv = 1.
builder.assert_product(&x, Expression::from(x_inv),
                       Expression::one());

// Non-deterministically generate x_inv = 1 / x.
builder.generator(
    x.dependencies(),
    move |values: &mut WireValues<F>| {
        let x_value = x.evaluate(values);
        let x_inv_value = x_value.multiplicative_inverse();
        values.set(x_inv, x_inv_value);
    },
);

// Output x_inv.
x_inv.into()

} ```

This is roughly equivalent to GadgetBuilder's built-in inverse method, with slight modifications for readability.

Disclaimer

This code has not been thoroughly reviewed or tested, and should not be used in any production systems.