A simple example
Let’s start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input , and will prove knowledge of two private inputs and such that
where is a fixed number (field element), such as
Define instructions
Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable.
For our circuit, we will use three instructions:
- Load a private number into the circuit.
- Multiply two numbers.
- Expose a number as a public input to the circuit.
We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals.
trait NumericInstructions<F: Field>: Chip<F> {
/// Variable representing a number.
type Num;
/// Loads a number into the circuit as a private input.
fn load_private(&self, layouter: impl Layouter<F>, a: Value<F>) -> Result<Self::Num, Error>;
/// Loads a number into the circuit as a fixed constant.
fn load_constant(&self, layouter: impl Layouter<F>, constant: F) -> Result<Self::Num, Error>;
/// Returns `c = a * b`.
fn mul(
&self,
layouter: impl Layouter<F>,
a: Self::Num,
b: Self::Num,
) -> Result<Self::Num, Error>;
/// Exposes a number as a public input to the circuit.
fn expose_public(
&self,
layouter: impl Layouter<F>,
num: Self::Num,
row: usize,
) -> Result<(), Error>;
}Define a chip implementation
For our circuit, we will build a chip that provides the above numeric instructions for a finite field.
/// The chip that will implement our instructions! Chips store their own
/// config, as well as type markers if necessary.
struct FieldChip<F: Field> {
config: FieldConfig,
_marker: PhantomData<F>,
}Every chip needs to implement the Chip trait. This defines the properties of the chip
that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial
state that the chip requires to be loaded into the circuit.
impl<F: Field> Chip<F> for FieldChip<F> {
type Config = FieldConfig;
type Loaded = ();
fn config(&self) -> &Self::Config {
&self.config
}
fn loaded(&self) -> &Self::Loaded {
&()
}
}Configure the chip
The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions.
/// Chip state is stored in a config struct. This is generated by the chip
/// during configuration, and then stored inside the chip.
#[derive(Clone, Debug)]
struct FieldConfig {
/// For this chip, we will use two advice columns to implement our instructions.
/// These are also the columns through which we communicate with other parts of
/// the circuit.
advice: [Column<Advice>; 2],
/// This is the public input (instance) column.
instance: Column<Instance>,
// We need a selector to enable the multiplication gate, so that we aren't placing
// any constraints on cells where `NumericInstructions::mul` is not being used.
// This is important when building larger circuits, where columns are used by
// multiple sets of instructions.
s_mul: Selector,
}
impl<F: Field> FieldChip<F> {
fn construct(config: <Self as Chip<F>>::Config) -> Self {
Self {
config,
_marker: PhantomData,
}
}
fn configure(
meta: &mut ConstraintSystem<F>,
advice: [Column<Advice>; 2],
instance: Column<Instance>,
constant: Column<Fixed>,
) -> <Self as Chip<F>>::Config {
meta.enable_equality(instance);
meta.enable_constant(constant);
for column in &advice {
meta.enable_equality(*column);
}
let s_mul = meta.selector();
// Define our multiplication gate!
meta.create_gate("mul", |meta| {
// To implement multiplication, we need three advice cells and a selector
// cell. We arrange them like so:
//
// | a0 | a1 | s_mul |
// |-----|-----|-------|
// | lhs | rhs | s_mul |
// | out | | |
//
// Gates may refer to any relative offsets we want, but each distinct
// offset adds a cost to the proof. The most common offsets are 0 (the
// current row), 1 (the next row), and -1 (the previous row), for which
// `Rotation` has specific constructors.
let lhs = meta.query_advice(advice[0], Rotation::cur());
let rhs = meta.query_advice(advice[1], Rotation::cur());
let out = meta.query_advice(advice[0], Rotation::next());
let s_mul = meta.query_selector(s_mul);
// Finally, we return the polynomial expressions that constrain this gate.
// For our multiplication gate, we only need a single polynomial constraint.
//
// The polynomial expressions returned from `create_gate` will be
// constrained by the proving system to equal zero. Our expression
// has the following properties:
// - When s_mul = 0, any value is allowed in lhs, rhs, and out.
// - When s_mul != 0, this constrains lhs * rhs = out.
vec![s_mul * (lhs * rhs - out)]
});
FieldConfig {
advice,
instance,
s_mul,
}
}
}Implement chip traits
/// A variable representing a number.
#[derive(Clone)]
struct Number<F: Field>(AssignedCell<F, F>);
impl<F: Field> NumericInstructions<F> for FieldChip<F> {
type Num = Number<F>;
fn load_private(
&self,
mut layouter: impl Layouter<F>,
value: Value<F>,
) -> Result<Self::Num, Error> {
let config = self.config();
layouter.assign_region(
|| "load private",
|mut region| {
region
.assign_advice(|| "private input", config.advice[0], 0, || value)
.map(Number)
},
)
}
fn load_constant(
&self,
mut layouter: impl Layouter<F>,
constant: F,
) -> Result<Self::Num, Error> {
let config = self.config();
layouter.assign_region(
|| "load constant",
|mut region| {
region
.assign_advice_from_constant(|| "constant value", config.advice[0], 0, constant)
.map(Number)
},
)
}
fn mul(
&self,
mut layouter: impl Layouter<F>,
a: Self::Num,
b: Self::Num,
) -> Result<Self::Num, Error> {
let config = self.config();
layouter.assign_region(
|| "mul",
|mut region: Region<'_, F>| {
// We only want to use a single multiplication gate in this region,
// so we enable it at region offset 0; this means it will constrain
// cells at offsets 0 and 1.
config.s_mul.enable(&mut region, 0)?;
// The inputs we've been given could be located anywhere in the circuit,
// but we can only rely on relative offsets inside this region. So we
// assign new cells inside the region and constrain them to have the
// same values as the inputs.
a.0.copy_advice(|| "lhs", &mut region, config.advice[0], 0)?;
b.0.copy_advice(|| "rhs", &mut region, config.advice[1], 0)?;
// Now we can assign the multiplication result, which is to be assigned
// into the output position.
let value = a.0.value().copied() * b.0.value();
// Finally, we do the assignment to the output, returning a
// variable to be used in another part of the circuit.
region
.assign_advice(|| "lhs * rhs", config.advice[0], 1, || value)
.map(Number)
},
)
}
fn expose_public(
&self,
mut layouter: impl Layouter<F>,
num: Self::Num,
row: usize,
) -> Result<(), Error> {
let config = self.config();
layouter.constrain_instance(num.0.cell(), config.instance, row)
}
}Build the circuit
Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit!
/// The full circuit implementation.
///
/// In this struct we store the private input variables. We use `Option<F>` because
/// they won't have any value during key generation. During proving, if any of these
/// were `None` we would get an error.
#[derive(Default)]
struct MyCircuit<F: Field> {
constant: F,
a: Value<F>,
b: Value<F>,
}
impl<F: Field> Circuit<F> for MyCircuit<F> {
// Since we are using a single chip for everything, we can just reuse its config.
type Config = FieldConfig;
type FloorPlanner = SimpleFloorPlanner;
fn without_witnesses(&self) -> Self {
Self::default()
}
fn configure(meta: &mut ConstraintSystem<F>) -> Self::Config {
// We create the two advice columns that FieldChip uses for I/O.
let advice = [meta.advice_column(), meta.advice_column()];
// We also need an instance column to store public inputs.
let instance = meta.instance_column();
// Create a fixed column to load constants.
let constant = meta.fixed_column();
FieldChip::configure(meta, advice, instance, constant)
}
fn synthesize(
&self,
config: Self::Config,
mut layouter: impl Layouter<F>,
) -> Result<(), Error> {
let field_chip = FieldChip::<F>::construct(config);
// Load our private values into the circuit.
let a = field_chip.load_private(layouter.namespace(|| "load a"), self.a)?;
let b = field_chip.load_private(layouter.namespace(|| "load b"), self.b)?;
// Load the constant factor into the circuit.
let constant =
field_chip.load_constant(layouter.namespace(|| "load constant"), self.constant)?;
// We only have access to plain multiplication.
// We could implement our circuit as:
// asq = a*a
// bsq = b*b
// absq = asq*bsq
// c = constant*asq*bsq
//
// but it's more efficient to implement it as:
// ab = a*b
// absq = ab^2
// c = constant*absq
let ab = field_chip.mul(layouter.namespace(|| "a * b"), a, b)?;
let absq = field_chip.mul(layouter.namespace(|| "ab * ab"), ab.clone(), ab)?;
let c = field_chip.mul(layouter.namespace(|| "constant * absq"), constant, absq)?;
// Expose the result as a public input to the circuit.
field_chip.expose_public(layouter.namespace(|| "expose c"), c, 0)
}
}Testing the circuit
halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The
private and public inputs to the circuit are constructed as we will do to create a proof,
but by passing them to MockProver::run we get an object that can test every constraint
in the circuit, and tell us exactly what is failing (if anything).
// The number of rows in our circuit cannot exceed 2^k. Since our example
// circuit is very small, we can pick a very small value here.
let k = 4;
// Prepare the private and public inputs to the circuit!
let constant = Fp::from(7);
let a = Fp::from(2);
let b = Fp::from(3);
let c = constant * a.square() * b.square();
// Instantiate the circuit with the private inputs.
let circuit = MyCircuit {
constant,
a: Value::known(a),
b: Value::known(b),
};
// Arrange the public input. We expose the multiplication result in row 0
// of the instance column, so we position it there in our public inputs.
let mut public_inputs = vec![c];
// Given the correct public input, our circuit will verify.
let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap();
assert_eq!(prover.verify(), Ok(()));
// If we try some other public input, the proof will fail!
public_inputs[0] += Fp::one();
let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap();
assert!(prover.verify().is_err());Full example
You can find the source code for this example here.