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< ol class = "chapter" > < li class = "chapter-item expanded affix " > < a href = "../index.html" > halo2< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts.html" > < strong aria-hidden = "true" > 1.< / strong > Concepts< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../concepts/proofs.html" > < strong aria-hidden = "true" > 1.1.< / strong > Proof systems< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/arithmetization.html" > < strong aria-hidden = "true" > 1.2.< / strong > UltraPLONK Arithmetization< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/cores.html" > < strong aria-hidden = "true" > 1.3.< / strong > Cores< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/chips.html" > < strong aria-hidden = "true" > 1.4.< / strong > Chips< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/gadgets.html" > < strong aria-hidden = "true" > 1.5.< / strong > Gadgets< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../user.html" > < strong aria-hidden = "true" > 2.< / strong > User Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../user/simple-example.html" class = "active" > < strong aria-hidden = "true" > 2.1.< / strong > A simple example< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/lookup-tables.html" > < strong aria-hidden = "true" > 2.2.< / strong > Lookup tables< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/gadgets.html" > < strong aria-hidden = "true" > 2.3.< / strong > Gadgets< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/tips-and-tricks.html" > < strong aria-hidden = "true" > 2.4.< / strong > Tips and tricks< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design.html" > < strong aria-hidden = "true" > 3.< / strong > Design< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/proving-system.html" > < strong aria-hidden = "true" > 3.1.< / strong > Proving system< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/proving-system/multipoint-opening.html" > < strong aria-hidden = "true" > 3.1.1.< / strong > Multipoint opening argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/permutation.html" > < strong aria-hidden = "true" > 3.1.2.< / strong > Permutation argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/lookup-argument.html" > < strong aria-hidden = "true" > 3.1.3.< / strong > Lookup argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/comparison.html" > < strong aria-hidden = "true" > 3.1.4.< / strong > Comparison to other work< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design/implementation.html" > < strong aria-hidden = "true" > 3.2.< / strong > Implementation< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/gadgets.html" > < strong aria-hidden = "true" > 3.3.< / strong > Gadgets< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/gadgets/sha256.html" > < strong aria-hidden = "true" > 3.3.1.< / strong > SHA-256< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/gadgets/sha256/table16.html" > < strong aria-hidden = "true" > 3.3.1.1.< / strong > 16-bit table chip< / a > < / li > < / ol > < / li > < / ol > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../background.html" > < strong aria-hidden = "true" > 4.< / strong > Background Material< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../background/fields.html" > < strong aria-hidden = "true" > 4.1.< / strong > Fields< / a > < / li > < li class = "chapter-item expanded " > < a href = "../background/polynomials.html" > < strong aria-hidden = "true" > 4.2.< / strong > Polynomials< / a > < / li > < li class = "chapter-item expanded " > < a href = "../background/groups.html" > < strong aria-hidden = "true" > 4.3.< / strong > Cryptographic groups< / a > < / li > < li class = "chapter-item expanded " > < a href = "../background/curves.html" > < strong aria-hidden = "true" > 4.4.< / strong > Elliptic curves< / a > < / li > < li class = "chapter-item expanded " > < a href = "../background/upa.html" > < strong aria-hidden = "true" > 4.5.< / strong > UltraPLONK arithmetisation< / a > < / li > < li class = "chapter-item exp
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< h1 class = "menu-title" > The halo2 Book< / h1 >
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< h1 > < a class = "header" href = "#a-simple-example" id = "a-simple-example" > A simple example< / a > < / h1 >
< p > 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 < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > c< / span > < / span > < / span > < / span > , and will prove knowledge of two private
inputs < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > a< / span > < / span > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > b< / span > < / span > < / span > < / span > such that< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8641079999999999em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8641079999999999em;" > < span style = "top:-3.113em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.8641079999999999em;vertical-align:0em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > b< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8641079999999999em;" > < span style = "top:-3.113em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mord" > .< / span > < / span > < / span > < / span > < / span > < / p >
< h2 > < a class = "header" href = "#define-instructions" id = "define-instructions" > Define instructions< / a > < / h2 >
< p > Firstly, we need to define the instructions that our circuit will rely on. Instructions
are the boundary between high-level < a href = "../concepts/gadgets.html" > gadgets< / a > 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.< / p >
< p > For our circuit, we will use three instructions:< / p >
< ul >
< li > Load a private number into the circuit.< / li >
< li > Multiply two numbers.< / li >
< li > Expose a number as a public input to the circuit.< / li >
< / ul >
< p > 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.< / p >
< pre > < code class = "language-rust ignore no_run" > trait NumericInstructions: Chip {
/// Variable representing a number.
type Num;
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/// Loads a number into the circuit as a private input.
fn load_private(
layouter: & mut impl Layouter< Self> ,
a: Option< Self::Field> ,
) -> Result< Self::Num, Error> ;
/// Returns `c = a * b`.
fn mul(
layouter: & mut impl Layouter< Self> ,
a: Self::Num,
b: Self::Num,
) -> Result< Self::Num, Error> ;
/// Exposes a number as a public input to the circuit.
fn expose_public(layouter: & mut impl Layouter< Self> , num: Self::Num) -> Result< (), Error> ;
}
< / code > < / pre >
< h2 > < a class = "header" href = "#define-a-chip-implementation" id = "define-a-chip-implementation" > Define a chip implementation< / a > < / h2 >
< p > For our circuit, we will build a < a href = "../concepts/chips.html" > chip< / a > that provides the above
numeric instructions for a finite field.< / p >
< pre > < code class = "language-rust ignore no_run" > /// The chip that will implement our instructions! Chips do not store any persistent state
/// themselves, and usually only contain type markers if necessary.
struct FieldChip< F: FieldExt> {
_marker: PhantomData< F> ,
}
< / code > < / pre >
< p > Every chip needs to implement the < code > Chip< / code > trait. This defines the properties of the chip
that a < code > Layouter< / code > may rely on when synthesizing a circuit, as well as enabling any initial
state that the chip requires to be loaded into the circuit.< / p >
< pre > < code class = "language-rust ignore no_run" > impl< F: FieldExt> Chip for FieldChip< F> {
type Config = FieldConfig;
type Field = F;
fn load(_layouter: & mut impl Layouter< Self> ) -> Result< (), halo2::plonk::Error> {
// None of the instructions implemented by this chip have any fixed state. But if
// we required e.g. a lookup table, this is where we would load it.
Ok(())
}
}
< / code > < / pre >
< h2 > < a class = "header" href = "#configure-the-chip" id = "configure-the-chip" > Configure the chip< / a > < / h2 >
< p > The chip needs to be configured with the columns, permutations, and gates that will be
required to implement all of the desired instructions.< / p >
< pre > < code class = "language-rust ignore no_run" > /// Chip state is stored in a separate config struct. This is generated by the chip during
/// configuration, and then handed to the `Layouter`, which makes it available to the chip
/// when it needs to implement its instructions.
#[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],
// We need to create a permutation between our advice columns. This allows us to
// copy numbers within these columns from arbitrary rows, which we can use to load
// inputs into our instruction regions.
perm: Permutation,
// 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: Column< Fixed> ,
// The selector for the public-input gate, which uses one of the advice columns.
s_pub: Column< Fixed> ,
}
impl< F: FieldExt> FieldChip< F> {
fn configure(
meta: & mut ConstraintSystem< F> ,
advice: [Column< Advice> ; 2],
aux: Column< Aux> ,
) -> FieldConfig {
let perm = Permutation::new(meta, & advice);
let s_mul = meta.fixed_column();
let s_pub = meta.fixed_column();
// 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_fixed(s_mul, Rotation::cur());
// The polynomial expression 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.
s_mul * (lhs * rhs + out * -F::one())
});
// Define our public-input gate!
meta.create_gate(" public input" , |meta| {
// We choose somewhat-arbitrarily that we will use the second advice column
// for exposing numbers as public inputs.
let a = meta.query_advice(advice[1], Rotation::cur());
let p = meta.query_aux(aux, Rotation::cur());
let s = meta.query_fixed(s_pub, Rotation::cur());
// We simply constrain the advice cell to be equal to the aux cell, when the
// selector is enabled.
s * (p + a * -F::one())
});
FieldConfig {
advice,
perm,
s_mul,
s_pub,
}
}
}
< / code > < / pre >
< h2 > < a class = "header" href = "#implement-chip-traits" id = "implement-chip-traits" > Implement chip traits< / a > < / h2 >
< pre > < code class = "language-rust ignore no_run" > /// A variable representing a number.
#[derive(Clone)]
struct Number< F: FieldExt> {
cell: Cell,
value: Option< F> ,
}
impl< F: FieldExt> NumericInstructions for FieldChip< F> {
type Num = Number< F> ;
fn load_private(
layouter: & mut impl Layouter< Self> ,
value: Option< Self::Field> ,
) -> Result< Self::Num, Error> {
let config = layouter.config().clone();
let mut num = None;
layouter.assign_region(
|| " load private" ,
|mut region| {
let cell = region.assign_advice(
|| " private input" ,
config.advice[0],
0,
|| value.ok_or(Error::SynthesisError),
)?;
num = Some(Number { cell, value });
Ok(())
},
)?;
Ok(num.unwrap())
}
fn mul(
layouter: & mut impl Layouter< Self> ,
a: Self::Num,
b: Self::Num,
) -> Result< Self::Num, Error> {
let config = layouter.config().clone();
let mut out = None;
layouter.assign_region(
|| " mul" ,
|mut region| {
// 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.
region.assign_fixed(|| " example mul" , config.s_mul, 0, || Ok(F::one()))?;
// 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.
let lhs = region.assign_advice(
|| " lhs" ,
config.advice[0],
0,
|| a.value.ok_or(Error::SynthesisError),
)?;
let rhs = region.assign_advice(
|| " rhs" ,
config.advice[1],
0,
|| b.value.ok_or(Error::SynthesisError),
)?;
region.constrain_equal(& config.perm, a.cell, lhs)?;
region.constrain_equal(& config.perm, b.cell, rhs)?;
// Now we can assign the multiplication result into the output position.
let value = a.value.and_then(|a| b.value.map(|b| a * b));
let cell = region.assign_advice(
|| " lhs * rhs" ,
config.advice[0],
1,
|| value.ok_or(Error::SynthesisError),
)?;
// Finally, we return a variable representing the output, to be used in
// another part of the circuit.
out = Some(Number { cell, value });
Ok(())
},
)?;
Ok(out.unwrap())
}
fn expose_public(layouter: & mut impl Layouter< Self> , num: Self::Num) -> Result< (), Error> {
let config = layouter.config().clone();
layouter.assign_region(
|| " expose public" ,
|mut region| {
// Enable the public-input gate.
region.assign_fixed(|| " public result" , config.s_pub, 0, || Ok(F::one()))?;
// Load the output into the correct advice column.
let out = region.assign_advice(
|| " public advice" ,
config.advice[1],
0,
|| num.value.ok_or(Error::SynthesisError),
)?;
region.constrain_equal(& config.perm, num.cell, out)?;
// We don't assign to the auxiliary column inside the circuit; the mapping
// of public inputs to cells is provided to the prover.
Ok(())
},
)
}
}
< / code > < / pre >
< h2 > < a class = "header" href = "#build-the-circuit" id = "build-the-circuit" > Build the circuit< / a > < / h2 >
< p > Now that we have the instructions we need, and a chip that implements them, we can finally
build our circuit!< / p >
< pre > < code class = "language-rust ignore no_run" > /// 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.
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struct MyCircuit< F: FieldExt> {
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a: Option< F> ,
b: Option< F> ,
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}
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impl< F: FieldExt> Circuit< F> for MyCircuit< F> {
// Since we are using a single chip for everything, we can just reuse its config.
type Config = FieldConfig;
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 auxiliary column to store public inputs.
let aux = meta.aux_column();
FieldChip::configure(meta, advice, aux)
}
fn synthesize(& self, cs: & mut impl Assignment< F> , config: Self::Config) -> Result< (), Error> {
let mut layouter = SingleChip::new(cs, config);
// Load our private values into the circuit.
let a = FieldChip::load_private(& mut layouter, self.a)?;
let b = FieldChip::load_private(& mut layouter, self.b)?;
// We only have access to plain multiplication. We could implement our circuit as:
// asq = a*a
// bsq = b*b
// c = asq*bsq
//
// but it's more efficient to implement it as:
// ab = a*b
// c = ab^2
let ab = FieldChip::mul(& mut layouter, a, b)?;
let c = FieldChip::mul(& mut layouter, ab.clone(), ab)?;
// Expose the result as a public input to the circuit.
FieldChip::expose_public(& mut layouter, c)
}
}
< / code > < / pre >
< h2 > < a class = "header" href = "#testing-the-circuit" id = "testing-the-circuit" > Testing the circuit< / a > < / h2 >
< p > < code > halo2::dev::MockProver< / code > 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 < code > MockProver::run< / code > we get an object that can test every constraint
in the circuit, and tell us exactly what is failing (if anything).< / p >
< pre > < code class = "language-rust ignore no_run" > // 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 = 3;
// Prepare the private and public inputs to the circuit!
let a = Fp::from(2);
let b = Fp::from(3);
let c = a.square() * b.square();
// Instantiate the circuit with the private inputs.
let circuit = MyCircuit {
a: Some(a),
b: Some(b),
};
// Arrange the public input. We expose the multiplication result in row 4 of the aux
// column, so we position it there in our public inputs.
let mut public_inputs = vec![Fp::zero(); 1 < < k];
public_inputs[6] = 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[6] += Fp::one();
let prover = MockProver::run(k, & circuit, vec![public_inputs]).unwrap();
assert_eq!(
prover.verify(),
Err(VerifyFailure::Gate {
gate_index: 1,
gate_name: " public input" ,
row: 6,
})
);
< / code > < / pre >
< h2 > < a class = "header" href = "#full-example" id = "full-example" > Full example< / a > < / h2 >
< p > You can find the source code for this example
< a href = "https://github.com/zcash/halo2/tree/main/examples/simple-example.rs" > here< / a > .< / p >
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