feat(backend): use custom gate for Schnorr signature verification (#397)

* schnorr signature prover optimization. TODO: implemented eval_unfiltered_circuit, serialize

* Use new gate(s) in MainPod circuit

* Formatting

* Clean-up

* Code review

* Code review

* Code review

* Formatting

* Remove unnecessary elements

---------

Co-authored-by: Linus Tang <linust@mit.edu>

src/backends/plonky2/primitives/ec/curve.rs

@@ -32,7 +32,7 @@ use crate::backends::plonky2::{
     primitives::ec::{
         bits::BigUInt320Target,
         field::{get_nnf_target, CircuitBuilderNNF, OEFTarget},
-        gates::{curve::ECAddHomogOffset, generic::SimpleGate},
+        gates::curve::{ECAddHomogOffsetGate, ECAddXuGate},
     },
     Error,
 };
@@ -305,6 +305,17 @@ pub(super) fn add_homog<const D: usize, F: ECFieldExt<D>>(x1: F, u1: F, x2: F, u
     [x, z, u, t]
 }
 
+/// Adds two elliptic curve points in affine coordinates.
+pub(super) fn add_xu<const D: usize, F: ECFieldExt<D> + std::ops::Div<Output = F>>(
+    x1: F,
+    u1: F,
+    x2: F,
+    u2: F,
+) -> [F; 2] {
+    let [x, z, u, t] = add_homog(x1, u1, x2, u2);
+    [x / z, u / t]
+}
+
 // See CircuitBuilderEllptic::add_point for an explanation of why we need this function.
 // cf. https://github.com/pornin/ecgfp5/blob/ce059c6d1e1662db437aecbf3db6bb67fe63c716/rust/src/curve.rs#L157
 pub(super) fn add_homog_offset<const D: usize, F: ECFieldExt<D>>(
@@ -341,7 +352,7 @@ static GROUP_ORDER_HALF_ROUND_UP: LazyLock<BigUint> =
 
 impl Point {
     const B1_U32: u32 = 263;
-    const B1: GoldilocksField = GoldilocksField(Self::B1_U32 as u64);
+    pub(crate) const B1: GoldilocksField = GoldilocksField(Self::B1_U32 as u64);
 
     pub fn b() -> ECField {
         ECField::from_basefield_array([
@@ -555,6 +566,75 @@ where
     }
 }
 
+pub trait CircuitBuilderSignature {
+    /// Computes `a*g + b*p`, where `g` is the generator of the curve.
+    fn linear_combination_point_gen(
+        &mut self,
+        a: &[BoolTarget; 320],
+        b: &[BoolTarget; 320],
+        p: &PointTarget,
+    ) -> PointTarget;
+}
+
+impl CircuitBuilderSignature for CircuitBuilder<GoldilocksField, 2> {
+    fn linear_combination_point_gen(
+        &mut self,
+        a: &[BoolTarget; 320],
+        b: &[BoolTarget; 320],
+        p: &PointTarget,
+    ) -> PointTarget {
+        let y = p;
+        let zero = self.identity_point();
+        let zero_target = self.zero();
+
+        let mut ans = zero.clone(); // accumulator
+
+        for x in 0..107 {
+            // prepare to apply gate
+            let mut inputs = Vec::with_capacity(26);
+
+            // scalar bits for g
+            if x == 0 {
+                inputs.push(zero_target);
+            } else {
+                inputs.push(a[320 - 3 * x].target);
+            }
+            inputs.push(a[319 - 3 * x].target);
+            inputs.push(a[318 - 3 * x].target);
+
+            // scalar bits for p (y)
+            if x == 0 {
+                inputs.push(zero_target);
+            } else {
+                inputs.push(b[320 - 3 * x].target);
+            }
+            inputs.push(b[319 - 3 * x].target);
+            inputs.push(b[318 - 3 * x].target);
+
+            // y point
+            inputs.extend_from_slice(&y.x.components);
+            inputs.extend_from_slice(&y.u.components);
+
+            // accumulator
+            inputs.extend_from_slice(&ans.x.components);
+            inputs.extend_from_slice(&ans.u.components);
+
+            // apply gate
+            let outputs = ECAddXuGate::apply(self, &inputs);
+            let x = FieldTarget::new(array::from_fn(|i| outputs[i]));
+            let u = FieldTarget::new(array::from_fn(|i| outputs[5 + i]));
+            ans = PointTarget {
+                x,
+                u,
+                checked_on_curve: true,
+                checked_in_subgroup: p.checked_in_subgroup,
+            };
+        }
+
+        ans
+    }
+}
+
 pub trait CircuitBuilderElliptic {
     fn add_virtual_point_target_unsafe(&mut self) -> PointTarget;
     fn add_virtual_point_target(&mut self) -> PointTarget;
@@ -617,7 +697,9 @@ impl CircuitBuilderElliptic for CircuitBuilder<GoldilocksField, 2> {
         inputs.extend_from_slice(&p1.u.components);
         inputs.extend_from_slice(&p2.x.components);
         inputs.extend_from_slice(&p2.u.components);
-        let outputs = ECAddHomogOffset::apply(self, &inputs);
+
+        let outputs = ECAddHomogOffsetGate::apply(self, &inputs);
+
         // plonky2 expects all gate constraints to be satisfied by the zero vector.
         // So our elliptic curve addition gate computes [x,z-b,u,t-b], and we have to add the b here.
         let [x, z, u, t] =
@@ -638,32 +720,6 @@ impl CircuitBuilderElliptic for CircuitBuilder<GoldilocksField, 2> {
 
     fn double_point(&mut self, p: &PointTarget) -> PointTarget {
         self.add_point(p, p)
-        /*
-        let t3 = self.nnf_mul(&p.u, &p.u);
-        let one = self.one();
-        let neg_one = self.neg_one();
-        let two = self.two();
-        let neg_four = self.constant(GoldilocksField::from_noncanonical_i64(-4));
-        let four_b = self.constant(GoldilocksField::from_canonical_u32(4 * Point::B1_U32));
-        let w1_1 = self.nnf_add_scalar_times_generator_power(one, 0, &p.x);
-        let w1_2 = self.nnf_add(&w1_1, &w1_1);
-        let w1_3 = self.nnf_mul(&w1_2, &t3);
-        let w1_4 = self.nnf_mul_scalar(neg_one, &w1_3);
-        let w1 = self.nnf_add_scalar_times_generator_power(one, 0, &w1_4);
-        let x_1 = self.nnf_mul_scalar(four_b, &t3);
-        let x = self.nnf_mul_generator(&x_1);
-        let z = self.nnf_mul(&w1, &w1);
-        let u_1 = self.nnf_add(&w1, &p.u);
-        let u_2 = self.nnf_mul(&u_1, &u_1);
-        let u_3 = self.nnf_sub(&u_2, &t3);
-        let u = self.nnf_sub(&u_3, &z);
-        let t_1 = self.nnf_mul_scalar(neg_four, &t3);
-        let t_2 = self.nnf_add_scalar_times_generator_power(two, 0, &t_1);
-        let t = self.nnf_sub(&t_2, &z);
-        let xq = self.nnf_div(&x, &z);
-        let uq = self.nnf_div(&u, &t);
-        PointTarget { x: xq, u: uq }
-        */
     }
 
     fn multiply_point(&mut self, p1_scalar: &[BoolTarget; 320], p1: &PointTarget) -> PointTarget {

src/backends/plonky2/primitives/ec/field.rs

@@ -1,4 +1,4 @@
-use std::{array, marker::PhantomData};
+use std::marker::PhantomData;
 
 use num::BigUint;
 use plonky2::{
@@ -19,10 +19,7 @@ use serde::{Deserialize, Serialize};
 
 //use super::gates::field::NNFMulGate;
 use crate::{
-    backends::plonky2::{
+    backends::plonky2::{basetypes::D, primitives::ec::gates::field::NNFMulGate},
-        basetypes::D,
-        primitives::ec::gates::{field::NNFMulSimple, generic::SimpleGate},
-    },
     middleware::F,
 };
 
@@ -219,11 +216,33 @@ impl<const DEG: usize, NNF: OEF<DEG> + FieldExtension<DEG, BaseField = F>>
         OEFTarget::new(std::array::from_fn(|i| sub_targets[i]))
     }
     fn nnf_mul(&mut self, x: &OEFTarget<DEG, NNF>, y: &OEFTarget<DEG, NNF>) -> OEFTarget<DEG, NNF> {
-        let mut inputs = Vec::with_capacity(10);
+        let gate = NNFMulGate::<D, DEG, NNF>::new_from_config(&self.config);
-        inputs.extend_from_slice(&x.components);
+        let (gate, i) = self.find_slot(gate, &[], &[]);
-        inputs.extend_from_slice(&y.components);
+        let wires_m0: OEFTarget<DEG, NNF> = OEFTarget::new(
-        let outputs = NNFMulSimple::<DEG, NNF>::apply(self, &inputs);
+            NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_0(i)
-        OEFTarget::new(array::from_fn(|i| outputs[i]))
+                .map(|i| Target::wire(gate, i))
+                .collect::<Vec<_>>()
+                .try_into()
+                .unwrap(),
+        );
+        let wires_m1: OEFTarget<DEG, NNF> = OEFTarget::new(
+            NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_1(i)
+                .map(|i| Target::wire(gate, i))
+                .collect::<Vec<_>>()
+                .try_into()
+                .unwrap(),
+        );
+        let output: OEFTarget<DEG, NNF> = OEFTarget::new(
+            NNFMulGate::<D, DEG, NNF>::wires_ith_output(i)
+                .map(|i| Target::wire(gate, i))
+                .collect::<Vec<_>>()
+                .try_into()
+                .unwrap(),
+        );
+        self.nnf_connect(&wires_m0, x);
+        self.nnf_connect(&wires_m1, y);
+
+        output
     }
     fn nnf_div(&mut self, x: &OEFTarget<DEG, NNF>, y: &OEFTarget<DEG, NNF>) -> OEFTarget<DEG, NNF> {
         let one = self.nnf_one();

src/backends/plonky2/primitives/ec/gates/curve.rs

@@ -1,10 +1,34 @@
-use std::array;
+use std::{array, ops::Range};
 
-use plonky2::field::goldilocks_field::GoldilocksField;
+use itertools::{zip_eq, Itertools};
+use plonky2::{
+    field::{
+        extension::{quintic::QuinticExtension, Extendable, FieldExtension, OEF},
+        goldilocks_field::GoldilocksField,
+        types::Field,
+    },
+    gates::gate::Gate,
+    hash::hash_types::RichField,
+    iop::{
+        ext_target::ExtensionTarget,
+        generator::{GeneratedValues, SimpleGenerator, WitnessGeneratorRef},
+        target::Target,
+        witness::{PartitionWitness, Witness, WitnessWrite},
+    },
+    plonk::{
+        circuit_builder::CircuitBuilder,
+        circuit_data::{CircuitConfig, CommonCircuitData},
+        vars::{EvaluationTargets, EvaluationVars},
+    },
+    util::serialization::{Buffer, IoResult, Read, Write},
+};
 
-use crate::backends::plonky2::primitives::ec::{
+use crate::backends::plonky2::{
-    curve::{add_homog_offset, ECFieldExt},
+    basetypes::F,
-    gates::{field::QuinticTensor, generic::SimpleGate},
+    primitives::ec::{
+        curve::{add_homog, add_homog_offset, add_xu, ECFieldExt, Point},
+        gates::field::{nnf_mul_ext, QuinticTensor},
+    },
 };
 
 /// Gate computing the addition of two elliptic curve points in
@@ -25,30 +49,835 @@ use crate::backends::plonky2::primitives::ec::{
 /// operation when all its witness wire values are zero (so that when the gate is partially used,
 /// the unused slots still pass the constraints). This is the reason why this gate doesn't add the
 /// final offset: if it did, the constraints wouldn't pass on the zero witness values.
-#[derive(Debug, Default, Clone)]
+#[derive(Debug, Clone)]
-pub struct ECAddHomogOffset;
+pub struct ECAddHomogOffsetGate {
+    /// Number of (offset) EC additions performed by the gate.
+    pub num_ops: usize,
+}
 
-impl SimpleGate for ECAddHomogOffset {
+impl ECAddHomogOffsetGate {
-    type F = GoldilocksField;
+    pub const fn new_from_config(config: &CircuitConfig) -> Self {
-    const INPUTS_PER_OP: usize = 20;
+        Self {
-    const OUTPUTS_PER_OP: usize = 20;
+            num_ops: Self::num_ops(config),
-    const DEGREE: usize = 4;
+        }
-    const ID: &'static str = "ECAddHomog";
+    }
-    fn eval<const D: usize>(
+
-        wires: &[<Self::F as plonky2::field::extension::Extendable<D>>::Extension],
+    /// Determine the maximum number of operations that can fit in one gate for the given config.
-    ) -> Vec<<Self::F as plonky2::field::extension::Extendable<D>>::Extension>
+    pub(crate) const fn num_ops(config: &CircuitConfig) -> usize {
+        let wires_per_op = 40;
+        config.num_routed_wires / wires_per_op
+    }
+
+    pub(crate) const fn wires_ith_addend_0(i: usize) -> Range<usize> {
+        40 * i..40 * i + 10
+    }
+    pub(crate) const fn wires_ith_addend_1(i: usize) -> Range<usize> {
+        40 * i + 10..40 * i + 20
+    }
+    pub(crate) const fn wires_ith_output(i: usize) -> Range<usize> {
+        40 * i + 20..40 * (i + 1)
+    }
+
+    pub fn apply<const D: usize>(
+        builder: &mut CircuitBuilder<F, D>,
+        targets: &[Target],
+    ) -> Vec<Target>
     where
-        Self::F: plonky2::field::extension::Extendable<D>,
+        F: Extendable<D>,
+    {
+        let gate = ECAddHomogOffsetGate::new_from_config(&builder.config);
+        let (row, op_num) = builder.find_slot(gate, &[], &[]);
+        let wires_a0 = Self::wires_ith_addend_0(op_num)
+            .map(|i| Target::wire(row, i))
+            .collect::<Vec<_>>();
+        let wires_a1 = Self::wires_ith_addend_1(op_num)
+            .map(|i| Target::wire(row, i))
+            .collect::<Vec<_>>();
+        let outputs = Self::wires_ith_output(op_num)
+            .map(|i| Target::wire(row, i))
+            .collect::<Vec<_>>();
+        zip_eq(targets, [wires_a0, wires_a1].concat()).for_each(|(i, w)| builder.connect(*i, w));
+
+        outputs
+    }
+}
+
+impl<const D: usize> Gate<F, D> for ECAddHomogOffsetGate
+where
+    F: Extendable<D>,
+{
+    fn id(&self) -> String {
+        format!("{self:?}")
+    }
+
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
+        dst.write_usize(self.num_ops)
+    }
+
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
+        let num_ops = src.read_usize()?;
+        Ok(Self { num_ops })
+    }
+
+    fn eval_unfiltered(&self, vars: EvaluationVars<F, D>) -> Vec<<F as Extendable<D>>::Extension> {
+        let mut constraints = Vec::with_capacity(self.num_ops * 20);
+        let extract_point = |range: Range<usize>| -> (QuinticTensor<D>, QuinticTensor<D>) {
+            let components = vars.local_wires[range].to_vec();
+            (
+                QuinticTensor::from_base(array::from_fn::<_, 5, _>(|i| components[i])),
+                QuinticTensor::from_base(array::from_fn::<_, 5, _>(|i| components[i + 5])),
+            )
+        };
+
+        for i in 0..self.num_ops {
+            let (a_0x, a_0u) = extract_point(Self::wires_ith_addend_0(i));
+            let (a_1x, a_1u) = extract_point(Self::wires_ith_addend_1(i));
+            let output_vec = vars.local_wires[Self::wires_ith_output(i)]
+                .iter()
+                .chunks(5)
+                .into_iter()
+                .map(|chunk| {
+                    let chunk_vec = chunk.collect::<Vec<_>>();
+                    QuinticTensor::from_base(array::from_fn(|i| *chunk_vec[i]))
+                })
+                .collect::<Vec<QuinticTensor<D>>>();
+            let computed_output = add_homog_offset(a_0x, a_0u, a_1x, a_1u);
+
+            let new_constraints =
+                zip_eq(output_vec, computed_output).flat_map(|(o, co)| (o - co).components);
+
+            constraints.extend(new_constraints);
+        }
+
+        constraints
+    }
+
+    fn eval_unfiltered_circuit(
+        &self,
+        builder: &mut CircuitBuilder<F, D>,
+        vars: EvaluationTargets<D>,
+    ) -> Vec<ExtensionTarget<D>> {
+        let mut constraints = Vec::with_capacity(self.num_ops * 20);
+
+        let extract_point =
+            |range: Range<usize>| -> ([ExtensionTarget<D>; 5], [ExtensionTarget<D>; 5]) {
+                let components = vars.local_wires[range].to_vec();
+                (
+                    array::from_fn::<_, 5, _>(|i| components[i]),
+                    array::from_fn::<_, 5, _>(|i| components[i + 5]),
+                )
+            };
+        for i in 0..self.num_ops {
+            let (x1, u1) = extract_point(Self::wires_ith_addend_0(i));
+            let (x2, u2) = extract_point(Self::wires_ith_addend_1(i));
+            let computed_output = ec_target_add_homog_offset(builder, &x1, &u1, &x2, &u2)
+                .into_iter()
+                .flatten();
+
+            let output: [ExtensionTarget<D>; 20] = vars.local_wires[Self::wires_ith_output(i)]
+                .try_into()
+                .unwrap();
+
+            let diffs = zip_eq(output, computed_output)
+                .map(|(o, co)| builder.sub_extension(o, co))
+                .collect::<Vec<_>>();
+            constraints.extend(diffs);
+        }
+
+        constraints
+    }
+
+    fn generators(&self, row: usize, _local_constants: &[F]) -> Vec<WitnessGeneratorRef<F, D>> {
+        (0..self.num_ops)
+            .map(|i| WitnessGeneratorRef::new(ECAddHomogOffsetGenerator { row, i }.adapter()))
+            .collect()
+    }
+
+    fn num_wires(&self) -> usize {
+        self.num_ops * 40
+    }
+
+    fn num_constants(&self) -> usize {
+        0
+    }
+
+    fn degree(&self) -> usize {
+        4
+    }
+
+    fn num_constraints(&self) -> usize {
+        self.num_ops * 20
+    }
+}
+
+#[derive(Clone, Debug, Default)]
+pub struct ECAddHomogOffsetGenerator {
+    row: usize,
+    i: usize,
+}
+
+impl<const D: usize> SimpleGenerator<F, D> for ECAddHomogOffsetGenerator
+where
+    F: Extendable<D>,
+{
+    fn id(&self) -> String {
+        "ECAddHomogOffsetGenerator".to_string()
+    }
+
+    fn dependencies(&self) -> Vec<Target> {
+        ECAddHomogOffsetGate::wires_ith_addend_0(self.i)
+            .chain(ECAddHomogOffsetGate::wires_ith_addend_1(self.i))
+            .map(|i| Target::wire(self.row, i))
+            .collect()
+    }
+
+    fn run_once(
+        &self,
+        witness: &PartitionWitness<F>,
+        out_buffer: &mut GeneratedValues<F>,
+    ) -> anyhow::Result<()> {
+        let extract_point = |range: Range<usize>| -> anyhow::Result<_> {
+            let components = range
+                .map(|i| witness.get_target(Target::wire(self.row, i)))
+                .collect::<Vec<_>>();
+            Ok((
+                QuinticExtension::from_basefield_array(array::from_fn::<_, 5, _>(|i| {
+                    components[i]
+                })),
+                QuinticExtension::from_basefield_array(array::from_fn::<_, 5, _>(|i| {
+                    components[i + 5]
+                })),
+            ))
+        };
+
+        let addend_0 = extract_point(ECAddHomogOffsetGate::wires_ith_addend_0(self.i))?;
+        let addend_1 = extract_point(ECAddHomogOffsetGate::wires_ith_addend_1(self.i))?;
+
+        let output_targets: [Target; 20] = ECAddHomogOffsetGate::wires_ith_output(self.i)
+            .map(|i| Target::wire(self.row, i))
+            .collect::<Vec<_>>()
+            .try_into()
+            .map_err(|e| anyhow::anyhow!("{:?}", e))?;
+
+        let computed_output = add_homog_offset(addend_0.0, addend_0.1, addend_1.0, addend_1.1)
+            .iter()
+            .flat_map(<QuinticExtension<F> as FieldExtension<5>>::to_basefield_array)
+            .collect::<Vec<_>>();
+
+        out_buffer.set_target_arr(&output_targets, &computed_output)
+    }
+
+    fn serialize(&self, dst: &mut Vec<u8>, _common_data: &CommonCircuitData<F, D>) -> IoResult<()> {
+        dst.write_usize(self.row)?;
+        dst.write_usize(self.i)
+    }
+
+    fn deserialize(src: &mut Buffer, _common_data: &CommonCircuitData<F, D>) -> IoResult<Self> {
+        let row = src.read_usize()?;
+        let i = src.read_usize()?;
+        Ok(Self { row, i })
+    }
+}
+
+#[derive(Clone, Debug, Default)]
+pub struct ECAddXuGenerator {
+    row: usize,
+}
+
+impl<const D: usize> SimpleGenerator<F, D> for ECAddXuGenerator
+where
+    F: Extendable<D>,
+{
+    fn serialize(
+        &self,
+        dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<GoldilocksField, D>,
+    ) -> IoResult<()> {
+        dst.write_usize(self.row)
+    }
+
+    fn deserialize(
+        src: &mut Buffer,
+        _common_data: &CommonCircuitData<GoldilocksField, D>,
+    ) -> IoResult<Self>
+    where
+        Self: Sized,
+    {
+        Ok(Self {
+            row: src.read_usize()?,
+        })
+    }
+
+    fn id(&self) -> String {
+        "ECAddXuGenerator".into()
+    }
+
+    fn dependencies(&self) -> Vec<Target> {
+        (0..26).map(|i| Target::wire(self.row, i)).collect()
+    }
+
+    fn run_once(
+        &self,
+        witness: &PartitionWitness<GoldilocksField>,
+        out_buffer: &mut GeneratedValues<GoldilocksField>,
+    ) -> anyhow::Result<()> {
+        let deps = self.dependencies();
+
+        let selectors_g: Vec<GoldilocksField> = deps[0..3]
+            .iter() // binary selectors for g
+            .map(|&target| witness.get_target(target))
+            .collect();
+
+        let selectors_y: Vec<GoldilocksField> = deps[3..6]
+            .iter() // binary selectors for y
+            .map(|&target| witness.get_target(target))
+            .collect();
+
+        // extract element of quintic extension of Goldilocks field from five consecutive targets
+        let extract_quintic = |start_idx: usize| -> QuinticExtension<GoldilocksField> {
+            QuinticExtension::<GoldilocksField>::from_basefield_array([
+                witness.get_target(deps[start_idx]),
+                witness.get_target(deps[start_idx + 1]),
+                witness.get_target(deps[start_idx + 2]),
+                witness.get_target(deps[start_idx + 3]),
+                witness.get_target(deps[start_idx + 4]),
+            ])
+        };
+
+        let g = Point::generator();
+
+        let g_x = g.x;
+        let g_u = g.u;
+        let y_x = extract_quintic(6);
+        let y_u = extract_quintic(11);
+
+        let mut p_x = extract_quintic(16);
+        let mut p_u = extract_quintic(21);
+
+        let mut write_quintic =
+            |start_wire: usize, value: &QuinticExtension<GoldilocksField>| -> anyhow::Result<()> {
+                let array: [GoldilocksField; 5] =
+                    QuinticExtension::<GoldilocksField>::to_basefield_array(value);
+                for (j, &num) in array.iter().enumerate() {
+                    out_buffer.set_target(Target::wire(self.row, start_wire + j), num)?;
+                }
+                Ok(())
+            };
+
+        // Double and add three times.
+        // Write points from right to left so that the result of the fifth add
+        // lies on a routable wire and thus can be copied to the next row.
+        (0..3).try_for_each(|i| {
+            // Double, write to wires [106-30*i..116-30*i]
+            [p_x, p_u] = add_xu::<1, QuinticExtension<GoldilocksField>>(p_x, p_u, p_x, p_u);
+            write_quintic(106 - 30 * i, &p_x)?;
+            write_quintic(111 - 30 * i, &p_u)?;
+
+            // Possibly add g, depending on selector. Write to wires  [96-30*i..106-30*i]
+            if selectors_g[i] == GoldilocksField::ONE {
+                [p_x, p_u] = add_xu::<1, QuinticExtension<GoldilocksField>>(p_x, p_u, g_x, g_u);
+            }
+            write_quintic(96 - 30 * i, &p_x)?;
+            write_quintic(101 - 30 * i, &p_u)?;
+
+            // Possibly add y, depending on selector. Write to wires  [86-30*i..96-30*i]
+            if selectors_y[i] == GoldilocksField::ONE {
+                [p_x, p_u] = add_xu::<1, QuinticExtension<GoldilocksField>>(p_x, p_u, y_x, y_u);
+            }
+            write_quintic(86 - 30 * i, &p_x)?;
+            write_quintic(91 - 30 * i, &p_u)
+        })
+    }
+}
+
+/// Gate that selectively carries out three rounds of the
+/// double-and-add algorithm loop applied to the curve generator and a
+/// given point.
+#[derive(Clone)]
+pub struct ECAddXuGate;
+
+impl ECAddXuGate {
+    const INPUTS_PER_OP: usize = 26;
+    const OUTPUTS_PER_OP: usize = 90;
+    const WIRES_PER_OP: usize = Self::INPUTS_PER_OP + Self::OUTPUTS_PER_OP;
+    const DEGREE: usize = 6;
+    const ID: &'static str = "ECAddXuGate";
+
+    pub fn apply<const D: usize>(
+        builder: &mut CircuitBuilder<F, D>,
+        targets: &[Target],
+    ) -> Vec<Target>
+    where
+        F: Extendable<D>,
+    {
+        assert!(targets.len() == Self::INPUTS_PER_OP);
+        let (row, _) = builder.find_slot(ECAddXuGate::new(), &[], &[]);
+        for (i, &t) in targets.iter().enumerate() {
+            builder.connect(t, Target::wire(row, i));
+        }
+
+        (0..10)
+            .map(|i| Target::wire(row, Self::INPUTS_PER_OP + i))
+            .collect()
+    }
+
+    pub fn new() -> Self {
+        Self
+    }
+
+    pub fn add_numerator_denominator<const D: usize>(
+        wires: &[<GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension],
+    ) -> Vec<<GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension>
+    where
+        GoldilocksField: plonky2::field::extension::Extendable<D>,
     {
         let mut ans = Vec::with_capacity(20);
-        let [x1, u1, x2, u2] =
+        let x1 = QuinticTensor::from_base(wires[0..5].try_into().unwrap());
-            array::from_fn(|j| QuinticTensor::from_base(array::from_fn(|i| wires[5 * j + i])));
+        let u1 = QuinticTensor::from_base(wires[5..10].try_into().unwrap());
-        let out = add_homog_offset(x1, u1, x2, u2);
+        let x2 = QuinticTensor::from_base(wires[10..15].try_into().unwrap());
+        let u2 = QuinticTensor::from_base(wires[15..20].try_into().unwrap());
+        let out = add_homog(x1, u1, x2, u2);
         for v in out {
             ans.extend(v.to_base());
         }
         ans
     }
+
+    pub fn convert<const D: usize>(
+        wires: &[<GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension],
+    ) -> Vec<<GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension>
+    where
+        GoldilocksField: plonky2::field::extension::Extendable<D>,
+    {
+        let mut ans = Vec::with_capacity(10);
+        let x1 = QuinticTensor::from_base(wires[0..5].try_into().unwrap());
+        let u1 = QuinticTensor::from_base(wires[5..10].try_into().unwrap());
+        ans.extend(x1.to_base());
+        ans.extend(u1.to_base());
+        ans
+    }
+}
+
+impl<const D: usize> Gate<F, D> for ECAddXuGate
+where
+    F: Extendable<D>,
+{
+    fn id(&self) -> String {
+        Self::ID.to_string()
+    }
+
+    fn serialize(
+        &self,
+        _dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<GoldilocksField, D>,
+    ) -> IoResult<()> {
+        Ok(())
+    }
+
+    fn deserialize(
+        _src: &mut Buffer<'_>,
+        _common_data: &CommonCircuitData<GoldilocksField, D>,
+    ) -> IoResult<Self>
+    where
+        Self: Sized,
+    {
+        Ok(Self)
+    }
+
+    fn eval_unfiltered(
+        &self,
+        vars: EvaluationVars<'_, GoldilocksField, D>,
+    ) -> Vec<<GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension> {
+        let mut constraints = Vec::new();
+
+        let g = Point::generator();
+
+        let double_constraint = |i: usize, j: usize| {
+            let x1 = QuinticTensor::from_base(vars.local_wires[i..i + 5].try_into().unwrap());
+            let u1 = QuinticTensor::from_base(vars.local_wires[i + 5..i + 10].try_into().unwrap());
+            let [x, z, u, t] = add_homog(x1, u1, x1, u1);
+            let mut new_constraints: Vec<
+                <GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension,
+            > = Vec::with_capacity(10);
+            let x2 = QuinticTensor::from_base(vars.local_wires[j..j + 5].try_into().unwrap());
+            let u2 = QuinticTensor::from_base(vars.local_wires[j + 5..j + 10].try_into().unwrap());
+
+            let first_constraints: Vec<_> = (x2 * z - x).components.to_vec();
+            let second_constraints: Vec<_> = (u2 * t - u).components.to_vec();
+
+            new_constraints.extend(&first_constraints);
+            new_constraints.extend(&second_constraints);
+            new_constraints
+        };
+
+        let select_and_add_constraint = |i: usize, j: usize, selector_index: usize, add_y: bool| {
+            let zero =
+                <GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension::ZERO;
+
+            let one = <GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension::ONE;
+            let one_full = QuinticTensor::from_base([one, zero, zero, zero, zero]);
+
+            let sel = vars.local_wires[selector_index];
+            let sel_full = QuinticTensor::from_base([sel, zero, zero, zero, zero]);
+
+            let x1 = QuinticTensor::from_base(vars.local_wires[i..i + 5].try_into().unwrap());
+            let u1 = QuinticTensor::from_base(vars.local_wires[i + 5..i + 10].try_into().unwrap());
+
+            let (x2, u2);
+            if add_y {
+                // (using hardcoded location of y)
+                x2 = QuinticTensor::from_base(vars.local_wires[6..11].try_into().unwrap());
+                u2 = QuinticTensor::from_base(vars.local_wires[11..16].try_into().unwrap());
+            } else {
+                let mut base_array: [GoldilocksField; 5] = g.x.to_basefield_array();
+                x2 = QuinticTensor::from_base(base_array.map(Into::into));
+                base_array = g.u.to_basefield_array();
+                u2 = QuinticTensor::from_base(base_array.map(Into::into));
+            }
+            let [x, z, u, t] = add_homog(x1, u1, x2, u2);
+
+            let mut new_constraints: Vec<
+                <GoldilocksField as plonky2::field::extension::Extendable<D>>::Extension,
+            > = Vec::with_capacity(10);
+            let x3 = QuinticTensor::from_base(vars.local_wires[j..j + 5].try_into().unwrap());
+            let u3 = QuinticTensor::from_base(vars.local_wires[j + 5..j + 10].try_into().unwrap());
+
+            let first_constraints: Vec<_> = (x3 * z - sel_full * x
+                + (sel_full - one_full) * x1 * z)
+                .components
+                .to_vec();
+            let second_constraints: Vec<_> = (u3 * t - sel_full * u
+                + (sel_full - one_full) * u1 * t)
+                .components
+                .to_vec();
+
+            new_constraints.extend_from_slice(&first_constraints[0..5]);
+            new_constraints.extend_from_slice(&second_constraints[0..5]);
+            new_constraints
+        };
+
+        constraints.extend(double_constraint(16, 106));
+        constraints.extend(select_and_add_constraint(106, 96, 0, false));
+        constraints.extend(select_and_add_constraint(96, 86, 3, true));
+
+        constraints.extend(double_constraint(86, 76));
+        constraints.extend(select_and_add_constraint(76, 66, 1, false));
+        constraints.extend(select_and_add_constraint(66, 56, 4, true));
+
+        constraints.extend(double_constraint(56, 46));
+        constraints.extend(select_and_add_constraint(46, 36, 2, false));
+        constraints.extend(select_and_add_constraint(36, 26, 5, true));
+
+        constraints
+    }
+
+    fn eval_unfiltered_circuit(
+        &self,
+        builder: &mut CircuitBuilder<GoldilocksField, D>,
+        vars: EvaluationTargets<'_, D>,
+    ) -> Vec<ExtensionTarget<D>> {
+        let mut constraints = Vec::new();
+
+        type Nnf = QuinticExtension<F>;
+
+        let zero = builder.zero_extension();
+
+        let g = Point::generator();
+        let [g_x, g_u]: [[F; 5]; 2] = [g.x.to_basefield_array(), g.u.to_basefield_array()];
+        let [g_x_ext_target, g_u_ext_target] = [
+            array::from_fn(|i| builder.add_const_extension(zero, g_x[i])),
+            array::from_fn(|i| builder.add_const_extension(zero, g_u[i])),
+        ];
+
+        let double_constraint =
+            |builder: &mut CircuitBuilder<GoldilocksField, D>, i: usize, j: usize| {
+                let x1 = array::from_fn(|k| vars.local_wires[i + k]);
+                let u1 = array::from_fn(|k| vars.local_wires[i + 5 + k]);
+                let [x, z, u, t] = homog_ec_target_add(builder, &x1, &u1, &x1, &u1);
+
+                let mut new_constraints = Vec::<ExtensionTarget<_>>::with_capacity(10);
+                let x2 = array::from_fn(|k| vars.local_wires[j + k]);
+                let u2 = array::from_fn(|k| vars.local_wires[j + 5 + k]);
+
+                let [expected_x, expected_u] = [
+                    nnf_mul_ext::<D, 5, Nnf>(builder, &x2, &z),
+                    nnf_mul_ext::<D, 5, Nnf>(builder, &u2, &t),
+                ];
+                let first_constraints = nnf_ext_target_sub::<D, 5, Nnf>(builder, &expected_x, &x);
+                let second_constraints = nnf_ext_target_sub::<D, 5, Nnf>(builder, &expected_u, &u);
+
+                new_constraints.extend(&first_constraints);
+                new_constraints.extend(&second_constraints);
+                new_constraints
+            };
+
+        let select_and_add_constraint = |builder: &mut CircuitBuilder<GoldilocksField, D>,
+                                         i: usize,
+                                         j: usize,
+                                         selector_index: usize,
+                                         add_y: bool| {
+            let one = builder.one_extension();
+            let sel = vars.local_wires[selector_index];
+
+            let x1 = array::from_fn(|k| vars.local_wires[i + k]);
+            let u1 = array::from_fn(|k| vars.local_wires[i + 5 + k]);
+
+            let (x2, u2) = if add_y {
+                // (using hardcoded location of y)
+                (
+                    array::from_fn(|k| vars.local_wires[6 + k]),
+                    array::from_fn(|k| vars.local_wires[11 + k]),
+                )
+            } else {
+                (g_x_ext_target, g_u_ext_target)
+            };
+
+            let [x, z, u, t] = homog_ec_target_add(builder, &x1, &u1, &x2, &u2);
+
+            let mut new_constraints = Vec::<ExtensionTarget<_>>::with_capacity(10);
+            let x3 = array::from_fn(|k| vars.local_wires[j + k]);
+            let u3 = array::from_fn(|k| vars.local_wires[j + 5 + k]);
+
+            let sel_minus_one = builder.sub_extension(sel, one);
+            let first_constraints = {
+                let term1 = nnf_mul_ext::<D, 5, Nnf>(builder, &x3, &z);
+                let term2 = array::from_fn(|i| builder.mul_extension(sel, x[i]));
+                let term3_1 = array::from_fn(|i| builder.mul_extension(sel_minus_one, x1[i]));
+                let term3 = nnf_mul_ext::<D, 5, Nnf>(builder, &term3_1, &z);
+                let partial_sum = nnf_ext_target_sub::<D, 5, Nnf>(builder, &term1, &term2);
+                nnf_ext_target_add::<D, 5, Nnf>(builder, &partial_sum, &term3)
+            };
+
+            let second_constraints = {
+                let term1 = nnf_mul_ext::<D, 5, Nnf>(builder, &u3, &t);
+                let term2 = array::from_fn(|i| builder.mul_extension(sel, u[i]));
+                let term3_1 = array::from_fn(|i| builder.mul_extension(sel_minus_one, u1[i]));
+                let term3 = nnf_mul_ext::<D, 5, Nnf>(builder, &term3_1, &t);
+                let partial_sum = nnf_ext_target_sub::<D, 5, Nnf>(builder, &term1, &term2);
+                nnf_ext_target_add::<D, 5, Nnf>(builder, &partial_sum, &term3)
+            };
+
+            new_constraints.extend(first_constraints);
+            new_constraints.extend(second_constraints);
+            new_constraints
+        };
+
+        constraints.extend(double_constraint(builder, 16, 106));
+        constraints.extend(select_and_add_constraint(builder, 106, 96, 0, false));
+        constraints.extend(select_and_add_constraint(builder, 96, 86, 3, true));
+
+        constraints.extend(double_constraint(builder, 86, 76));
+        constraints.extend(select_and_add_constraint(builder, 76, 66, 1, false));
+        constraints.extend(select_and_add_constraint(builder, 66, 56, 4, true));
+
+        constraints.extend(double_constraint(builder, 56, 46));
+        constraints.extend(select_and_add_constraint(builder, 46, 36, 2, false));
+        constraints.extend(select_and_add_constraint(builder, 36, 26, 5, true));
+
+        constraints
+    }
+
+    fn generators(
+        &self,
+        row: usize,
+        _local_constants: &[GoldilocksField],
+    ) -> Vec<WitnessGeneratorRef<GoldilocksField, D>> {
+        vec![WitnessGeneratorRef::new(ECAddXuGenerator { row }.adapter())]
+    }
+
+    fn num_wires(&self) -> usize {
+        Self::WIRES_PER_OP
+    }
+
+    fn degree(&self) -> usize {
+        Self::DEGREE
+    }
+
+    fn num_ops(&self) -> usize {
+        1
+    }
+
+    fn num_constants(&self) -> usize {
+        0
+    }
+
+    fn num_constraints(&self) -> usize {
+        90
+    }
+}
+
+// Useful auxiliary methods for defining the above gate follow.
+fn nnf_ext_target_add<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    y: &[ExtensionTarget<D>; DEG],
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    let sum_target = zip_eq(x, y)
+        .map(|(a, b)| builder.add_extension(*a, *b))
+        .collect::<Vec<_>>();
+    array::from_fn(|i| sum_target[i])
+}
+
+fn nnf_ext_target_sub<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    y: &[ExtensionTarget<D>; DEG],
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    let diff_target = zip_eq(x, y)
+        .map(|(a, b)| builder.sub_extension(*a, *b))
+        .collect::<Vec<_>>();
+    array::from_fn(|i| diff_target[i])
+}
+
+fn nnf_ext_target_add_field_gen<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    factor: NNF::BaseField,
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    array::from_fn(|i| {
+        if i == 1 {
+            builder.add_const_extension(x[1], factor)
+        } else {
+            x[i]
+        }
+    })
+}
+
+fn nnf_ext_target_mul_field_gen<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    factor: NNF::BaseField,
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    array::from_fn(|i| {
+        if i == 0 {
+            builder.mul_const_extension(factor * NNF::W, x[DEG - 1])
+        } else {
+            builder.mul_const_extension(factor, x[i - 1])
+        }
+    })
+}
+
+fn nnf_ext_target_add_scalar<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    scal: NNF::BaseField,
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    array::from_fn(|i| {
+        if i == 0 {
+            builder.add_const_extension(x[0], scal)
+        } else {
+            x[i]
+        }
+    })
+}
+
+fn homog_ec_target_addition_terms<const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    x1: &[ExtensionTarget<D>; 5],
+    u1: &[ExtensionTarget<D>; 5],
+    x2: &[ExtensionTarget<D>; 5],
+    u2: &[ExtensionTarget<D>; 5],
+) -> [[ExtensionTarget<D>; 5]; 5]
+where
+    F: Extendable<D>,
+{
+    type Nnf = QuinticExtension<F>;
+
+    let t1 = nnf_mul_ext::<D, 5, Nnf>(builder, x1, x2);
+    let t3 = nnf_mul_ext::<D, 5, Nnf>(builder, u1, u2);
+    let t5 = nnf_ext_target_add::<D, 5, Nnf>(builder, x1, x2);
+    let t6 = nnf_ext_target_add::<D, 5, Nnf>(builder, u1, u2);
+    let t7 = nnf_ext_target_add_field_gen::<D, 5, Nnf>(builder, &t1, Point::B1);
+
+    let twice_t7 = nnf_ext_target_add::<D, 5, Nnf>(builder, &t7, &t7);
+    let t5_mul_fg2b =
+        nnf_ext_target_mul_field_gen::<D, 5, Nnf>(builder, &t5, Point::B1 + Point::B1);
+    let t5_mul_fg2b_plus_twice_t7 =
+        nnf_ext_target_add::<D, 5, Nnf>(builder, &t5_mul_fg2b, &twice_t7);
+    let t9 = nnf_mul_ext::<D, 5, Nnf>(builder, &t3, &t5_mul_fg2b_plus_twice_t7);
+
+    let t5_plus_t7 = nnf_ext_target_add::<D, 5, Nnf>(builder, &t5, &t7);
+    let twice_t3 = nnf_ext_target_add::<D, 5, Nnf>(builder, &t3, &t3);
+    let twice_t3_plus_one =
+        nnf_ext_target_add_scalar::<D, 5, Nnf>(builder, &twice_t3, GoldilocksField::ONE);
+    let t10 = nnf_mul_ext::<D, 5, Nnf>(builder, &twice_t3_plus_one, &t5_plus_t7);
+    [t1, t6, t7, t9, t10]
+}
+
+// TODO: Make this more generic?
+/// Analogue of `add_homog` for extension targets.
+fn homog_ec_target_add<const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    x1: &[ExtensionTarget<D>; 5],
+    u1: &[ExtensionTarget<D>; 5],
+    x2: &[ExtensionTarget<D>; 5],
+    u2: &[ExtensionTarget<D>; 5],
+) -> [[ExtensionTarget<D>; 5]; 4]
+where
+    F: Extendable<D>,
+{
+    type Nnf = QuinticExtension<F>;
+
+    let [t1, t6, t7, t9, t10] = homog_ec_target_addition_terms(builder, x1, u1, x2, u2);
+
+    let t10_minus_t7 = nnf_ext_target_sub::<D, 5, Nnf>(builder, &t10, &t7);
+    let x = nnf_ext_target_mul_field_gen::<D, 5, Nnf>(builder, &t10_minus_t7, Point::B1);
+
+    let z = nnf_ext_target_sub::<D, 5, Nnf>(builder, &t7, &t9);
+
+    let minus_t1 = array::from_fn(|i| builder.mul_const_extension(-F::ONE, t1[i]));
+    let minus_t1_plus_fgpb =
+        nnf_ext_target_add_field_gen::<D, 5, Nnf>(builder, &minus_t1, Point::B1);
+    let u = nnf_mul_ext::<D, 5, Nnf>(builder, &t6, &minus_t1_plus_fgpb);
+
+    let t = nnf_ext_target_add::<D, 5, Nnf>(builder, &t7, &t9);
+
+    [x, z, u, t]
+}
+
+/// Analogue of `add_homog_offset` for extension targets.
+fn ec_target_add_homog_offset<const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    x1: &[ExtensionTarget<D>; 5],
+    u1: &[ExtensionTarget<D>; 5],
+    x2: &[ExtensionTarget<D>; 5],
+    u2: &[ExtensionTarget<D>; 5],
+) -> [[ExtensionTarget<D>; 5]; 4]
+where
+    F: Extendable<D>,
+{
+    type Nnf = QuinticExtension<F>;
+
+    let [t1, t6, t7, t9, t10] = homog_ec_target_addition_terms(builder, x1, u1, x2, u2);
+
+    let t10_minus_t7 = nnf_ext_target_sub::<D, 5, Nnf>(builder, &t10, &t7);
+    let x = nnf_ext_target_mul_field_gen::<D, 5, Nnf>(builder, &t10_minus_t7, Point::B1);
+
+    let z = nnf_ext_target_sub::<D, 5, Nnf>(builder, &t1, &t9);
+
+    let minus_t1 = array::from_fn(|i| builder.mul_const_extension(-F::ONE, t1[i]));
+    let minus_t1_plus_fgpb =
+        nnf_ext_target_add_field_gen::<D, 5, Nnf>(builder, &minus_t1, Point::B1);
+    let u = nnf_mul_ext::<D, 5, Nnf>(builder, &t6, &minus_t1_plus_fgpb);
+
+    let t = nnf_ext_target_add::<D, 5, Nnf>(builder, &t1, &t9);
+
+    [x, z, u, t]
 }
 
 #[cfg(test)]
@@ -58,40 +887,39 @@ mod test {
         plonk::{circuit_data::CircuitConfig, config::PoseidonGoldilocksConfig},
     };
 
-    use crate::backends::plonky2::primitives::ec::gates::{
+    use crate::backends::plonky2::primitives::ec::gates::curve::{
-        curve::ECAddHomogOffset, generic::GateAdapter,
+        ECAddHomogOffsetGate, ECAddXuGate,
     };
 
     #[test]
-    fn test_recursion() -> Result<(), anyhow::Error> {
+    fn test_ec_add_gate() -> Result<(), anyhow::Error> {
         let config = CircuitConfig::standard_recursion_config();
-        let gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+        let gate = ECAddHomogOffsetGate::new_from_config(&config);
 
         test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(gate)
     }
 
     #[test]
-    fn test_low_degree_orig() -> Result<(), anyhow::Error> {
+    fn test_ec_add_xu_gate() -> Result<(), anyhow::Error> {
+        let gate = ECAddXuGate::new();
+
+        test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(gate)
+    }
+
+    #[test]
+    fn test_ec_add_gate_low_degree() -> Result<(), anyhow::Error> {
         let config = CircuitConfig::standard_recursion_config();
-        let gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
+        let gate = ECAddHomogOffsetGate::new_from_config(&config);
 
         test_low_degree::<_, _, 2>(gate);
         Ok(())
     }
 
     #[test]
-    fn test_low_degree_recursive() -> Result<(), anyhow::Error> {
+    fn test_ec_add_xu_gate_low_degree() -> Result<(), anyhow::Error> {
-        let config = CircuitConfig::standard_recursion_config();
+        let gate = ECAddXuGate::new();
-        let orig_gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
 
-        test_low_degree::<_, _, 2>(orig_gate.recursive_gate());
+        test_low_degree::<_, _, 2>(gate);
         Ok(())
     }
-
-    #[test]
-    fn test_double_recursion() -> Result<(), anyhow::Error> {
-        let config = CircuitConfig::standard_recursion_config();
-        let orig_gate = GateAdapter::<ECAddHomogOffset>::new_from_config(&config);
-        test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(orig_gate.recursive_gate())
-    }
 }

src/backends/plonky2/primitives/ec/gates/field.rs

@@ -1,19 +1,33 @@
 use std::{
     array,
     marker::PhantomData,
-    ops::{Add, Mul, Neg, Sub},
+    ops::{Add, Mul, Neg, Range, Sub},
 };
 
+use itertools::zip_eq;
 use plonky2::{
     field::{
         extension::{quintic::QuinticExtension, Extendable, FieldExtension, OEF},
         goldilocks_field::GoldilocksField,
         types::Field,
     },
+    gates::gate::Gate,
     hash::hash_types::RichField,
+    iop::{
+        ext_target::ExtensionTarget,
+        generator::{GeneratedValues, SimpleGenerator, WitnessGeneratorRef},
+        target::Target,
+        witness::{PartitionWitness, Witness, WitnessWrite},
+    },
+    plonk::{
+        circuit_builder::CircuitBuilder,
+        circuit_data::{CircuitConfig, CommonCircuitData},
+        vars::{EvaluationTargets, EvaluationVars},
+    },
+    util::serialization::{Buffer, IoResult, Read, Write},
 };
 
-use crate::backends::plonky2::primitives::ec::{curve::ECFieldExt, gates::generic::SimpleGate};
+use crate::backends::plonky2::primitives::ec::curve::ECFieldExt;
 
 #[derive(Clone, Copy, PartialEq, Eq, Debug)]
 pub struct TensorProduct<const D1: usize, const D2: usize, F1, F2>
@@ -137,42 +151,284 @@ where
     }
 }
 
-#[derive(Debug, Clone, Copy, Default)]
+/// A gate which can perform a multiplication on OEF.
-pub struct NNFMulSimple<const DEG: usize, NNF: OEF<DEG>> {
+/// If the config has enough routed wires, it can support several such operations in one gate.
-    _phantom_data: PhantomData<fn(NNF) -> NNF>,
+#[derive(Debug, Clone)]
+pub struct NNFMulGate<const D: usize, const DEG: usize, NNF: OEF<DEG>> {
+    /// Number of multiplications performed by the gate.
+    pub num_ops: usize,
+    _phantom_data: PhantomData<NNF>,
 }
 
-impl<const DEG: usize, NNF: OEF<DEG>> NNFMulSimple<DEG, NNF> {
+impl<const D: usize, const DEG: usize, NNF: OEF<DEG>> NNFMulGate<D, DEG, NNF> {
-    pub fn new() -> Self {
+    pub const fn new_from_config(config: &CircuitConfig) -> Self {
         Self {
+            num_ops: Self::num_ops(config),
             _phantom_data: PhantomData,
         }
     }
+
+    /// Determine the maximum number of operations that can fit in one gate for the given config.
+    pub(crate) const fn num_ops(config: &CircuitConfig) -> usize {
+        let wires_per_op = 3 * DEG;
+        config.num_routed_wires / wires_per_op
+    }
+
+    pub(crate) const fn wires_ith_multiplicand_0(i: usize) -> Range<usize> {
+        3 * DEG * i..3 * DEG * i + DEG
+    }
+    pub(crate) const fn wires_ith_multiplicand_1(i: usize) -> Range<usize> {
+        3 * DEG * i + DEG..3 * DEG * i + 2 * DEG
+    }
+    pub(crate) const fn wires_ith_output(i: usize) -> Range<usize> {
+        3 * DEG * i + 2 * DEG..3 * DEG * (i + 1)
+    }
 }
 
-impl<NNF, const NNF_DEG: usize> SimpleGate for NNFMulSimple<NNF_DEG, NNF>
+impl<const D: usize, const DEG: usize, NNF: OEF<DEG>> Gate<NNF::BaseField, D>
+    for NNFMulGate<D, DEG, NNF>
 where
-    NNF: OEF<NNF_DEG>,
+    NNF::BaseField: RichField + Extendable<D>,
-    NNF::BaseField: RichField + Extendable<1>,
 {
-    type F = NNF::BaseField;
+    fn id(&self) -> String {
-    const INPUTS_PER_OP: usize = 2 * NNF_DEG;
+        format!("{self:?}")
-    const OUTPUTS_PER_OP: usize = NNF_DEG;
-    const DEGREE: usize = 2;
-    const ID: &'static str = "NNFSimpleGate";
-
-    fn eval<const D: usize>(
-        wires: &[<Self::F as Extendable<D>>::Extension],
-    ) -> Vec<<Self::F as Extendable<D>>::Extension>
-    where
-        Self::F: Extendable<D>,
-    {
-        let x: TensorProduct<NNF_DEG, D, NNF, <Self::F as Extendable<D>>::Extension> =
-            TensorProduct::new(array::from_fn(|i| wires[i]));
-        let y = TensorProduct::new(array::from_fn(|i| wires[NNF_DEG + i]));
-        let prod = x * y;
-        prod.components.into()
     }
+
+    fn serialize(
+        &self,
+        dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<NNF::BaseField, D>,
+    ) -> IoResult<()> {
+        dst.write_usize(self.num_ops)
+    }
+
+    fn deserialize(
+        src: &mut Buffer,
+        _common_data: &CommonCircuitData<NNF::BaseField, D>,
+    ) -> IoResult<Self> {
+        let num_ops = src.read_usize()?;
+        Ok(Self {
+            num_ops,
+            _phantom_data: PhantomData,
+        })
+    }
+
+    fn eval_unfiltered(
+        &self,
+        vars: EvaluationVars<NNF::BaseField, D>,
+    ) -> Vec<<NNF::BaseField as Extendable<D>>::Extension> {
+        let mut constraints = Vec::with_capacity(self.num_ops * DEG);
+        for i in 0..self.num_ops {
+            let multiplicand_0: TensorProduct<DEG, D, NNF, _> = TensorProduct::new(
+                vars.local_wires[Self::wires_ith_multiplicand_0(i)]
+                    .try_into()
+                    .unwrap(),
+            );
+            let multiplicand_1 = TensorProduct::new(
+                vars.local_wires[Self::wires_ith_multiplicand_1(i)]
+                    .try_into()
+                    .unwrap(),
+            );
+            let output = TensorProduct::new(
+                vars.local_wires[Self::wires_ith_output(i)]
+                    .try_into()
+                    .unwrap(),
+            );
+            let computed_output = multiplicand_0 * multiplicand_1;
+
+            constraints.extend((output - computed_output).components);
+        }
+
+        constraints
+    }
+
+    fn eval_unfiltered_circuit(
+        &self,
+        builder: &mut CircuitBuilder<NNF::BaseField, D>,
+        vars: EvaluationTargets<D>,
+    ) -> Vec<ExtensionTarget<D>> {
+        let mut constraints = Vec::with_capacity(self.num_ops * DEG);
+        for i in 0..self.num_ops {
+            let multiplicand_0: [ExtensionTarget<D>; DEG] = vars.local_wires
+                [Self::wires_ith_multiplicand_0(i)]
+            .try_into()
+            .unwrap();
+            let multiplicand_1: [ExtensionTarget<D>; DEG] = vars.local_wires
+                [Self::wires_ith_multiplicand_1(i)]
+            .try_into()
+            .unwrap();
+            let output: [ExtensionTarget<D>; DEG] = vars.local_wires[Self::wires_ith_output(i)]
+                .try_into()
+                .unwrap();
+            let computed_output =
+                nnf_mul_ext::<_, DEG, NNF>(builder, &multiplicand_0, &multiplicand_1);
+
+            let diffs = zip_eq(output, computed_output)
+                .map(|(o, co)| builder.sub_extension(o, co))
+                .collect::<Vec<_>>();
+            constraints.extend(diffs);
+        }
+
+        constraints
+    }
+
+    fn generators(
+        &self,
+        row: usize,
+        _local_constants: &[NNF::BaseField],
+    ) -> Vec<WitnessGeneratorRef<NNF::BaseField, D>> {
+        (0..self.num_ops)
+            .map(|i| {
+                WitnessGeneratorRef::new(
+                    NNFMulGenerator::<D, DEG, NNF> {
+                        row,
+                        i,
+                        phantom_data: PhantomData,
+                    }
+                    .adapter(),
+                )
+            })
+            .collect()
+    }
+
+    fn num_wires(&self) -> usize {
+        self.num_ops * 3 * DEG
+    }
+
+    fn num_constants(&self) -> usize {
+        0
+    }
+
+    fn degree(&self) -> usize {
+        3
+    }
+
+    fn num_constraints(&self) -> usize {
+        self.num_ops * DEG
+    }
+}
+
+#[derive(Clone, Debug, Default)]
+pub struct NNFMulGenerator<const D: usize, const DEG: usize, NNF: OEF<DEG>> {
+    row: usize,
+    i: usize,
+    phantom_data: PhantomData<NNF>,
+}
+
+impl<const D: usize, const DEG: usize, NNF: OEF<DEG>> SimpleGenerator<NNF::BaseField, D>
+    for NNFMulGenerator<D, DEG, NNF>
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    fn id(&self) -> String {
+        "NNFMulGenerator".to_string()
+    }
+
+    fn dependencies(&self) -> Vec<Target> {
+        NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_0(self.i)
+            .chain(NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_1(self.i))
+            .map(|i| Target::wire(self.row, i))
+            .collect()
+    }
+
+    fn run_once(
+        &self,
+        witness: &PartitionWitness<NNF::BaseField>,
+        out_buffer: &mut GeneratedValues<NNF::BaseField>,
+    ) -> anyhow::Result<()> {
+        let extract_nnf = |range: Range<usize>| -> anyhow::Result<NNF> {
+            let components: [NNF::BaseField; DEG] = range
+                .map(|i| witness.get_target(Target::wire(self.row, i)))
+                .collect::<Vec<_>>()
+                .try_into()
+                .map_err(|e| anyhow::anyhow!("{:?}", e))?;
+            Ok(NNF::from_basefield_array(components))
+        };
+
+        let multiplicand_0 =
+            extract_nnf(NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_0(self.i))?;
+        let multiplicand_1 =
+            extract_nnf(NNFMulGate::<D, DEG, NNF>::wires_ith_multiplicand_1(self.i))?;
+
+        let output_targets: [Target; DEG] = NNFMulGate::<D, DEG, NNF>::wires_ith_output(self.i)
+            .map(|i| Target::wire(self.row, i))
+            .collect::<Vec<_>>()
+            .try_into()
+            .map_err(|e| anyhow::anyhow!("{:?}", e))?;
+
+        let computed_output = multiplicand_0 * multiplicand_1;
+
+        out_buffer.set_target_arr(&output_targets, &computed_output.to_basefield_array())
+    }
+
+    fn serialize(
+        &self,
+        dst: &mut Vec<u8>,
+        _common_data: &CommonCircuitData<NNF::BaseField, D>,
+    ) -> IoResult<()> {
+        dst.write_usize(self.row)?;
+        dst.write_usize(self.i)
+    }
+
+    fn deserialize(
+        src: &mut Buffer,
+        _common_data: &CommonCircuitData<NNF::BaseField, D>,
+    ) -> IoResult<Self> {
+        let row = src.read_usize()?;
+        let i = src.read_usize()?;
+        Ok(Self {
+            row,
+            i,
+            phantom_data: PhantomData,
+        })
+    }
+}
+
+pub(crate) fn nnf_mul_ext<const D: usize, const DEG: usize, NNF: OEF<DEG>>(
+    builder: &mut CircuitBuilder<NNF::BaseField, D>,
+    x: &[ExtensionTarget<D>; DEG],
+    y: &[ExtensionTarget<D>; DEG],
+) -> [ExtensionTarget<D>; DEG]
+where
+    NNF::BaseField: RichField + Extendable<D>,
+{
+    let zero = builder.zero_extension();
+    let mul_targets = (0..DEG - 1)
+        .map(|k| {
+            let term1 = (0..=k)
+                .map(|i| builder.mul_extension(x[i], y[k - i]))
+                .collect::<Vec<_>>()
+                .into_iter()
+                .reduce(|sum, summand| builder.add_extension(sum, summand))
+                .expect("Missing summands");
+            let term2 = (k + 1..DEG)
+                .map(|i| {
+                    builder.arithmetic_extension(
+                        NNF::W,
+                        NNF::BaseField::ZERO,
+                        x[i],
+                        y[DEG + k - i],
+                        zero,
+                    )
+                })
+                .collect::<Vec<_>>()
+                .into_iter()
+                .reduce(|sum, summand| builder.add_extension(sum, summand))
+                .expect("Missing summands");
+            builder.add_extension(term1, term2)
+        })
+        .collect::<Vec<_>>()
+        .into_iter()
+        .chain(std::iter::once(
+            (0..DEG)
+                .map(|i| builder.mul_extension(x[i], y[DEG - 1 - i]))
+                .collect::<Vec<_>>()
+                .into_iter()
+                .reduce(|sum, summand| builder.add_extension(sum, summand))
+                .expect("Missing summands"),
+        ))
+        .collect::<Vec<_>>();
+    std::array::from_fn(|i| mul_targets[i])
 }
 
 #[cfg(test)]
@@ -183,52 +439,22 @@ mod test {
         plonk::{circuit_data::CircuitConfig, config::PoseidonGoldilocksConfig},
     };
 
-    use crate::backends::plonky2::primitives::ec::gates::{
+    use crate::backends::plonky2::{basetypes::D, primitives::ec::gates::field::NNFMulGate};
-        field::NNFMulSimple, generic::GateAdapter,
-    };
 
     #[test]
-    fn test_recursion() -> Result<(), anyhow::Error> {
+    fn test_nnf_mul_gate() -> Result<(), anyhow::Error> {
         let config = CircuitConfig::standard_recursion_config();
-        let gate =
+        let gate = NNFMulGate::<D, 5, QuinticExtension<GoldilocksField>>::new_from_config(&config);
-            GateAdapter::<NNFMulSimple<5, QuinticExtension<GoldilocksField>>>::new_from_config(
-                &config,
-            );
 
         test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(gate)
     }
 
     #[test]
-    fn test_low_degree_orig() -> Result<(), anyhow::Error> {
+    fn test_nnf_mul_gate_low_degree() -> Result<(), anyhow::Error> {
         let config = CircuitConfig::standard_recursion_config();
-        let gate =
+        let gate = NNFMulGate::<D, 5, QuinticExtension<GoldilocksField>>::new_from_config(&config);
-            GateAdapter::<NNFMulSimple<5, QuinticExtension<GoldilocksField>>>::new_from_config(
-                &config,
-            );
 
         test_low_degree::<_, _, 2>(gate);
         Ok(())
     }
-
-    #[test]
-    fn test_low_degree_recursive() -> Result<(), anyhow::Error> {
-        let config = CircuitConfig::standard_recursion_config();
-        let orig_gate =
-            GateAdapter::<NNFMulSimple<5, QuinticExtension<GoldilocksField>>>::new_from_config(
-                &config,
-            );
-
-        test_low_degree::<_, _, 2>(orig_gate.recursive_gate());
-        Ok(())
-    }
-
-    #[test]
-    fn test_double_recursion() -> Result<(), anyhow::Error> {
-        let config = CircuitConfig::standard_recursion_config();
-        let orig_gate =
-            GateAdapter::<NNFMulSimple<5, QuinticExtension<GoldilocksField>>>::new_from_config(
-                &config,
-            );
-        test_eval_fns::<_, PoseidonGoldilocksConfig, _, 2>(orig_gate.recursive_gate())
-    }
 }

src/backends/plonky2/primitives/ec/schnorr.rs

@@ -30,7 +30,7 @@ use crate::{
         deserialize_bytes,
         primitives::ec::{
             bits::{BigUInt320Target, CircuitBuilderBits},
-            curve::{CircuitBuilderElliptic, PointTarget, WitnessWriteCurve, GROUP_ORDER},
+            curve::{CircuitBuilderSignature, PointTarget, WitnessWriteCurve, GROUP_ORDER},
         },
         serialize_bytes, Error,
     },
@@ -155,14 +155,14 @@ impl SignatureTarget {
         msg: HashOutTarget,
         public_key: &PointTarget,
     ) -> BoolTarget {
-        let g = builder.constant_point(Point::generator());
         let sig1_bits = self.s.bits;
         let sig2_bits = self.e.bits;
-        let r = builder.linear_combination_points(&sig1_bits, &sig2_bits, &g, public_key);
+        let r = builder.linear_combination_point_gen(&sig1_bits, &sig2_bits, public_key);
         let u_arr = r.u.components;
         let inputs = u_arr.into_iter().chain(msg.elements).collect::<Vec<_>>();
         let e_hash = hash_array_circuit(builder, &inputs);
         let e = builder.field_elements_to_biguint(&e_hash);
+
         builder.is_equal_slice(&self.e.limbs, &e.limbs)
     }
 }
@@ -405,22 +405,38 @@ mod test {
 
     #[test]
     fn test_verify_signature_circuit() -> Result<(), anyhow::Error> {
-        let (public_key, msg, sig) = gen_signed_message();
+        let (public_key, msg, sig) = gen_signed_message(); // gets signed message
         let config = CircuitConfig::standard_recursion_config();
         let mut builder = CircuitBuilder::<GoldilocksField, 2>::new(config);
+
         let key_t = builder.add_virtual_point_target();
         let msg_t = builder.add_virtual_hash();
         let sig_t = SignatureTarget::add_virtual_target(&mut builder);
+
         let verified = sig_t.verify(&mut builder, msg_t, &key_t);
+
         builder.assert_one(verified.target);
         let mut pw = PartialWitness::new();
         pw.set_point_target(&key_t, &public_key)?;
         pw.set_hash_target(msg_t, msg.0.into())?;
         pw.set_biguint320_target(&sig_t.s, &sig.s)?;
         pw.set_biguint320_target(&sig_t.e, &sig.e)?;
+
+        println!("Building circuit...");
+        let start = std::time::Instant::now();
         let data = builder.build::<PoseidonGoldilocksConfig>();
+        println!("Circuit built in {:?}", start.elapsed());
+
+        println!("Generating proof...");
+        let start = std::time::Instant::now();
         let proof = data.prove(pw)?;
+        println!("Proof generated in {:?}", start.elapsed());
+
+        println!("Verifying proof...");
+        let start = std::time::Instant::now();
         data.verify(proof)?;
+        println!("Proof verified in {:?}", start.elapsed());
+
         Ok(())
     }
 

src/backends/plonky2/recursion/circuit.rs

@@ -38,9 +38,6 @@ use crate::{
     backends::plonky2::{
         basetypes::{C, D},
         error::Result,
-        primitives::ec::gates::{
-            curve::ECAddHomogOffset, field::NNFMulSimple, generic::GateAdapter,
-        },
     },
     measure_gates_begin, measure_gates_end,
     middleware::F,
@@ -364,9 +361,6 @@ fn coset_interpolation_gate(
 /// NOTE: The overhead between verifying any proof with just the `NoopGate` and verifying a proof
 /// with all these standard gates is about 400 num_gates (rows), no matter the circuit size.
 fn standard_gates(config: &CircuitConfig) -> Vec<GateRef<F, D>> {
-    let nnf_mul_simple =
-        GateAdapter::<NNFMulSimple<5, QuinticExtension<F>>>::new_from_config(config);
-    let ec_add_homog_offset = GateAdapter::<ECAddHomogOffset>::new_from_config(config);
     vec![
         GateRef::new(plonky2::gates::noop::NoopGate {}),
         GateRef::new(plonky2::gates::constant::ConstantGate::new(
@@ -391,10 +385,19 @@ fn standard_gates(config: &CircuitConfig) -> Vec<GateRef<F, D>> {
         GateRef::new(plonky2::gates::random_access::RandomAccessGate::new_from_config(config, 4)),
         GateRef::new(plonky2::gates::random_access::RandomAccessGate::new_from_config(config, 5)),
         GateRef::new(plonky2::gates::random_access::RandomAccessGate::new_from_config(config, 6)),
-        GateRef::new(nnf_mul_simple.recursive_gate()),
+        GateRef::new(
-        GateRef::new(nnf_mul_simple),
+            crate::backends::plonky2::primitives::ec::gates::field::NNFMulGate::<
-        GateRef::new(ec_add_homog_offset.recursive_gate()),
+                D,
-        GateRef::new(ec_add_homog_offset),
+                5,
+                QuinticExtension<F>,
+            >::new_from_config(config),
+        ),
+        GateRef::new(
+            crate::backends::plonky2::primitives::ec::gates::curve::ECAddXuGate::new(),
+        ),
+        GateRef::new(
+            crate::backends::plonky2::primitives::ec::gates::curve::ECAddHomogOffsetGate::new_from_config(config),
+        ),
         GateRef::new(plonky2::gates::exponentiation::ExponentiationGate::new_from_config(config)),
         // It would be better do `CosetInterpolationGate::with_max_degree(4, 6)` but unfortunately
         // that plonk2 method is `pub(crate)`, so we need to get around that somehow.

src/backends/plonky2/serialization.rs

@@ -27,9 +27,10 @@ use crate::backends::plonky2::{
         curve::PointSquareRootGenerator,
         field::QuotientGeneratorOEF,
         gates::{
-            curve::ECAddHomogOffset,
+            curve::{
-            field::NNFMulSimple,
+                ECAddHomogOffsetGate, ECAddHomogOffsetGenerator, ECAddXuGate, ECAddXuGenerator,
-            generic::{GateAdapter, RecursiveGateAdapter, RecursiveGenerator},
+            },
+            field::{NNFMulGate, NNFMulGenerator},
         },
     },
 };
@@ -56,10 +57,9 @@ impl GateSerializer<F, D> for Pod2GateSerializer {
         ReducingExtensionGate<D>,
         ReducingGate<D>,
         // pod2 custom gates
-        GateAdapter::<NNFMulSimple<5, QuinticExtension<F>>>,
+        NNFMulGate::<D, 5, QuinticExtension<F>>,
-        RecursiveGateAdapter::<D, NNFMulSimple<5, QuinticExtension<F>>>,
+        ECAddXuGate,
-        GateAdapter::<ECAddHomogOffset>,
+        ECAddHomogOffsetGate,
-        RecursiveGateAdapter::<D, ECAddHomogOffset>,
         ComparisonGate::<F, D>
     }
 }
@@ -127,10 +127,9 @@ impl WitnessGeneratorSerializer<F, D> for Pod2GeneratorSerializer {
         QuotientGeneratorOEF<5, QuinticExtension<F>>,
         PointSquareRootGenerator,
         ConditionalZeroGenerator<F, D>,
-        RecursiveGenerator<D, NNFMulSimple<5, QuinticExtension<F>>>,
+        NNFMulGenerator::<D, 5, QuinticExtension<F>>,
-        RecursiveGenerator<1, NNFMulSimple<5, QuinticExtension<F>>>,
+        ECAddXuGenerator,
-        RecursiveGenerator<D, ECAddHomogOffset>,
+        ECAddHomogOffsetGenerator,
-        RecursiveGenerator<1, ECAddHomogOffset>,
         ComparisonGenerator<F, D>,
         TableGetGenerator
     }