first commit

.gitignore

@@ -0,0 +1,11 @@
+# Cargo build
+/target
+Cargo.lock
+
+# Profile-guided optimization
+/tmp
+pgo-data.profdata
+
+# MacOS nuisances
+.DS_Store
+

Cargo.toml

@@ -0,0 +1,21 @@
+[package]
+name = "plonky2_ecdsa"
+description = "ECDSA gadget for Plonky2"
+version = "0.1.0"
+license = "MIT OR Apache-2.0"
+edition = "2021"
+
+[features]
+parallel = ["plonky2_maybe_rayon/parallel", "plonky2/parallel"]
+
+[dependencies]
+anyhow = { version = "1.0.40", default-features = false }
+itertools = { version = "0.10.0", default-features = false }
+plonky2_maybe_rayon = { version = "0.1.0", default-features = false }
+num = { version = "0.4.0", default-features = false }
+plonky2 = { version = "0.1.2", default-features = false }
+plonky2_u32 = { version = "0.1.0", default-features = false }
+serde = { version = "1.0", default-features = false, features = ["derive"] }
+
+[dev-dependencies]
+rand = { version = "0.8.4", default-features = false, features = ["getrandom"] }

LICENSE-APACHE

@@ -0,0 +1,202 @@
+                              Apache License
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+                     http://www.apache.org/licenses/
+
+TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
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+Copyright [yyyy] [name of copyright owner]
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+	http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
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+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+

LICENSE-MIT

@@ -0,0 +1,21 @@
+The MIT License (MIT)
+
+Copyright (c) 2022 The Plonky2 Authors
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
+THE SOFTWARE.

README.md

@@ -0,0 +1,13 @@
+## License
+
+Licensed under either of
+
+* Apache License, Version 2.0, ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0)
+* MIT license ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT)
+
+at your option.
+
+
+### Contribution
+
+Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

src/curve/curve_adds.rs

@@ -0,0 +1,158 @@
+use core::ops::Add;
+
+use plonky2::field::ops::Square;
+use plonky2::field::types::Field;
+
+use crate::curve::curve_types::{AffinePoint, Curve, ProjectivePoint};
+
+impl<C: Curve> Add<ProjectivePoint<C>> for ProjectivePoint<C> {
+    type Output = ProjectivePoint<C>;
+
+    fn add(self, rhs: ProjectivePoint<C>) -> Self::Output {
+        let ProjectivePoint {
+            x: x1,
+            y: y1,
+            z: z1,
+        } = self;
+        let ProjectivePoint {
+            x: x2,
+            y: y2,
+            z: z2,
+        } = rhs;
+
+        if z1 == C::BaseField::ZERO {
+            return rhs;
+        }
+        if z2 == C::BaseField::ZERO {
+            return self;
+        }
+
+        let x1z2 = x1 * z2;
+        let y1z2 = y1 * z2;
+        let x2z1 = x2 * z1;
+        let y2z1 = y2 * z1;
+
+        // Check if we're doubling or adding inverses.
+        if x1z2 == x2z1 {
+            if y1z2 == y2z1 {
+                // TODO: inline to avoid redundant muls.
+                return self.double();
+            }
+            if y1z2 == -y2z1 {
+                return ProjectivePoint::ZERO;
+            }
+        }
+
+        // From https://www.hyperelliptic.org/EFD/g1p/data/shortw/projective/addition/add-1998-cmo-2
+        let z1z2 = z1 * z2;
+        let u = y2z1 - y1z2;
+        let uu = u.square();
+        let v = x2z1 - x1z2;
+        let vv = v.square();
+        let vvv = v * vv;
+        let r = vv * x1z2;
+        let a = uu * z1z2 - vvv - r.double();
+        let x3 = v * a;
+        let y3 = u * (r - a) - vvv * y1z2;
+        let z3 = vvv * z1z2;
+        ProjectivePoint::nonzero(x3, y3, z3)
+    }
+}
+
+impl<C: Curve> Add<AffinePoint<C>> for ProjectivePoint<C> {
+    type Output = ProjectivePoint<C>;
+
+    fn add(self, rhs: AffinePoint<C>) -> Self::Output {
+        let ProjectivePoint {
+            x: x1,
+            y: y1,
+            z: z1,
+        } = self;
+        let AffinePoint {
+            x: x2,
+            y: y2,
+            zero: zero2,
+        } = rhs;
+
+        if z1 == C::BaseField::ZERO {
+            return rhs.to_projective();
+        }
+        if zero2 {
+            return self;
+        }
+
+        let x2z1 = x2 * z1;
+        let y2z1 = y2 * z1;
+
+        // Check if we're doubling or adding inverses.
+        if x1 == x2z1 {
+            if y1 == y2z1 {
+                // TODO: inline to avoid redundant muls.
+                return self.double();
+            }
+            if y1 == -y2z1 {
+                return ProjectivePoint::ZERO;
+            }
+        }
+
+        // From https://www.hyperelliptic.org/EFD/g1p/data/shortw/projective/addition/madd-1998-cmo
+        let u = y2z1 - y1;
+        let uu = u.square();
+        let v = x2z1 - x1;
+        let vv = v.square();
+        let vvv = v * vv;
+        let r = vv * x1;
+        let a = uu * z1 - vvv - r.double();
+        let x3 = v * a;
+        let y3 = u * (r - a) - vvv * y1;
+        let z3 = vvv * z1;
+        ProjectivePoint::nonzero(x3, y3, z3)
+    }
+}
+
+impl<C: Curve> Add<AffinePoint<C>> for AffinePoint<C> {
+    type Output = ProjectivePoint<C>;
+
+    fn add(self, rhs: AffinePoint<C>) -> Self::Output {
+        let AffinePoint {
+            x: x1,
+            y: y1,
+            zero: zero1,
+        } = self;
+        let AffinePoint {
+            x: x2,
+            y: y2,
+            zero: zero2,
+        } = rhs;
+
+        if zero1 {
+            return rhs.to_projective();
+        }
+        if zero2 {
+            return self.to_projective();
+        }
+
+        // Check if we're doubling or adding inverses.
+        if x1 == x2 {
+            if y1 == y2 {
+                return self.to_projective().double();
+            }
+            if y1 == -y2 {
+                return ProjectivePoint::ZERO;
+            }
+        }
+
+        // From https://www.hyperelliptic.org/EFD/g1p/data/shortw/projective/addition/mmadd-1998-cmo
+        let u = y2 - y1;
+        let uu = u.square();
+        let v = x2 - x1;
+        let vv = v.square();
+        let vvv = v * vv;
+        let r = vv * x1;
+        let a = uu - vvv - r.double();
+        let x3 = v * a;
+        let y3 = u * (r - a) - vvv * y1;
+        let z3 = vvv;
+        ProjectivePoint::nonzero(x3, y3, z3)
+    }
+}

src/curve/curve_msm.rs

@@ -0,0 +1,265 @@
+use alloc::vec::Vec;
+
+use itertools::Itertools;
+use plonky2::field::types::{Field, PrimeField};
+use plonky2_maybe_rayon::*;
+
+use crate::curve::curve_summation::affine_multisummation_best;
+use crate::curve::curve_types::{AffinePoint, Curve, ProjectivePoint};
+
+/// In Yao's method, we compute an affine summation for each digit. In a parallel setting, it would
+/// be easiest to assign individual summations to threads, but this would be sub-optimal because
+/// multi-summations can be more efficient than repeating individual summations (see
+/// `affine_multisummation_best`). Thus we divide digits into large chunks, and assign chunks of
+/// digits to threads. Note that there is a delicate balance here, as large chunks can result in
+/// uneven distributions of work among threads.
+const DIGITS_PER_CHUNK: usize = 80;
+
+#[derive(Clone, Debug)]
+pub struct MsmPrecomputation<C: Curve> {
+    /// For each generator (in the order they were passed to `msm_precompute`), contains a vector
+    /// of powers, i.e. [(2^w)^i] for i < DIGITS.
+    // TODO: Use compressed coordinates here.
+    powers_per_generator: Vec<Vec<AffinePoint<C>>>,
+
+    /// The window size.
+    w: usize,
+}
+
+pub fn msm_precompute<C: Curve>(
+    generators: &[ProjectivePoint<C>],
+    w: usize,
+) -> MsmPrecomputation<C> {
+    MsmPrecomputation {
+        powers_per_generator: generators
+            .into_par_iter()
+            .map(|&g| precompute_single_generator(g, w))
+            .collect(),
+        w,
+    }
+}
+
+fn precompute_single_generator<C: Curve>(g: ProjectivePoint<C>, w: usize) -> Vec<AffinePoint<C>> {
+    let digits = (C::ScalarField::BITS + w - 1) / w;
+    let mut powers: Vec<ProjectivePoint<C>> = Vec::with_capacity(digits);
+    powers.push(g);
+    for i in 1..digits {
+        let mut power_i_proj = powers[i - 1];
+        for _j in 0..w {
+            power_i_proj = power_i_proj.double();
+        }
+        powers.push(power_i_proj);
+    }
+    ProjectivePoint::batch_to_affine(&powers)
+}
+
+pub fn msm_parallel<C: Curve>(
+    scalars: &[C::ScalarField],
+    generators: &[ProjectivePoint<C>],
+    w: usize,
+) -> ProjectivePoint<C> {
+    let precomputation = msm_precompute(generators, w);
+    msm_execute_parallel(&precomputation, scalars)
+}
+
+pub fn msm_execute<C: Curve>(
+    precomputation: &MsmPrecomputation<C>,
+    scalars: &[C::ScalarField],
+) -> ProjectivePoint<C> {
+    assert_eq!(precomputation.powers_per_generator.len(), scalars.len());
+    let w = precomputation.w;
+    let digits = (C::ScalarField::BITS + w - 1) / w;
+    let base = 1 << w;
+
+    // This is a variant of Yao's method, adapted to the multi-scalar setting. Because we use
+    // extremely large windows, the repeated scans in Yao's method could be more expensive than the
+    // actual group operations. To avoid this, we store a multimap from each possible digit to the
+    // positions in which that digit occurs in the scalars. These positions have the form (i, j),
+    // where i is the index of the generator and j is an index into the digits of the scalar
+    // associated with that generator.
+    let mut digit_occurrences: Vec<Vec<(usize, usize)>> = Vec::with_capacity(digits);
+    for _i in 0..base {
+        digit_occurrences.push(Vec::new());
+    }
+    for (i, scalar) in scalars.iter().enumerate() {
+        let digits = to_digits::<C>(scalar, w);
+        for (j, &digit) in digits.iter().enumerate() {
+            digit_occurrences[digit].push((i, j));
+        }
+    }
+
+    let mut y = ProjectivePoint::ZERO;
+    let mut u = ProjectivePoint::ZERO;
+
+    for digit in (1..base).rev() {
+        for &(i, j) in &digit_occurrences[digit] {
+            u = u + precomputation.powers_per_generator[i][j];
+        }
+        y = y + u;
+    }
+
+    y
+}
+
+pub fn msm_execute_parallel<C: Curve>(
+    precomputation: &MsmPrecomputation<C>,
+    scalars: &[C::ScalarField],
+) -> ProjectivePoint<C> {
+    assert_eq!(precomputation.powers_per_generator.len(), scalars.len());
+    let w = precomputation.w;
+    let digits = (C::ScalarField::BITS + w - 1) / w;
+    let base = 1 << w;
+
+    // This is a variant of Yao's method, adapted to the multi-scalar setting. Because we use
+    // extremely large windows, the repeated scans in Yao's method could be more expensive than the
+    // actual group operations. To avoid this, we store a multimap from each possible digit to the
+    // positions in which that digit occurs in the scalars. These positions have the form (i, j),
+    // where i is the index of the generator and j is an index into the digits of the scalar
+    // associated with that generator.
+    let mut digit_occurrences: Vec<Vec<(usize, usize)>> = Vec::with_capacity(digits);
+    for _i in 0..base {
+        digit_occurrences.push(Vec::new());
+    }
+    for (i, scalar) in scalars.iter().enumerate() {
+        let digits = to_digits::<C>(scalar, w);
+        for (j, &digit) in digits.iter().enumerate() {
+            digit_occurrences[digit].push((i, j));
+        }
+    }
+
+    // For each digit, we add up the powers associated with all occurrences that digit.
+    let digits: Vec<usize> = (0..base).collect();
+    let digit_acc: Vec<ProjectivePoint<C>> = digits
+        .par_chunks(DIGITS_PER_CHUNK)
+        .flat_map(|chunk| {
+            let summations: Vec<Vec<AffinePoint<C>>> = chunk
+                .iter()
+                .map(|&digit| {
+                    digit_occurrences[digit]
+                        .iter()
+                        .map(|&(i, j)| precomputation.powers_per_generator[i][j])
+                        .collect()
+                })
+                .collect();
+            affine_multisummation_best(summations)
+        })
+        .collect();
+    // println!("Computing the per-digit summations (in parallel) took {}s", start.elapsed().as_secs_f64());
+
+    let mut y = ProjectivePoint::ZERO;
+    let mut u = ProjectivePoint::ZERO;
+    for digit in (1..base).rev() {
+        u = u + digit_acc[digit];
+        y = y + u;
+    }
+    // println!("Final summation (sequential) {}s", start.elapsed().as_secs_f64());
+    y
+}
+
+pub(crate) fn to_digits<C: Curve>(x: &C::ScalarField, w: usize) -> Vec<usize> {
+    let scalar_bits = C::ScalarField::BITS;
+    let num_digits = (scalar_bits + w - 1) / w;
+
+    // Convert x to a bool array.
+    let x_canonical: Vec<_> = x
+        .to_canonical_biguint()
+        .to_u64_digits()
+        .iter()
+        .cloned()
+        .pad_using(scalar_bits / 64, |_| 0)
+        .collect();
+    let mut x_bits = Vec::with_capacity(scalar_bits);
+    for i in 0..scalar_bits {
+        x_bits.push((x_canonical[i / 64] >> (i as u64 % 64) & 1) != 0);
+    }
+
+    let mut digits = Vec::with_capacity(num_digits);
+    for i in 0..num_digits {
+        let mut digit = 0;
+        for j in ((i * w)..((i + 1) * w).min(scalar_bits)).rev() {
+            digit <<= 1;
+            digit |= x_bits[j] as usize;
+        }
+        digits.push(digit);
+    }
+    digits
+}
+
+#[cfg(test)]
+mod tests {
+    use alloc::vec;
+
+    use num::BigUint;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+
+    use super::*;
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_to_digits() {
+        let x_canonical = [
+            0b10101010101010101010101010101010,
+            0b10101010101010101010101010101010,
+            0b11001100110011001100110011001100,
+            0b11001100110011001100110011001100,
+            0b11110000111100001111000011110000,
+            0b11110000111100001111000011110000,
+            0b00001111111111111111111111111111,
+            0b11111111111111111111111111111111,
+        ];
+        let x = Secp256K1Scalar::from_noncanonical_biguint(BigUint::from_slice(&x_canonical));
+        assert_eq!(x.to_canonical_biguint().to_u32_digits(), x_canonical);
+        assert_eq!(
+            to_digits::<Secp256K1>(&x, 17),
+            vec![
+                0b01010101010101010,
+                0b10101010101010101,
+                0b01010101010101010,
+                0b11001010101010101,
+                0b01100110011001100,
+                0b00110011001100110,
+                0b10011001100110011,
+                0b11110000110011001,
+                0b01111000011110000,
+                0b00111100001111000,
+                0b00011110000111100,
+                0b11111111111111110,
+                0b01111111111111111,
+                0b11111111111111000,
+                0b11111111111111111,
+                0b1,
+            ]
+        );
+    }
+
+    #[test]
+    fn test_msm() {
+        let w = 5;
+
+        let generator_1 = Secp256K1::GENERATOR_PROJECTIVE;
+        let generator_2 = generator_1 + generator_1;
+        let generator_3 = generator_1 + generator_2;
+
+        let scalar_1 = Secp256K1Scalar::from_noncanonical_biguint(BigUint::from_slice(&[
+            11111111, 22222222, 33333333, 44444444,
+        ]));
+        let scalar_2 = Secp256K1Scalar::from_noncanonical_biguint(BigUint::from_slice(&[
+            22222222, 22222222, 33333333, 44444444,
+        ]));
+        let scalar_3 = Secp256K1Scalar::from_noncanonical_biguint(BigUint::from_slice(&[
+            33333333, 22222222, 33333333, 44444444,
+        ]));
+
+        let generators = vec![generator_1, generator_2, generator_3];
+        let scalars = vec![scalar_1, scalar_2, scalar_3];
+
+        let precomputation = msm_precompute(&generators, w);
+        let result_msm = msm_execute(&precomputation, &scalars);
+
+        let result_naive = Secp256K1::convert(scalar_1) * generator_1
+            + Secp256K1::convert(scalar_2) * generator_2
+            + Secp256K1::convert(scalar_3) * generator_3;
+
+        assert_eq!(result_msm, result_naive);
+    }
+}

src/curve/curve_multiplication.rs

@@ -0,0 +1,100 @@
+use alloc::vec::Vec;
+use core::ops::Mul;
+
+use plonky2::field::types::{Field, PrimeField};
+
+use crate::curve::curve_types::{Curve, CurveScalar, ProjectivePoint};
+
+const WINDOW_BITS: usize = 4;
+const BASE: usize = 1 << WINDOW_BITS;
+
+fn digits_per_scalar<C: Curve>() -> usize {
+    (C::ScalarField::BITS + WINDOW_BITS - 1) / WINDOW_BITS
+}
+
+/// Precomputed state used for scalar x ProjectivePoint multiplications,
+/// specific to a particular generator.
+#[derive(Clone)]
+pub struct MultiplicationPrecomputation<C: Curve> {
+    /// [(2^w)^i] g for each i < digits_per_scalar.
+    powers: Vec<ProjectivePoint<C>>,
+}
+
+impl<C: Curve> ProjectivePoint<C> {
+    pub fn mul_precompute(&self) -> MultiplicationPrecomputation<C> {
+        let num_digits = digits_per_scalar::<C>();
+        let mut powers = Vec::with_capacity(num_digits);
+        powers.push(*self);
+        for i in 1..num_digits {
+            let mut power_i = powers[i - 1];
+            for _j in 0..WINDOW_BITS {
+                power_i = power_i.double();
+            }
+            powers.push(power_i);
+        }
+
+        MultiplicationPrecomputation { powers }
+    }
+
+    #[must_use]
+    pub fn mul_with_precomputation(
+        &self,
+        scalar: C::ScalarField,
+        precomputation: MultiplicationPrecomputation<C>,
+    ) -> Self {
+        // Yao's method; see https://koclab.cs.ucsb.edu/teaching/ecc/eccPapers/Doche-ch09.pdf
+        let precomputed_powers = precomputation.powers;
+
+        let digits = to_digits::<C>(&scalar);
+
+        let mut y = ProjectivePoint::ZERO;
+        let mut u = ProjectivePoint::ZERO;
+        let mut all_summands = Vec::new();
+        for j in (1..BASE).rev() {
+            let mut u_summands = Vec::new();
+            for (i, &digit) in digits.iter().enumerate() {
+                if digit == j as u64 {
+                    u_summands.push(precomputed_powers[i]);
+                }
+            }
+            all_summands.push(u_summands);
+        }
+
+        let all_sums: Vec<ProjectivePoint<C>> = all_summands
+            .iter()
+            .cloned()
+            .map(|vec| vec.iter().fold(ProjectivePoint::ZERO, |a, &b| a + b))
+            .collect();
+        for i in 0..all_sums.len() {
+            u = u + all_sums[i];
+            y = y + u;
+        }
+        y
+    }
+}
+
+impl<C: Curve> Mul<ProjectivePoint<C>> for CurveScalar<C> {
+    type Output = ProjectivePoint<C>;
+
+    fn mul(self, rhs: ProjectivePoint<C>) -> Self::Output {
+        let precomputation = rhs.mul_precompute();
+        rhs.mul_with_precomputation(self.0, precomputation)
+    }
+}
+
+#[allow(clippy::assertions_on_constants)]
+fn to_digits<C: Curve>(x: &C::ScalarField) -> Vec<u64> {
+    debug_assert!(
+        64 % WINDOW_BITS == 0,
+        "For simplicity, only power-of-two window sizes are handled for now"
+    );
+    let digits_per_u64 = 64 / WINDOW_BITS;
+    let mut digits = Vec::with_capacity(digits_per_scalar::<C>());
+    for limb in x.to_canonical_biguint().to_u64_digits() {
+        for j in 0..digits_per_u64 {
+            digits.push((limb >> (j * WINDOW_BITS) as u64) % BASE as u64);
+        }
+    }
+
+    digits
+}

src/curve/curve_summation.rs

@@ -0,0 +1,238 @@
+use alloc::vec;
+use alloc::vec::Vec;
+use core::iter::Sum;
+
+use plonky2::field::ops::Square;
+use plonky2::field::types::Field;
+
+use crate::curve::curve_types::{AffinePoint, Curve, ProjectivePoint};
+
+impl<C: Curve> Sum<AffinePoint<C>> for ProjectivePoint<C> {
+    fn sum<I: Iterator<Item = AffinePoint<C>>>(iter: I) -> ProjectivePoint<C> {
+        let points: Vec<_> = iter.collect();
+        affine_summation_best(points)
+    }
+}
+
+impl<C: Curve> Sum for ProjectivePoint<C> {
+    fn sum<I: Iterator<Item = ProjectivePoint<C>>>(iter: I) -> ProjectivePoint<C> {
+        iter.fold(ProjectivePoint::ZERO, |acc, x| acc + x)
+    }
+}
+
+pub fn affine_summation_best<C: Curve>(summation: Vec<AffinePoint<C>>) -> ProjectivePoint<C> {
+    let result = affine_multisummation_best(vec![summation]);
+    debug_assert_eq!(result.len(), 1);
+    result[0]
+}
+
+pub fn affine_multisummation_best<C: Curve>(
+    summations: Vec<Vec<AffinePoint<C>>>,
+) -> Vec<ProjectivePoint<C>> {
+    let pairwise_sums: usize = summations.iter().map(|summation| summation.len() / 2).sum();
+
+    // This threshold is chosen based on data from the summation benchmarks.
+    if pairwise_sums < 70 {
+        affine_multisummation_pairwise(summations)
+    } else {
+        affine_multisummation_batch_inversion(summations)
+    }
+}
+
+/// Adds each pair of points using an affine + affine = projective formula, then adds up the
+/// intermediate sums using a projective formula.
+pub fn affine_multisummation_pairwise<C: Curve>(
+    summations: Vec<Vec<AffinePoint<C>>>,
+) -> Vec<ProjectivePoint<C>> {
+    summations
+        .into_iter()
+        .map(affine_summation_pairwise)
+        .collect()
+}
+
+/// Adds each pair of points using an affine + affine = projective formula, then adds up the
+/// intermediate sums using a projective formula.
+pub fn affine_summation_pairwise<C: Curve>(points: Vec<AffinePoint<C>>) -> ProjectivePoint<C> {
+    let mut reduced_points: Vec<ProjectivePoint<C>> = Vec::new();
+    for chunk in points.chunks(2) {
+        match chunk.len() {
+            1 => reduced_points.push(chunk[0].to_projective()),
+            2 => reduced_points.push(chunk[0] + chunk[1]),
+            _ => panic!(),
+        }
+    }
+    // TODO: Avoid copying (deref)
+    reduced_points
+        .iter()
+        .fold(ProjectivePoint::ZERO, |sum, x| sum + *x)
+}
+
+/// Computes several summations of affine points by applying an affine group law, except that the
+/// divisions are batched via Montgomery's trick.
+pub fn affine_summation_batch_inversion<C: Curve>(
+    summation: Vec<AffinePoint<C>>,
+) -> ProjectivePoint<C> {
+    let result = affine_multisummation_batch_inversion(vec![summation]);
+    debug_assert_eq!(result.len(), 1);
+    result[0]
+}
+
+/// Computes several summations of affine points by applying an affine group law, except that the
+/// divisions are batched via Montgomery's trick.
+pub fn affine_multisummation_batch_inversion<C: Curve>(
+    summations: Vec<Vec<AffinePoint<C>>>,
+) -> Vec<ProjectivePoint<C>> {
+    let mut elements_to_invert = Vec::new();
+
+    // For each pair of points, (x1, y1) and (x2, y2), that we're going to add later, we want to
+    // invert either y (if the points are equal) or x1 - x2 (otherwise). We will use these later.
+    for summation in &summations {
+        let n = summation.len();
+        // The special case for n=0 is to avoid underflow.
+        let range_end = if n == 0 { 0 } else { n - 1 };
+
+        for i in (0..range_end).step_by(2) {
+            let p1 = summation[i];
+            let p2 = summation[i + 1];
+            let AffinePoint {
+                x: x1,
+                y: y1,
+                zero: zero1,
+            } = p1;
+            let AffinePoint {
+                x: x2,
+                y: _y2,
+                zero: zero2,
+            } = p2;
+
+            if zero1 || zero2 || p1 == -p2 {
+                // These are trivial cases where we won't need any inverse.
+            } else if p1 == p2 {
+                elements_to_invert.push(y1.double());
+            } else {
+                elements_to_invert.push(x1 - x2);
+            }
+        }
+    }
+
+    let inverses: Vec<C::BaseField> =
+        C::BaseField::batch_multiplicative_inverse(&elements_to_invert);
+
+    let mut all_reduced_points = Vec::with_capacity(summations.len());
+    let mut inverse_index = 0;
+    for summation in summations {
+        let n = summation.len();
+        let mut reduced_points = Vec::with_capacity((n + 1) / 2);
+
+        // The special case for n=0 is to avoid underflow.
+        let range_end = if n == 0 { 0 } else { n - 1 };
+
+        for i in (0..range_end).step_by(2) {
+            let p1 = summation[i];
+            let p2 = summation[i + 1];
+            let AffinePoint {
+                x: x1,
+                y: y1,
+                zero: zero1,
+            } = p1;
+            let AffinePoint {
+                x: x2,
+                y: y2,
+                zero: zero2,
+            } = p2;
+
+            let sum = if zero1 {
+                p2
+            } else if zero2 {
+                p1
+            } else if p1 == -p2 {
+                AffinePoint::ZERO
+            } else {
+                // It's a non-trivial case where we need one of the inverses we computed earlier.
+                let inverse = inverses[inverse_index];
+                inverse_index += 1;
+
+                if p1 == p2 {
+                    // This is the doubling case.
+                    let mut numerator = x1.square().triple();
+                    if C::A.is_nonzero() {
+                        numerator += C::A;
+                    }
+                    let quotient = numerator * inverse;
+                    let x3 = quotient.square() - x1.double();
+                    let y3 = quotient * (x1 - x3) - y1;
+                    AffinePoint::nonzero(x3, y3)
+                } else {
+                    // This is the general case. We use the incomplete addition formulas 4.3 and 4.4.
+                    let quotient = (y1 - y2) * inverse;
+                    let x3 = quotient.square() - x1 - x2;
+                    let y3 = quotient * (x1 - x3) - y1;
+                    AffinePoint::nonzero(x3, y3)
+                }
+            };
+            reduced_points.push(sum);
+        }
+
+        // If n is odd, the last point was not part of a pair.
+        if n % 2 == 1 {
+            reduced_points.push(summation[n - 1]);
+        }
+
+        all_reduced_points.push(reduced_points);
+    }
+
+    // We should have consumed all of the inverses from the batch computation.
+    debug_assert_eq!(inverse_index, inverses.len());
+
+    // Recurse with our smaller set of points.
+    affine_multisummation_best(all_reduced_points)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_pairwise_affine_summation() {
+        let g_affine = Secp256K1::GENERATOR_AFFINE;
+        let g2_affine = (g_affine + g_affine).to_affine();
+        let g3_affine = (g_affine + g_affine + g_affine).to_affine();
+        let g2_proj = g2_affine.to_projective();
+        let g3_proj = g3_affine.to_projective();
+        assert_eq!(
+            affine_summation_pairwise::<Secp256K1>(vec![g_affine, g_affine]),
+            g2_proj
+        );
+        assert_eq!(
+            affine_summation_pairwise::<Secp256K1>(vec![g_affine, g2_affine]),
+            g3_proj
+        );
+        assert_eq!(
+            affine_summation_pairwise::<Secp256K1>(vec![g_affine, g_affine, g_affine]),
+            g3_proj
+        );
+        assert_eq!(
+            affine_summation_pairwise::<Secp256K1>(vec![]),
+            ProjectivePoint::ZERO
+        );
+    }
+
+    #[test]
+    fn test_pairwise_affine_summation_batch_inversion() {
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let g_proj = g.to_projective();
+        assert_eq!(
+            affine_summation_batch_inversion::<Secp256K1>(vec![g, g]),
+            g_proj + g_proj
+        );
+        assert_eq!(
+            affine_summation_batch_inversion::<Secp256K1>(vec![g, g, g]),
+            g_proj + g_proj + g_proj
+        );
+        assert_eq!(
+            affine_summation_batch_inversion::<Secp256K1>(vec![]),
+            ProjectivePoint::ZERO
+        );
+    }
+}

src/curve/curve_types.rs

@@ -0,0 +1,286 @@
+use alloc::vec::Vec;
+use core::fmt::Debug;
+use core::hash::{Hash, Hasher};
+use core::ops::Neg;
+
+use plonky2::field::ops::Square;
+use plonky2::field::types::{Field, PrimeField};
+use serde::{Deserialize, Serialize};
+
+// To avoid implementation conflicts from associated types,
+// see https://github.com/rust-lang/rust/issues/20400
+pub struct CurveScalar<C: Curve>(pub <C as Curve>::ScalarField);
+
+/// A short Weierstrass curve.
+pub trait Curve: 'static + Sync + Sized + Copy + Debug {
+    type BaseField: PrimeField;
+    type ScalarField: PrimeField;
+
+    const A: Self::BaseField;
+    const B: Self::BaseField;
+
+    const GENERATOR_AFFINE: AffinePoint<Self>;
+
+    const GENERATOR_PROJECTIVE: ProjectivePoint<Self> = ProjectivePoint {
+        x: Self::GENERATOR_AFFINE.x,
+        y: Self::GENERATOR_AFFINE.y,
+        z: Self::BaseField::ONE,
+    };
+
+    fn convert(x: Self::ScalarField) -> CurveScalar<Self> {
+        CurveScalar(x)
+    }
+
+    fn is_safe_curve() -> bool {
+        // Added additional check to prevent using vulnerabilties in case a discriminant is equal to 0.
+        (Self::A.cube().double().double() + Self::B.square().triple().triple().triple())
+            .is_nonzero()
+    }
+}
+
+/// A point on a short Weierstrass curve, represented in affine coordinates.
+#[derive(Copy, Clone, Debug, Deserialize, Serialize)]
+pub struct AffinePoint<C: Curve> {
+    pub x: C::BaseField,
+    pub y: C::BaseField,
+    pub zero: bool,
+}
+
+impl<C: Curve> AffinePoint<C> {
+    pub const ZERO: Self = Self {
+        x: C::BaseField::ZERO,
+        y: C::BaseField::ZERO,
+        zero: true,
+    };
+
+    pub fn nonzero(x: C::BaseField, y: C::BaseField) -> Self {
+        let point = Self { x, y, zero: false };
+        debug_assert!(point.is_valid());
+        point
+    }
+
+    pub fn is_valid(&self) -> bool {
+        let Self { x, y, zero } = *self;
+        zero || y.square() == x.cube() + C::A * x + C::B
+    }
+
+    pub fn to_projective(&self) -> ProjectivePoint<C> {
+        let Self { x, y, zero } = *self;
+        let z = if zero {
+            C::BaseField::ZERO
+        } else {
+            C::BaseField::ONE
+        };
+
+        ProjectivePoint { x, y, z }
+    }
+
+    pub fn batch_to_projective(affine_points: &[Self]) -> Vec<ProjectivePoint<C>> {
+        affine_points.iter().map(Self::to_projective).collect()
+    }
+
+    #[must_use]
+    pub fn double(&self) -> Self {
+        let AffinePoint { x: x1, y: y1, zero } = *self;
+
+        if zero {
+            return AffinePoint::ZERO;
+        }
+
+        let double_y = y1.double();
+        let inv_double_y = double_y.inverse(); // (2y)^(-1)
+        let triple_xx = x1.square().triple(); // 3x^2
+        let lambda = (triple_xx + C::A) * inv_double_y;
+        let x3 = lambda.square() - self.x.double();
+        let y3 = lambda * (x1 - x3) - y1;
+
+        Self {
+            x: x3,
+            y: y3,
+            zero: false,
+        }
+    }
+}
+
+impl<C: Curve> PartialEq for AffinePoint<C> {
+    fn eq(&self, other: &Self) -> bool {
+        let AffinePoint {
+            x: x1,
+            y: y1,
+            zero: zero1,
+        } = *self;
+        let AffinePoint {
+            x: x2,
+            y: y2,
+            zero: zero2,
+        } = *other;
+        if zero1 || zero2 {
+            return zero1 == zero2;
+        }
+        x1 == x2 && y1 == y2
+    }
+}
+
+impl<C: Curve> Eq for AffinePoint<C> {}
+
+impl<C: Curve> Hash for AffinePoint<C> {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        if self.zero {
+            self.zero.hash(state);
+        } else {
+            self.x.hash(state);
+            self.y.hash(state);
+        }
+    }
+}
+
+/// A point on a short Weierstrass curve, represented in projective coordinates.
+#[derive(Copy, Clone, Debug)]
+pub struct ProjectivePoint<C: Curve> {
+    pub x: C::BaseField,
+    pub y: C::BaseField,
+    pub z: C::BaseField,
+}
+
+impl<C: Curve> ProjectivePoint<C> {
+    pub const ZERO: Self = Self {
+        x: C::BaseField::ZERO,
+        y: C::BaseField::ONE,
+        z: C::BaseField::ZERO,
+    };
+
+    pub fn nonzero(x: C::BaseField, y: C::BaseField, z: C::BaseField) -> Self {
+        let point = Self { x, y, z };
+        debug_assert!(point.is_valid());
+        point
+    }
+
+    pub fn is_valid(&self) -> bool {
+        let Self { x, y, z } = *self;
+        z.is_zero() || y.square() * z == x.cube() + C::A * x * z.square() + C::B * z.cube()
+    }
+
+    pub fn to_affine(&self) -> AffinePoint<C> {
+        let Self { x, y, z } = *self;
+        if z == C::BaseField::ZERO {
+            AffinePoint::ZERO
+        } else {
+            let z_inv = z.inverse();
+            AffinePoint::nonzero(x * z_inv, y * z_inv)
+        }
+    }
+
+    pub fn batch_to_affine(proj_points: &[Self]) -> Vec<AffinePoint<C>> {
+        let n = proj_points.len();
+        let zs: Vec<C::BaseField> = proj_points.iter().map(|pp| pp.z).collect();
+        let z_invs = C::BaseField::batch_multiplicative_inverse(&zs);
+
+        let mut result = Vec::with_capacity(n);
+        for i in 0..n {
+            let Self { x, y, z } = proj_points[i];
+            result.push(if z == C::BaseField::ZERO {
+                AffinePoint::ZERO
+            } else {
+                let z_inv = z_invs[i];
+                AffinePoint::nonzero(x * z_inv, y * z_inv)
+            });
+        }
+        result
+    }
+
+    // From https://www.hyperelliptic.org/EFD/g1p/data/shortw/projective/doubling/dbl-2007-bl
+    #[must_use]
+    pub fn double(&self) -> Self {
+        let Self { x, y, z } = *self;
+        if z == C::BaseField::ZERO {
+            return ProjectivePoint::ZERO;
+        }
+
+        let xx = x.square();
+        let zz = z.square();
+        let mut w = xx.triple();
+        if C::A.is_nonzero() {
+            w += C::A * zz;
+        }
+        let s = y.double() * z;
+        let r = y * s;
+        let rr = r.square();
+        let b = (x + r).square() - (xx + rr);
+        let h = w.square() - b.double();
+        let x3 = h * s;
+        let y3 = w * (b - h) - rr.double();
+        let z3 = s.cube();
+        Self {
+            x: x3,
+            y: y3,
+            z: z3,
+        }
+    }
+
+    pub fn add_slices(a: &[Self], b: &[Self]) -> Vec<Self> {
+        assert_eq!(a.len(), b.len());
+        a.iter()
+            .zip(b.iter())
+            .map(|(&a_i, &b_i)| a_i + b_i)
+            .collect()
+    }
+
+    #[must_use]
+    pub fn neg(&self) -> Self {
+        Self {
+            x: self.x,
+            y: -self.y,
+            z: self.z,
+        }
+    }
+}
+
+impl<C: Curve> PartialEq for ProjectivePoint<C> {
+    fn eq(&self, other: &Self) -> bool {
+        let ProjectivePoint {
+            x: x1,
+            y: y1,
+            z: z1,
+        } = *self;
+        let ProjectivePoint {
+            x: x2,
+            y: y2,
+            z: z2,
+        } = *other;
+        if z1 == C::BaseField::ZERO || z2 == C::BaseField::ZERO {
+            return z1 == z2;
+        }
+
+        // We want to compare (x1/z1, y1/z1) == (x2/z2, y2/z2).
+        // But to avoid field division, it is better to compare (x1*z2, y1*z2) == (x2*z1, y2*z1).
+        x1 * z2 == x2 * z1 && y1 * z2 == y2 * z1
+    }
+}
+
+impl<C: Curve> Eq for ProjectivePoint<C> {}
+
+impl<C: Curve> Neg for AffinePoint<C> {
+    type Output = AffinePoint<C>;
+
+    fn neg(self) -> Self::Output {
+        let AffinePoint { x, y, zero } = self;
+        AffinePoint { x, y: -y, zero }
+    }
+}
+
+impl<C: Curve> Neg for ProjectivePoint<C> {
+    type Output = ProjectivePoint<C>;
+
+    fn neg(self) -> Self::Output {
+        let ProjectivePoint { x, y, z } = self;
+        ProjectivePoint { x, y: -y, z }
+    }
+}
+
+pub fn base_to_scalar<C: Curve>(x: C::BaseField) -> C::ScalarField {
+    C::ScalarField::from_noncanonical_biguint(x.to_canonical_biguint())
+}
+
+pub fn scalar_to_base<C: Curve>(x: C::ScalarField) -> C::BaseField {
+    C::BaseField::from_noncanonical_biguint(x.to_canonical_biguint())
+}

src/curve/ecdsa.rs

@@ -0,0 +1,84 @@
+use plonky2::field::types::{Field, Sample};
+use serde::{Deserialize, Serialize};
+
+use crate::curve::curve_msm::msm_parallel;
+use crate::curve::curve_types::{base_to_scalar, AffinePoint, Curve, CurveScalar};
+
+#[derive(Copy, Clone, Debug, Deserialize, Eq, Hash, PartialEq, Serialize)]
+pub struct ECDSASignature<C: Curve> {
+    pub r: C::ScalarField,
+    pub s: C::ScalarField,
+}
+
+#[derive(Copy, Clone, Debug, Deserialize, Eq, Hash, PartialEq, Serialize)]
+pub struct ECDSASecretKey<C: Curve>(pub C::ScalarField);
+
+impl<C: Curve> ECDSASecretKey<C> {
+    pub fn to_public(&self) -> ECDSAPublicKey<C> {
+        ECDSAPublicKey((CurveScalar(self.0) * C::GENERATOR_PROJECTIVE).to_affine())
+    }
+}
+
+#[derive(Copy, Clone, Debug, Deserialize, Eq, Hash, PartialEq, Serialize)]
+pub struct ECDSAPublicKey<C: Curve>(pub AffinePoint<C>);
+
+pub fn sign_message<C: Curve>(msg: C::ScalarField, sk: ECDSASecretKey<C>) -> ECDSASignature<C> {
+    let (k, rr) = {
+        let mut k = C::ScalarField::rand();
+        let mut rr = (CurveScalar(k) * C::GENERATOR_PROJECTIVE).to_affine();
+        while rr.x == C::BaseField::ZERO {
+            k = C::ScalarField::rand();
+            rr = (CurveScalar(k) * C::GENERATOR_PROJECTIVE).to_affine();
+        }
+        (k, rr)
+    };
+    let r = base_to_scalar::<C>(rr.x);
+
+    let s = k.inverse() * (msg + r * sk.0);
+
+    ECDSASignature { r, s }
+}
+
+pub fn verify_message<C: Curve>(
+    msg: C::ScalarField,
+    sig: ECDSASignature<C>,
+    pk: ECDSAPublicKey<C>,
+) -> bool {
+    let ECDSASignature { r, s } = sig;
+
+    assert!(pk.0.is_valid());
+
+    let c = s.inverse();
+    let u1 = msg * c;
+    let u2 = r * c;
+
+    let g = C::GENERATOR_PROJECTIVE;
+    let w = 5; // Experimentally fastest
+    let point_proj = msm_parallel(&[u1, u2], &[g, pk.0.to_projective()], w);
+    let point = point_proj.to_affine();
+
+    let x = base_to_scalar::<C>(point.x);
+    r == x
+}
+
+#[cfg(test)]
+mod tests {
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::Sample;
+
+    use crate::curve::ecdsa::{sign_message, verify_message, ECDSASecretKey};
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_ecdsa_native() {
+        type C = Secp256K1;
+
+        let msg = Secp256K1Scalar::rand();
+        let sk = ECDSASecretKey::<C>(Secp256K1Scalar::rand());
+        let pk = sk.to_public();
+
+        let sig = sign_message(msg, sk);
+        let result = verify_message(msg, sig, pk);
+        assert!(result);
+    }
+}

src/curve/glv.rs

@@ -0,0 +1,140 @@
+use num::rational::Ratio;
+use num::BigUint;
+use plonky2::field::secp256k1_base::Secp256K1Base;
+use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+use plonky2::field::types::{Field, PrimeField};
+
+use crate::curve::curve_msm::msm_parallel;
+use crate::curve::curve_types::{AffinePoint, ProjectivePoint};
+use crate::curve::secp256k1::Secp256K1;
+
+pub const GLV_BETA: Secp256K1Base = Secp256K1Base([
+    13923278643952681454,
+    11308619431505398165,
+    7954561588662645993,
+    8856726876819556112,
+]);
+
+pub const GLV_S: Secp256K1Scalar = Secp256K1Scalar([
+    16069571880186789234,
+    1310022930574435960,
+    11900229862571533402,
+    6008836872998760672,
+]);
+
+const A1: Secp256K1Scalar = Secp256K1Scalar([16747920425669159701, 3496713202691238861, 0, 0]);
+
+const MINUS_B1: Secp256K1Scalar =
+    Secp256K1Scalar([8022177200260244675, 16448129721693014056, 0, 0]);
+
+const A2: Secp256K1Scalar = Secp256K1Scalar([6323353552219852760, 1498098850674701302, 1, 0]);
+
+const B2: Secp256K1Scalar = Secp256K1Scalar([16747920425669159701, 3496713202691238861, 0, 0]);
+
+/// Algorithm 15.41 in Handbook of Elliptic and Hyperelliptic Curve Cryptography.
+/// Decompose a scalar `k` into two small scalars `k1, k2` with `|k1|, |k2| < √p` that satisfy
+/// `k1 + s * k2 = k`.
+/// Returns `(|k1|, |k2|, k1 < 0, k2 < 0)`.
+pub fn decompose_secp256k1_scalar(
+    k: Secp256K1Scalar,
+) -> (Secp256K1Scalar, Secp256K1Scalar, bool, bool) {
+    let p = Secp256K1Scalar::order();
+    let c1_biguint = Ratio::new(
+        B2.to_canonical_biguint() * k.to_canonical_biguint(),
+        p.clone(),
+    )
+    .round()
+    .to_integer();
+    let c1 = Secp256K1Scalar::from_noncanonical_biguint(c1_biguint);
+    let c2_biguint = Ratio::new(
+        MINUS_B1.to_canonical_biguint() * k.to_canonical_biguint(),
+        p.clone(),
+    )
+    .round()
+    .to_integer();
+    let c2 = Secp256K1Scalar::from_noncanonical_biguint(c2_biguint);
+
+    let k1_raw = k - c1 * A1 - c2 * A2;
+    let k2_raw = c1 * MINUS_B1 - c2 * B2;
+    debug_assert!(k1_raw + GLV_S * k2_raw == k);
+
+    let two = BigUint::from_slice(&[2]);
+    let k1_neg = k1_raw.to_canonical_biguint() > p.clone() / two.clone();
+    let k1 = if k1_neg {
+        Secp256K1Scalar::from_noncanonical_biguint(p.clone() - k1_raw.to_canonical_biguint())
+    } else {
+        k1_raw
+    };
+    let k2_neg = k2_raw.to_canonical_biguint() > p.clone() / two;
+    let k2 = if k2_neg {
+        Secp256K1Scalar::from_noncanonical_biguint(p - k2_raw.to_canonical_biguint())
+    } else {
+        k2_raw
+    };
+
+    (k1, k2, k1_neg, k2_neg)
+}
+
+/// See Section 15.2.1 in Handbook of Elliptic and Hyperelliptic Curve Cryptography.
+/// GLV scalar multiplication `k * P = k1 * P + k2 * psi(P)`, where `k = k1 + s * k2` is the
+/// decomposition computed in `decompose_secp256k1_scalar(k)` and `psi` is the Secp256k1
+/// endomorphism `psi: (x, y) |-> (beta * x, y)` equivalent to scalar multiplication by `s`.
+pub fn glv_mul(p: ProjectivePoint<Secp256K1>, k: Secp256K1Scalar) -> ProjectivePoint<Secp256K1> {
+    let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
+
+    let p_affine = p.to_affine();
+    let sp = AffinePoint::<Secp256K1> {
+        x: p_affine.x * GLV_BETA,
+        y: p_affine.y,
+        zero: p_affine.zero,
+    };
+
+    let first = if k1_neg { p.neg() } else { p };
+    let second = if k2_neg {
+        sp.to_projective().neg()
+    } else {
+        sp.to_projective()
+    };
+
+    msm_parallel(&[k1, k2], &[first, second], 5)
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::{Field, Sample};
+
+    use crate::curve::curve_types::{Curve, CurveScalar};
+    use crate::curve::glv::{decompose_secp256k1_scalar, glv_mul, GLV_S};
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_glv_decompose() -> Result<()> {
+        let k = Secp256K1Scalar::rand();
+        let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
+        let one = Secp256K1Scalar::ONE;
+        let m1 = if k1_neg { -one } else { one };
+        let m2 = if k2_neg { -one } else { one };
+
+        assert!(k1 * m1 + GLV_S * k2 * m2 == k);
+
+        Ok(())
+    }
+
+    #[test]
+    fn test_glv_mul() -> Result<()> {
+        for _ in 0..20 {
+            let k = Secp256K1Scalar::rand();
+
+            let p = CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE;
+
+            let kp = CurveScalar(k) * p;
+            let glv = glv_mul(p, k);
+
+            assert!(kp == glv);
+        }
+
+        Ok(())
+    }
+}

src/curve/mod.rs

@@ -0,0 +1,8 @@
+pub mod curve_adds;
+pub mod curve_msm;
+pub mod curve_multiplication;
+pub mod curve_summation;
+pub mod curve_types;
+pub mod ecdsa;
+pub mod glv;
+pub mod secp256k1;

src/curve/secp256k1.rs

@@ -0,0 +1,100 @@
+use plonky2::field::secp256k1_base::Secp256K1Base;
+use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+use plonky2::field::types::Field;
+use serde::{Deserialize, Serialize};
+
+use crate::curve::curve_types::{AffinePoint, Curve};
+
+#[derive(Debug, Copy, Clone, Deserialize, Eq, Hash, PartialEq, Serialize)]
+pub struct Secp256K1;
+
+impl Curve for Secp256K1 {
+    type BaseField = Secp256K1Base;
+    type ScalarField = Secp256K1Scalar;
+
+    const A: Secp256K1Base = Secp256K1Base::ZERO;
+    const B: Secp256K1Base = Secp256K1Base([7, 0, 0, 0]);
+    const GENERATOR_AFFINE: AffinePoint<Self> = AffinePoint {
+        x: SECP256K1_GENERATOR_X,
+        y: SECP256K1_GENERATOR_Y,
+        zero: false,
+    };
+}
+
+// 55066263022277343669578718895168534326250603453777594175500187360389116729240
+const SECP256K1_GENERATOR_X: Secp256K1Base = Secp256K1Base([
+    0x59F2815B16F81798,
+    0x029BFCDB2DCE28D9,
+    0x55A06295CE870B07,
+    0x79BE667EF9DCBBAC,
+]);
+
+/// 32670510020758816978083085130507043184471273380659243275938904335757337482424
+const SECP256K1_GENERATOR_Y: Secp256K1Base = Secp256K1Base([
+    0x9C47D08FFB10D4B8,
+    0xFD17B448A6855419,
+    0x5DA4FBFC0E1108A8,
+    0x483ADA7726A3C465,
+]);
+
+#[cfg(test)]
+mod tests {
+    use num::BigUint;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::{Field, PrimeField};
+
+    use crate::curve::curve_types::{AffinePoint, Curve, ProjectivePoint};
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_generator() {
+        let g = Secp256K1::GENERATOR_AFFINE;
+        assert!(g.is_valid());
+
+        let neg_g = AffinePoint::<Secp256K1> {
+            x: g.x,
+            y: -g.y,
+            zero: g.zero,
+        };
+        assert!(neg_g.is_valid());
+    }
+
+    #[test]
+    fn test_naive_multiplication() {
+        let g = Secp256K1::GENERATOR_PROJECTIVE;
+        let ten = Secp256K1Scalar::from_canonical_u64(10);
+        let product = mul_naive(ten, g);
+        let sum = g + g + g + g + g + g + g + g + g + g;
+        assert_eq!(product, sum);
+    }
+
+    #[test]
+    fn test_g1_multiplication() {
+        let lhs = Secp256K1Scalar::from_noncanonical_biguint(BigUint::from_slice(&[
+            1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888,
+        ]));
+        assert_eq!(
+            Secp256K1::convert(lhs) * Secp256K1::GENERATOR_PROJECTIVE,
+            mul_naive(lhs, Secp256K1::GENERATOR_PROJECTIVE)
+        );
+    }
+
+    /// A simple, somewhat inefficient implementation of multiplication which is used as a reference
+    /// for correctness.
+    fn mul_naive(
+        lhs: Secp256K1Scalar,
+        rhs: ProjectivePoint<Secp256K1>,
+    ) -> ProjectivePoint<Secp256K1> {
+        let mut g = rhs;
+        let mut sum = ProjectivePoint::ZERO;
+        for limb in lhs.to_canonical_biguint().to_u64_digits().iter() {
+            for j in 0..64 {
+                if (limb >> j & 1u64) != 0u64 {
+                    sum = sum + g;
+                }
+                g = g.double();
+            }
+        }
+        sum
+    }
+}

src/gadgets/biguint.rs

@@ -0,0 +1,508 @@
+use alloc::vec;
+use alloc::vec::Vec;
+use core::marker::PhantomData;
+
+use num::{BigUint, Integer, Zero};
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::{PrimeField, PrimeField64};
+use plonky2::hash::hash_types::RichField;
+use plonky2::iop::generator::{GeneratedValues, SimpleGenerator};
+use plonky2::iop::target::{BoolTarget, Target};
+use plonky2::iop::witness::{PartitionWitness, Witness};
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2_u32::gadgets::arithmetic_u32::{CircuitBuilderU32, U32Target};
+use plonky2_u32::gadgets::multiple_comparison::list_le_u32_circuit;
+use plonky2_u32::witness::{GeneratedValuesU32, WitnessU32};
+
+#[derive(Clone, Debug)]
+pub struct BigUintTarget {
+    pub limbs: Vec<U32Target>,
+}
+
+impl BigUintTarget {
+    pub fn num_limbs(&self) -> usize {
+        self.limbs.len()
+    }
+
+    pub fn get_limb(&self, i: usize) -> U32Target {
+        self.limbs[i]
+    }
+}
+
+pub trait CircuitBuilderBiguint<F: RichField + Extendable<D>, const D: usize> {
+    fn constant_biguint(&mut self, value: &BigUint) -> BigUintTarget;
+
+    fn zero_biguint(&mut self) -> BigUintTarget;
+
+    fn connect_biguint(&mut self, lhs: &BigUintTarget, rhs: &BigUintTarget);
+
+    fn pad_biguints(
+        &mut self,
+        a: &BigUintTarget,
+        b: &BigUintTarget,
+    ) -> (BigUintTarget, BigUintTarget);
+
+    fn cmp_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BoolTarget;
+
+    fn add_virtual_biguint_target(&mut self, num_limbs: usize) -> BigUintTarget;
+
+    /// Add two `BigUintTarget`s.
+    fn add_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget;
+
+    /// Subtract two `BigUintTarget`s. We assume that the first is larger than the second.
+    fn sub_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget;
+
+    fn mul_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget;
+
+    fn mul_biguint_by_bool(&mut self, a: &BigUintTarget, b: BoolTarget) -> BigUintTarget;
+
+    /// Returns x * y + z. This is no more efficient than mul-then-add; it's purely for convenience (only need to call one CircuitBuilder function).
+    fn mul_add_biguint(
+        &mut self,
+        x: &BigUintTarget,
+        y: &BigUintTarget,
+        z: &BigUintTarget,
+    ) -> BigUintTarget;
+
+    fn div_rem_biguint(
+        &mut self,
+        a: &BigUintTarget,
+        b: &BigUintTarget,
+    ) -> (BigUintTarget, BigUintTarget);
+
+    fn div_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget;
+
+    fn rem_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderBiguint<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn constant_biguint(&mut self, value: &BigUint) -> BigUintTarget {
+        let limb_values = value.to_u32_digits();
+        let limbs = limb_values.iter().map(|&l| self.constant_u32(l)).collect();
+
+        BigUintTarget { limbs }
+    }
+
+    fn zero_biguint(&mut self) -> BigUintTarget {
+        self.constant_biguint(&BigUint::zero())
+    }
+
+    fn connect_biguint(&mut self, lhs: &BigUintTarget, rhs: &BigUintTarget) {
+        let min_limbs = lhs.num_limbs().min(rhs.num_limbs());
+        for i in 0..min_limbs {
+            self.connect_u32(lhs.get_limb(i), rhs.get_limb(i));
+        }
+
+        for i in min_limbs..lhs.num_limbs() {
+            self.assert_zero_u32(lhs.get_limb(i));
+        }
+        for i in min_limbs..rhs.num_limbs() {
+            self.assert_zero_u32(rhs.get_limb(i));
+        }
+    }
+
+    fn pad_biguints(
+        &mut self,
+        a: &BigUintTarget,
+        b: &BigUintTarget,
+    ) -> (BigUintTarget, BigUintTarget) {
+        if a.num_limbs() > b.num_limbs() {
+            let mut padded_b = b.clone();
+            for _ in b.num_limbs()..a.num_limbs() {
+                padded_b.limbs.push(self.zero_u32());
+            }
+
+            (a.clone(), padded_b)
+        } else {
+            let mut padded_a = a.clone();
+            for _ in a.num_limbs()..b.num_limbs() {
+                padded_a.limbs.push(self.zero_u32());
+            }
+
+            (padded_a, b.clone())
+        }
+    }
+
+    fn cmp_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BoolTarget {
+        let (a, b) = self.pad_biguints(a, b);
+
+        list_le_u32_circuit(self, a.limbs, b.limbs)
+    }
+
+    fn add_virtual_biguint_target(&mut self, num_limbs: usize) -> BigUintTarget {
+        let limbs = self.add_virtual_u32_targets(num_limbs);
+
+        BigUintTarget { limbs }
+    }
+
+    fn add_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget {
+        let num_limbs = a.num_limbs().max(b.num_limbs());
+
+        let mut combined_limbs = vec![];
+        let mut carry = self.zero_u32();
+        for i in 0..num_limbs {
+            let a_limb = (i < a.num_limbs())
+                .then(|| a.limbs[i])
+                .unwrap_or_else(|| self.zero_u32());
+            let b_limb = (i < b.num_limbs())
+                .then(|| b.limbs[i])
+                .unwrap_or_else(|| self.zero_u32());
+
+            let (new_limb, new_carry) = self.add_many_u32(&[carry, a_limb, b_limb]);
+            carry = new_carry;
+            combined_limbs.push(new_limb);
+        }
+        combined_limbs.push(carry);
+
+        BigUintTarget {
+            limbs: combined_limbs,
+        }
+    }
+
+    fn sub_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget {
+        let (a, b) = self.pad_biguints(a, b);
+        let num_limbs = a.limbs.len();
+
+        let mut result_limbs = vec![];
+
+        let mut borrow = self.zero_u32();
+        for i in 0..num_limbs {
+            let (result, new_borrow) = self.sub_u32(a.limbs[i], b.limbs[i], borrow);
+            result_limbs.push(result);
+            borrow = new_borrow;
+        }
+        // Borrow should be zero here.
+
+        BigUintTarget {
+            limbs: result_limbs,
+        }
+    }
+
+    fn mul_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget {
+        let total_limbs = a.limbs.len() + b.limbs.len();
+
+        let mut to_add = vec![vec![]; total_limbs];
+        for i in 0..a.limbs.len() {
+            for j in 0..b.limbs.len() {
+                let (product, carry) = self.mul_u32(a.limbs[i], b.limbs[j]);
+                to_add[i + j].push(product);
+                to_add[i + j + 1].push(carry);
+            }
+        }
+
+        let mut combined_limbs = vec![];
+        let mut carry = self.zero_u32();
+        for summands in &mut to_add {
+            let (new_result, new_carry) = self.add_u32s_with_carry(summands, carry);
+            combined_limbs.push(new_result);
+            carry = new_carry;
+        }
+        combined_limbs.push(carry);
+
+        BigUintTarget {
+            limbs: combined_limbs,
+        }
+    }
+
+    fn mul_biguint_by_bool(&mut self, a: &BigUintTarget, b: BoolTarget) -> BigUintTarget {
+        let t = b.target;
+
+        BigUintTarget {
+            limbs: a
+                .limbs
+                .iter()
+                .map(|&l| U32Target(self.mul(l.0, t)))
+                .collect(),
+        }
+    }
+
+    fn mul_add_biguint(
+        &mut self,
+        x: &BigUintTarget,
+        y: &BigUintTarget,
+        z: &BigUintTarget,
+    ) -> BigUintTarget {
+        let prod = self.mul_biguint(x, y);
+        self.add_biguint(&prod, z)
+    }
+
+    fn div_rem_biguint(
+        &mut self,
+        a: &BigUintTarget,
+        b: &BigUintTarget,
+    ) -> (BigUintTarget, BigUintTarget) {
+        let a_len = a.limbs.len();
+        let b_len = b.limbs.len();
+        let div_num_limbs = if b_len > a_len + 1 {
+            0
+        } else {
+            a_len - b_len + 1
+        };
+        let div = self.add_virtual_biguint_target(div_num_limbs);
+        let rem = self.add_virtual_biguint_target(b_len);
+
+        self.add_simple_generator(BigUintDivRemGenerator::<F, D> {
+            a: a.clone(),
+            b: b.clone(),
+            div: div.clone(),
+            rem: rem.clone(),
+            _phantom: PhantomData,
+        });
+
+        let div_b = self.mul_biguint(&div, b);
+        let div_b_plus_rem = self.add_biguint(&div_b, &rem);
+        self.connect_biguint(a, &div_b_plus_rem);
+
+        let cmp_rem_b = self.cmp_biguint(&rem, b);
+        self.assert_one(cmp_rem_b.target);
+
+        (div, rem)
+    }
+
+    fn div_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget {
+        let (div, _rem) = self.div_rem_biguint(a, b);
+        div
+    }
+
+    fn rem_biguint(&mut self, a: &BigUintTarget, b: &BigUintTarget) -> BigUintTarget {
+        let (_div, rem) = self.div_rem_biguint(a, b);
+        rem
+    }
+}
+
+pub trait WitnessBigUint<F: PrimeField64>: Witness<F> {
+    fn get_biguint_target(&self, target: BigUintTarget) -> BigUint;
+    fn set_biguint_target(&mut self, target: &BigUintTarget, value: &BigUint);
+}
+
+impl<T: Witness<F>, F: PrimeField64> WitnessBigUint<F> for T {
+    fn get_biguint_target(&self, target: BigUintTarget) -> BigUint {
+        target
+            .limbs
+            .into_iter()
+            .rev()
+            .fold(BigUint::zero(), |acc, limb| {
+                (acc << 32) + self.get_target(limb.0).to_canonical_biguint()
+            })
+    }
+
+    fn set_biguint_target(&mut self, target: &BigUintTarget, value: &BigUint) {
+        let mut limbs = value.to_u32_digits();
+        assert!(target.num_limbs() >= limbs.len());
+        limbs.resize(target.num_limbs(), 0);
+        for i in 0..target.num_limbs() {
+            self.set_u32_target(target.limbs[i], limbs[i]);
+        }
+    }
+}
+
+pub trait GeneratedValuesBigUint<F: PrimeField> {
+    fn set_biguint_target(&mut self, target: &BigUintTarget, value: &BigUint);
+}
+
+impl<F: PrimeField> GeneratedValuesBigUint<F> for GeneratedValues<F> {
+    fn set_biguint_target(&mut self, target: &BigUintTarget, value: &BigUint) {
+        let mut limbs = value.to_u32_digits();
+        assert!(target.num_limbs() >= limbs.len());
+        limbs.resize(target.num_limbs(), 0);
+        for i in 0..target.num_limbs() {
+            self.set_u32_target(target.get_limb(i), limbs[i]);
+        }
+    }
+}
+
+#[derive(Debug)]
+struct BigUintDivRemGenerator<F: RichField + Extendable<D>, const D: usize> {
+    a: BigUintTarget,
+    b: BigUintTarget,
+    div: BigUintTarget,
+    rem: BigUintTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
+    for BigUintDivRemGenerator<F, D>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.a
+            .limbs
+            .iter()
+            .chain(&self.b.limbs)
+            .map(|&l| l.0)
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let a = witness.get_biguint_target(self.a.clone());
+        let b = witness.get_biguint_target(self.b.clone());
+        let (div, rem) = a.div_rem(&b);
+
+        out_buffer.set_biguint_target(&self.div, &div);
+        out_buffer.set_biguint_target(&self.rem, &rem);
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use num::{BigUint, FromPrimitive, Integer};
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+    use rand::rngs::OsRng;
+    use rand::Rng;
+
+    use crate::gadgets::biguint::{CircuitBuilderBiguint, WitnessBigUint};
+
+    #[test]
+    fn test_biguint_add() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let mut rng = OsRng;
+
+        let x_value = BigUint::from_u128(rng.gen()).unwrap();
+        let y_value = BigUint::from_u128(rng.gen()).unwrap();
+        let expected_z_value = &x_value + &y_value;
+
+        let config = CircuitConfig::standard_recursion_config();
+        let mut pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.add_virtual_biguint_target(x_value.to_u32_digits().len());
+        let y = builder.add_virtual_biguint_target(y_value.to_u32_digits().len());
+        let z = builder.add_biguint(&x, &y);
+        let expected_z = builder.add_virtual_biguint_target(expected_z_value.to_u32_digits().len());
+        builder.connect_biguint(&z, &expected_z);
+
+        pw.set_biguint_target(&x, &x_value);
+        pw.set_biguint_target(&y, &y_value);
+        pw.set_biguint_target(&expected_z, &expected_z_value);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_biguint_sub() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let mut rng = OsRng;
+
+        let mut x_value = BigUint::from_u128(rng.gen()).unwrap();
+        let mut y_value = BigUint::from_u128(rng.gen()).unwrap();
+        if y_value > x_value {
+            (x_value, y_value) = (y_value, x_value);
+        }
+        let expected_z_value = &x_value - &y_value;
+
+        let config = CircuitConfig::standard_recursion_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_biguint(&x_value);
+        let y = builder.constant_biguint(&y_value);
+        let z = builder.sub_biguint(&x, &y);
+        let expected_z = builder.constant_biguint(&expected_z_value);
+
+        builder.connect_biguint(&z, &expected_z);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_biguint_mul() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let mut rng = OsRng;
+
+        let x_value = BigUint::from_u128(rng.gen()).unwrap();
+        let y_value = BigUint::from_u128(rng.gen()).unwrap();
+        let expected_z_value = &x_value * &y_value;
+
+        let config = CircuitConfig::standard_recursion_config();
+        let mut pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.add_virtual_biguint_target(x_value.to_u32_digits().len());
+        let y = builder.add_virtual_biguint_target(y_value.to_u32_digits().len());
+        let z = builder.mul_biguint(&x, &y);
+        let expected_z = builder.add_virtual_biguint_target(expected_z_value.to_u32_digits().len());
+        builder.connect_biguint(&z, &expected_z);
+
+        pw.set_biguint_target(&x, &x_value);
+        pw.set_biguint_target(&y, &y_value);
+        pw.set_biguint_target(&expected_z, &expected_z_value);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_biguint_cmp() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let mut rng = OsRng;
+
+        let x_value = BigUint::from_u128(rng.gen()).unwrap();
+        let y_value = BigUint::from_u128(rng.gen()).unwrap();
+
+        let config = CircuitConfig::standard_recursion_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_biguint(&x_value);
+        let y = builder.constant_biguint(&y_value);
+        let cmp = builder.cmp_biguint(&x, &y);
+        let expected_cmp = builder.constant_bool(x_value <= y_value);
+
+        builder.connect(cmp.target, expected_cmp.target);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_biguint_div_rem() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let mut rng = OsRng;
+
+        let mut x_value = BigUint::from_u128(rng.gen()).unwrap();
+        let mut y_value = BigUint::from_u128(rng.gen()).unwrap();
+        if y_value > x_value {
+            (x_value, y_value) = (y_value, x_value);
+        }
+        let (expected_div_value, expected_rem_value) = x_value.div_rem(&y_value);
+
+        let config = CircuitConfig::standard_recursion_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_biguint(&x_value);
+        let y = builder.constant_biguint(&y_value);
+        let (div, rem) = builder.div_rem_biguint(&x, &y);
+
+        let expected_div = builder.constant_biguint(&expected_div_value);
+        let expected_rem = builder.constant_biguint(&expected_rem_value);
+
+        builder.connect_biguint(&div, &expected_div);
+        builder.connect_biguint(&rem, &expected_rem);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+}

src/gadgets/curve.rs

@@ -0,0 +1,486 @@
+use alloc::vec;
+use alloc::vec::Vec;
+
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::Sample;
+use plonky2::hash::hash_types::RichField;
+use plonky2::iop::target::BoolTarget;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+
+use crate::curve::curve_types::{AffinePoint, Curve, CurveScalar};
+use crate::gadgets::nonnative::{CircuitBuilderNonNative, NonNativeTarget};
+
+/// A Target representing an affine point on the curve `C`. We use incomplete arithmetic for efficiency,
+/// so we assume these points are not zero.
+#[derive(Clone, Debug)]
+pub struct AffinePointTarget<C: Curve> {
+    pub x: NonNativeTarget<C::BaseField>,
+    pub y: NonNativeTarget<C::BaseField>,
+}
+
+impl<C: Curve> AffinePointTarget<C> {
+    pub fn to_vec(&self) -> Vec<NonNativeTarget<C::BaseField>> {
+        vec![self.x.clone(), self.y.clone()]
+    }
+}
+
+pub trait CircuitBuilderCurve<F: RichField + Extendable<D>, const D: usize> {
+    fn constant_affine_point<C: Curve>(&mut self, point: AffinePoint<C>) -> AffinePointTarget<C>;
+
+    fn connect_affine_point<C: Curve>(
+        &mut self,
+        lhs: &AffinePointTarget<C>,
+        rhs: &AffinePointTarget<C>,
+    );
+
+    fn add_virtual_affine_point_target<C: Curve>(&mut self) -> AffinePointTarget<C>;
+
+    fn curve_assert_valid<C: Curve>(&mut self, p: &AffinePointTarget<C>);
+
+    fn curve_neg<C: Curve>(&mut self, p: &AffinePointTarget<C>) -> AffinePointTarget<C>;
+
+    fn curve_conditional_neg<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        b: BoolTarget,
+    ) -> AffinePointTarget<C>;
+
+    fn curve_double<C: Curve>(&mut self, p: &AffinePointTarget<C>) -> AffinePointTarget<C>;
+
+    fn curve_repeated_double<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: usize,
+    ) -> AffinePointTarget<C>;
+
+    /// Add two points, which are assumed to be non-equal.
+    fn curve_add<C: Curve>(
+        &mut self,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+    ) -> AffinePointTarget<C>;
+
+    fn curve_conditional_add<C: Curve>(
+        &mut self,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+        b: BoolTarget,
+    ) -> AffinePointTarget<C>;
+
+    fn curve_scalar_mul<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: &NonNativeTarget<C::ScalarField>,
+    ) -> AffinePointTarget<C>;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderCurve<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn constant_affine_point<C: Curve>(&mut self, point: AffinePoint<C>) -> AffinePointTarget<C> {
+        debug_assert!(!point.zero);
+        AffinePointTarget {
+            x: self.constant_nonnative(point.x),
+            y: self.constant_nonnative(point.y),
+        }
+    }
+
+    fn connect_affine_point<C: Curve>(
+        &mut self,
+        lhs: &AffinePointTarget<C>,
+        rhs: &AffinePointTarget<C>,
+    ) {
+        self.connect_nonnative(&lhs.x, &rhs.x);
+        self.connect_nonnative(&lhs.y, &rhs.y);
+    }
+
+    fn add_virtual_affine_point_target<C: Curve>(&mut self) -> AffinePointTarget<C> {
+        let x = self.add_virtual_nonnative_target();
+        let y = self.add_virtual_nonnative_target();
+
+        AffinePointTarget { x, y }
+    }
+
+    fn curve_assert_valid<C: Curve>(&mut self, p: &AffinePointTarget<C>) {
+        let a = self.constant_nonnative(C::A);
+        let b = self.constant_nonnative(C::B);
+
+        let y_squared = self.mul_nonnative(&p.y, &p.y);
+        let x_squared = self.mul_nonnative(&p.x, &p.x);
+        let x_cubed = self.mul_nonnative(&x_squared, &p.x);
+        let a_x = self.mul_nonnative(&a, &p.x);
+        let a_x_plus_b = self.add_nonnative(&a_x, &b);
+        let rhs = self.add_nonnative(&x_cubed, &a_x_plus_b);
+
+        self.connect_nonnative(&y_squared, &rhs);
+    }
+
+    fn curve_neg<C: Curve>(&mut self, p: &AffinePointTarget<C>) -> AffinePointTarget<C> {
+        let neg_y = self.neg_nonnative(&p.y);
+        AffinePointTarget {
+            x: p.x.clone(),
+            y: neg_y,
+        }
+    }
+
+    fn curve_conditional_neg<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        b: BoolTarget,
+    ) -> AffinePointTarget<C> {
+        AffinePointTarget {
+            x: p.x.clone(),
+            y: self.nonnative_conditional_neg(&p.y, b),
+        }
+    }
+
+    fn curve_double<C: Curve>(&mut self, p: &AffinePointTarget<C>) -> AffinePointTarget<C> {
+        let AffinePointTarget { x, y } = p;
+        let double_y = self.add_nonnative(y, y);
+        let inv_double_y = self.inv_nonnative(&double_y);
+        let x_squared = self.mul_nonnative(x, x);
+        let double_x_squared = self.add_nonnative(&x_squared, &x_squared);
+        let triple_x_squared = self.add_nonnative(&double_x_squared, &x_squared);
+
+        let a = self.constant_nonnative(C::A);
+        let triple_xx_a = self.add_nonnative(&triple_x_squared, &a);
+        let lambda = self.mul_nonnative(&triple_xx_a, &inv_double_y);
+        let lambda_squared = self.mul_nonnative(&lambda, &lambda);
+        let x_double = self.add_nonnative(x, x);
+
+        let x3 = self.sub_nonnative(&lambda_squared, &x_double);
+
+        let x_diff = self.sub_nonnative(x, &x3);
+        let lambda_x_diff = self.mul_nonnative(&lambda, &x_diff);
+
+        let y3 = self.sub_nonnative(&lambda_x_diff, y);
+
+        AffinePointTarget { x: x3, y: y3 }
+    }
+
+    fn curve_repeated_double<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: usize,
+    ) -> AffinePointTarget<C> {
+        let mut result = p.clone();
+
+        for _ in 0..n {
+            result = self.curve_double(&result);
+        }
+
+        result
+    }
+
+    fn curve_add<C: Curve>(
+        &mut self,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+    ) -> AffinePointTarget<C> {
+        let AffinePointTarget { x: x1, y: y1 } = p1;
+        let AffinePointTarget { x: x2, y: y2 } = p2;
+
+        let u = self.sub_nonnative(y2, y1);
+        let v = self.sub_nonnative(x2, x1);
+        let v_inv = self.inv_nonnative(&v);
+        let s = self.mul_nonnative(&u, &v_inv);
+        let s_squared = self.mul_nonnative(&s, &s);
+        let x_sum = self.add_nonnative(x2, x1);
+        let x3 = self.sub_nonnative(&s_squared, &x_sum);
+        let x_diff = self.sub_nonnative(x1, &x3);
+        let prod = self.mul_nonnative(&s, &x_diff);
+        let y3 = self.sub_nonnative(&prod, y1);
+
+        AffinePointTarget { x: x3, y: y3 }
+    }
+
+    fn curve_conditional_add<C: Curve>(
+        &mut self,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+        b: BoolTarget,
+    ) -> AffinePointTarget<C> {
+        let not_b = self.not(b);
+        let sum = self.curve_add(p1, p2);
+        let x_if_true = self.mul_nonnative_by_bool(&sum.x, b);
+        let y_if_true = self.mul_nonnative_by_bool(&sum.y, b);
+        let x_if_false = self.mul_nonnative_by_bool(&p1.x, not_b);
+        let y_if_false = self.mul_nonnative_by_bool(&p1.y, not_b);
+
+        let x = self.add_nonnative(&x_if_true, &x_if_false);
+        let y = self.add_nonnative(&y_if_true, &y_if_false);
+
+        AffinePointTarget { x, y }
+    }
+
+    fn curve_scalar_mul<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: &NonNativeTarget<C::ScalarField>,
+    ) -> AffinePointTarget<C> {
+        let bits = self.split_nonnative_to_bits(n);
+
+        let rando = (CurveScalar(C::ScalarField::rand()) * C::GENERATOR_PROJECTIVE).to_affine();
+        let randot = self.constant_affine_point(rando);
+        // Result starts at `rando`, which is later subtracted, because we don't support arithmetic with the zero point.
+        let mut result = self.add_virtual_affine_point_target();
+        self.connect_affine_point(&randot, &result);
+
+        let mut two_i_times_p = self.add_virtual_affine_point_target();
+        self.connect_affine_point(p, &two_i_times_p);
+
+        for &bit in bits.iter() {
+            let not_bit = self.not(bit);
+
+            let result_plus_2_i_p = self.curve_add(&result, &two_i_times_p);
+
+            let new_x_if_bit = self.mul_nonnative_by_bool(&result_plus_2_i_p.x, bit);
+            let new_x_if_not_bit = self.mul_nonnative_by_bool(&result.x, not_bit);
+            let new_y_if_bit = self.mul_nonnative_by_bool(&result_plus_2_i_p.y, bit);
+            let new_y_if_not_bit = self.mul_nonnative_by_bool(&result.y, not_bit);
+
+            let new_x = self.add_nonnative(&new_x_if_bit, &new_x_if_not_bit);
+            let new_y = self.add_nonnative(&new_y_if_bit, &new_y_if_not_bit);
+
+            result = AffinePointTarget { x: new_x, y: new_y };
+
+            two_i_times_p = self.curve_double(&two_i_times_p);
+        }
+
+        // Subtract off result's intial value of `rando`.
+        let neg_r = self.curve_neg(&randot);
+        result = self.curve_add(&result, &neg_r);
+
+        result
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use core::ops::Neg;
+
+    use anyhow::Result;
+    use plonky2::field::secp256k1_base::Secp256K1Base;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::{Field, Sample};
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use crate::curve::curve_types::{AffinePoint, Curve, CurveScalar};
+    use crate::curve::secp256k1::Secp256K1;
+    use crate::gadgets::curve::CircuitBuilderCurve;
+    use crate::gadgets::nonnative::CircuitBuilderNonNative;
+
+    #[test]
+    fn test_curve_point_is_valid() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let g_target = builder.constant_affine_point(g);
+        let neg_g_target = builder.curve_neg(&g_target);
+
+        builder.curve_assert_valid(&g_target);
+        builder.curve_assert_valid(&neg_g_target);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    #[should_panic]
+    fn test_curve_point_is_not_valid() {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let not_g = AffinePoint::<Secp256K1> {
+            x: g.x,
+            y: g.y + Secp256K1Base::ONE,
+            zero: g.zero,
+        };
+        let not_g_target = builder.constant_affine_point(not_g);
+
+        builder.curve_assert_valid(&not_g_target);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof).unwrap()
+    }
+
+    #[test]
+    fn test_curve_double() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let g_target = builder.constant_affine_point(g);
+        let neg_g_target = builder.curve_neg(&g_target);
+
+        let double_g = g.double();
+        let double_g_expected = builder.constant_affine_point(double_g);
+        builder.curve_assert_valid(&double_g_expected);
+
+        let double_neg_g = (-g).double();
+        let double_neg_g_expected = builder.constant_affine_point(double_neg_g);
+        builder.curve_assert_valid(&double_neg_g_expected);
+
+        let double_g_actual = builder.curve_double(&g_target);
+        let double_neg_g_actual = builder.curve_double(&neg_g_target);
+        builder.curve_assert_valid(&double_g_actual);
+        builder.curve_assert_valid(&double_neg_g_actual);
+
+        builder.connect_affine_point(&double_g_expected, &double_g_actual);
+        builder.connect_affine_point(&double_neg_g_expected, &double_neg_g_actual);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_curve_add() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let double_g = g.double();
+        let g_plus_2g = (g + double_g).to_affine();
+        let g_plus_2g_expected = builder.constant_affine_point(g_plus_2g);
+        builder.curve_assert_valid(&g_plus_2g_expected);
+
+        let g_target = builder.constant_affine_point(g);
+        let double_g_target = builder.curve_double(&g_target);
+        let g_plus_2g_actual = builder.curve_add(&g_target, &double_g_target);
+        builder.curve_assert_valid(&g_plus_2g_actual);
+
+        builder.connect_affine_point(&g_plus_2g_expected, &g_plus_2g_actual);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_curve_conditional_add() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let double_g = g.double();
+        let g_plus_2g = (g + double_g).to_affine();
+        let g_plus_2g_expected = builder.constant_affine_point(g_plus_2g);
+
+        let g_expected = builder.constant_affine_point(g);
+        let double_g_target = builder.curve_double(&g_expected);
+        let t = builder._true();
+        let f = builder._false();
+        let g_plus_2g_actual = builder.curve_conditional_add(&g_expected, &double_g_target, t);
+        let g_actual = builder.curve_conditional_add(&g_expected, &double_g_target, f);
+
+        builder.connect_affine_point(&g_plus_2g_expected, &g_plus_2g_actual);
+        builder.connect_affine_point(&g_expected, &g_actual);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    #[ignore]
+    fn test_curve_mul() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_PROJECTIVE.to_affine();
+        let five = Secp256K1Scalar::from_canonical_usize(5);
+        let neg_five = five.neg();
+        let neg_five_scalar = CurveScalar::<Secp256K1>(neg_five);
+        let neg_five_g = (neg_five_scalar * g.to_projective()).to_affine();
+        let neg_five_g_expected = builder.constant_affine_point(neg_five_g);
+        builder.curve_assert_valid(&neg_five_g_expected);
+
+        let g_target = builder.constant_affine_point(g);
+        let neg_five_target = builder.constant_nonnative(neg_five);
+        let neg_five_g_actual = builder.curve_scalar_mul(&g_target, &neg_five_target);
+        builder.curve_assert_valid(&neg_five_g_actual);
+
+        builder.connect_affine_point(&neg_five_g_expected, &neg_five_g_actual);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    #[ignore]
+    fn test_curve_random() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let rando =
+            (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE).to_affine();
+        let randot = builder.constant_affine_point(rando);
+
+        let two_target = builder.constant_nonnative(Secp256K1Scalar::TWO);
+        let randot_doubled = builder.curve_double(&randot);
+        let randot_times_two = builder.curve_scalar_mul(&randot, &two_target);
+        builder.connect_affine_point(&randot_doubled, &randot_times_two);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+}

src/gadgets/curve_fixed_base.rs

@@ -0,0 +1,118 @@
+use alloc::vec::Vec;
+
+use num::BigUint;
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::Field;
+use plonky2::hash::hash_types::RichField;
+use plonky2::hash::keccak::KeccakHash;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2::plonk::config::{GenericHashOut, Hasher};
+
+use crate::curve::curve_types::{AffinePoint, Curve, CurveScalar};
+use crate::gadgets::curve::{AffinePointTarget, CircuitBuilderCurve};
+use crate::gadgets::curve_windowed_mul::CircuitBuilderWindowedMul;
+use crate::gadgets::nonnative::NonNativeTarget;
+use crate::gadgets::split_nonnative::CircuitBuilderSplit;
+
+/// Compute windowed fixed-base scalar multiplication, using a 4-bit window.
+pub fn fixed_base_curve_mul_circuit<C: Curve, F: RichField + Extendable<D>, const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    base: AffinePoint<C>,
+    scalar: &NonNativeTarget<C::ScalarField>,
+) -> AffinePointTarget<C> {
+    // Holds `(16^i) * base` for `i=0..scalar.value.limbs.len() * 8`.
+    let scaled_base = (0..scalar.value.limbs.len() * 8).scan(base, |acc, _| {
+        let tmp = *acc;
+        for _ in 0..4 {
+            *acc = acc.double();
+        }
+        Some(tmp)
+    });
+
+    let limbs = builder.split_nonnative_to_4_bit_limbs(scalar);
+
+    let hash_0 = KeccakHash::<32>::hash_no_pad(&[F::ZERO]);
+    let hash_0_scalar = C::ScalarField::from_noncanonical_biguint(BigUint::from_bytes_le(
+        &GenericHashOut::<F>::to_bytes(&hash_0),
+    ));
+    let rando = (CurveScalar(hash_0_scalar) * C::GENERATOR_PROJECTIVE).to_affine();
+
+    let zero = builder.zero();
+    let mut result = builder.constant_affine_point(rando);
+    // `s * P = sum s_i * P_i` with `P_i = (16^i) * P` and `s = sum s_i * (16^i)`.
+    for (limb, point) in limbs.into_iter().zip(scaled_base) {
+        // `muls_point[t] = t * P_i` for `t=0..16`.
+        let mut muls_point = (0..16)
+            .scan(AffinePoint::ZERO, |acc, _| {
+                let tmp = *acc;
+                *acc = (point + *acc).to_affine();
+                Some(tmp)
+            })
+            // First element if zero, so we skip it since `constant_affine_point` takes non-zero input.
+            .skip(1)
+            .map(|p| builder.constant_affine_point(p))
+            .collect::<Vec<_>>();
+        // We add back a point in position 0. `limb == zero` is checked below, so this point can be arbitrary.
+        muls_point.insert(0, muls_point[0].clone());
+        let is_zero = builder.is_equal(limb, zero);
+        let should_add = builder.not(is_zero);
+        // `r = s_i * P_i`
+        let r = builder.random_access_curve_points(limb, muls_point);
+        result = builder.curve_conditional_add(&result, &r, should_add);
+    }
+
+    let to_add = builder.constant_affine_point(-rando);
+    builder.curve_add(&result, &to_add)
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::{PrimeField, Sample};
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use crate::curve::curve_types::{Curve, CurveScalar};
+    use crate::curve::secp256k1::Secp256K1;
+    use crate::gadgets::biguint::WitnessBigUint;
+    use crate::gadgets::curve::CircuitBuilderCurve;
+    use crate::gadgets::curve_fixed_base::fixed_base_curve_mul_circuit;
+    use crate::gadgets::nonnative::CircuitBuilderNonNative;
+
+    #[test]
+    #[ignore]
+    fn test_fixed_base() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let mut pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g = Secp256K1::GENERATOR_AFFINE;
+        let n = Secp256K1Scalar::rand();
+
+        let res = (CurveScalar(n) * g.to_projective()).to_affine();
+        let res_expected = builder.constant_affine_point(res);
+        builder.curve_assert_valid(&res_expected);
+
+        let n_target = builder.add_virtual_nonnative_target::<Secp256K1Scalar>();
+        pw.set_biguint_target(&n_target.value, &n.to_canonical_biguint());
+
+        let res_target = fixed_base_curve_mul_circuit(&mut builder, g, &n_target);
+        builder.curve_assert_valid(&res_target);
+
+        builder.connect_affine_point(&res_target, &res_expected);
+
+        dbg!(builder.num_gates());
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+}

src/gadgets/curve_msm.rs

@@ -0,0 +1,138 @@
+use alloc::vec;
+
+use num::BigUint;
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::Field;
+use plonky2::hash::hash_types::RichField;
+use plonky2::hash::keccak::KeccakHash;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2::plonk::config::{GenericHashOut, Hasher};
+
+use crate::curve::curve_types::{Curve, CurveScalar};
+use crate::gadgets::curve::{AffinePointTarget, CircuitBuilderCurve};
+use crate::gadgets::curve_windowed_mul::CircuitBuilderWindowedMul;
+use crate::gadgets::nonnative::NonNativeTarget;
+use crate::gadgets::split_nonnative::CircuitBuilderSplit;
+
+/// Computes `n*p + m*q` using windowed MSM, with a 2-bit window.
+/// See Algorithm 9.23 in Handbook of Elliptic and Hyperelliptic Curve Cryptography for a
+/// description.
+/// Note: Doesn't work if `p == q`.
+pub fn curve_msm_circuit<C: Curve, F: RichField + Extendable<D>, const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    p: &AffinePointTarget<C>,
+    q: &AffinePointTarget<C>,
+    n: &NonNativeTarget<C::ScalarField>,
+    m: &NonNativeTarget<C::ScalarField>,
+) -> AffinePointTarget<C> {
+    let limbs_n = builder.split_nonnative_to_2_bit_limbs(n);
+    let limbs_m = builder.split_nonnative_to_2_bit_limbs(m);
+    assert_eq!(limbs_n.len(), limbs_m.len());
+    let num_limbs = limbs_n.len();
+
+    let hash_0 = KeccakHash::<32>::hash_no_pad(&[F::ZERO]);
+    let hash_0_scalar = C::ScalarField::from_noncanonical_biguint(BigUint::from_bytes_le(
+        &GenericHashOut::<F>::to_bytes(&hash_0),
+    ));
+    let rando = (CurveScalar(hash_0_scalar) * C::GENERATOR_PROJECTIVE).to_affine();
+    let rando_t = builder.constant_affine_point(rando);
+    let neg_rando = builder.constant_affine_point(-rando);
+
+    // Precomputes `precomputation[i + 4*j] = i*p + j*q` for `i,j=0..4`.
+    let mut precomputation = vec![p.clone(); 16];
+    let mut cur_p = rando_t.clone();
+    let mut cur_q = rando_t.clone();
+    for i in 0..4 {
+        precomputation[i] = cur_p.clone();
+        precomputation[4 * i] = cur_q.clone();
+        cur_p = builder.curve_add(&cur_p, p);
+        cur_q = builder.curve_add(&cur_q, q);
+    }
+    for i in 1..4 {
+        precomputation[i] = builder.curve_add(&precomputation[i], &neg_rando);
+        precomputation[4 * i] = builder.curve_add(&precomputation[4 * i], &neg_rando);
+    }
+    for i in 1..4 {
+        for j in 1..4 {
+            precomputation[i + 4 * j] =
+                builder.curve_add(&precomputation[i], &precomputation[4 * j]);
+        }
+    }
+
+    let four = builder.constant(F::from_canonical_usize(4));
+
+    let zero = builder.zero();
+    let mut result = rando_t;
+    for (limb_n, limb_m) in limbs_n.into_iter().zip(limbs_m).rev() {
+        result = builder.curve_repeated_double(&result, 2);
+        let index = builder.mul_add(four, limb_m, limb_n);
+        let r = builder.random_access_curve_points(index, precomputation.clone());
+        let is_zero = builder.is_equal(index, zero);
+        let should_add = builder.not(is_zero);
+        result = builder.curve_conditional_add(&result, &r, should_add);
+    }
+    let starting_point_multiplied = (0..2 * num_limbs).fold(rando, |acc, _| acc.double());
+    let to_add = builder.constant_affine_point(-starting_point_multiplied);
+    result = builder.curve_add(&result, &to_add);
+
+    result
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::Sample;
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use crate::curve::curve_types::{Curve, CurveScalar};
+    use crate::curve::secp256k1::Secp256K1;
+    use crate::gadgets::curve::CircuitBuilderCurve;
+    use crate::gadgets::curve_msm::curve_msm_circuit;
+    use crate::gadgets::nonnative::CircuitBuilderNonNative;
+
+    #[test]
+    #[ignore]
+    fn test_curve_msm() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let p =
+            (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE).to_affine();
+        let q =
+            (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE).to_affine();
+        let n = Secp256K1Scalar::rand();
+        let m = Secp256K1Scalar::rand();
+
+        let res =
+            (CurveScalar(n) * p.to_projective() + CurveScalar(m) * q.to_projective()).to_affine();
+        let res_expected = builder.constant_affine_point(res);
+        builder.curve_assert_valid(&res_expected);
+
+        let p_target = builder.constant_affine_point(p);
+        let q_target = builder.constant_affine_point(q);
+        let n_target = builder.constant_nonnative(n);
+        let m_target = builder.constant_nonnative(m);
+
+        let res_target =
+            curve_msm_circuit(&mut builder, &p_target, &q_target, &n_target, &m_target);
+        builder.curve_assert_valid(&res_target);
+
+        builder.connect_affine_point(&res_target, &res_expected);
+
+        dbg!(builder.num_gates());
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+}

src/gadgets/curve_windowed_mul.rs

@@ -0,0 +1,254 @@
+use alloc::vec;
+use alloc::vec::Vec;
+use core::marker::PhantomData;
+
+use num::BigUint;
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::{Field, Sample};
+use plonky2::hash::hash_types::RichField;
+use plonky2::hash::keccak::KeccakHash;
+use plonky2::iop::target::{BoolTarget, Target};
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2::plonk::config::{GenericHashOut, Hasher};
+use plonky2_u32::gadgets::arithmetic_u32::{CircuitBuilderU32, U32Target};
+
+use crate::curve::curve_types::{Curve, CurveScalar};
+use crate::gadgets::biguint::BigUintTarget;
+use crate::gadgets::curve::{AffinePointTarget, CircuitBuilderCurve};
+use crate::gadgets::nonnative::{CircuitBuilderNonNative, NonNativeTarget};
+use crate::gadgets::split_nonnative::CircuitBuilderSplit;
+
+const WINDOW_SIZE: usize = 4;
+
+pub trait CircuitBuilderWindowedMul<F: RichField + Extendable<D>, const D: usize> {
+    fn precompute_window<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+    ) -> Vec<AffinePointTarget<C>>;
+
+    fn random_access_curve_points<C: Curve>(
+        &mut self,
+        access_index: Target,
+        v: Vec<AffinePointTarget<C>>,
+    ) -> AffinePointTarget<C>;
+
+    fn if_affine_point<C: Curve>(
+        &mut self,
+        b: BoolTarget,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+    ) -> AffinePointTarget<C>;
+
+    fn curve_scalar_mul_windowed<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: &NonNativeTarget<C::ScalarField>,
+    ) -> AffinePointTarget<C>;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderWindowedMul<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn precompute_window<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+    ) -> Vec<AffinePointTarget<C>> {
+        let g = (CurveScalar(C::ScalarField::rand()) * C::GENERATOR_PROJECTIVE).to_affine();
+        let neg = {
+            let mut neg = g;
+            neg.y = -neg.y;
+            self.constant_affine_point(neg)
+        };
+
+        let mut multiples = vec![self.constant_affine_point(g)];
+        for i in 1..1 << WINDOW_SIZE {
+            multiples.push(self.curve_add(p, &multiples[i - 1]));
+        }
+        for i in 1..1 << WINDOW_SIZE {
+            multiples[i] = self.curve_add(&neg, &multiples[i]);
+        }
+        multiples
+    }
+
+    fn random_access_curve_points<C: Curve>(
+        &mut self,
+        access_index: Target,
+        v: Vec<AffinePointTarget<C>>,
+    ) -> AffinePointTarget<C> {
+        let num_limbs = C::BaseField::BITS / 32;
+        let zero = self.zero_u32();
+        let x_limbs: Vec<Vec<_>> = (0..num_limbs)
+            .map(|i| {
+                v.iter()
+                    .map(|p| p.x.value.limbs.get(i).unwrap_or(&zero).0)
+                    .collect()
+            })
+            .collect();
+        let y_limbs: Vec<Vec<_>> = (0..num_limbs)
+            .map(|i| {
+                v.iter()
+                    .map(|p| p.y.value.limbs.get(i).unwrap_or(&zero).0)
+                    .collect()
+            })
+            .collect();
+
+        let selected_x_limbs: Vec<_> = x_limbs
+            .iter()
+            .map(|limbs| U32Target(self.random_access(access_index, limbs.clone())))
+            .collect();
+        let selected_y_limbs: Vec<_> = y_limbs
+            .iter()
+            .map(|limbs| U32Target(self.random_access(access_index, limbs.clone())))
+            .collect();
+
+        let x = NonNativeTarget {
+            value: BigUintTarget {
+                limbs: selected_x_limbs,
+            },
+            _phantom: PhantomData,
+        };
+        let y = NonNativeTarget {
+            value: BigUintTarget {
+                limbs: selected_y_limbs,
+            },
+            _phantom: PhantomData,
+        };
+        AffinePointTarget { x, y }
+    }
+
+    fn if_affine_point<C: Curve>(
+        &mut self,
+        b: BoolTarget,
+        p1: &AffinePointTarget<C>,
+        p2: &AffinePointTarget<C>,
+    ) -> AffinePointTarget<C> {
+        let new_x = self.if_nonnative(b, &p1.x, &p2.x);
+        let new_y = self.if_nonnative(b, &p1.y, &p2.y);
+        AffinePointTarget { x: new_x, y: new_y }
+    }
+
+    fn curve_scalar_mul_windowed<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        n: &NonNativeTarget<C::ScalarField>,
+    ) -> AffinePointTarget<C> {
+        let hash_0 = KeccakHash::<25>::hash_no_pad(&[F::ZERO]);
+        let hash_0_scalar = C::ScalarField::from_noncanonical_biguint(BigUint::from_bytes_le(
+            &GenericHashOut::<F>::to_bytes(&hash_0),
+        ));
+        let starting_point = CurveScalar(hash_0_scalar) * C::GENERATOR_PROJECTIVE;
+        let starting_point_multiplied = {
+            let mut cur = starting_point;
+            for _ in 0..C::ScalarField::BITS {
+                cur = cur.double();
+            }
+            cur
+        };
+
+        let mut result = self.constant_affine_point(starting_point.to_affine());
+
+        let precomputation = self.precompute_window(p);
+        let zero = self.zero();
+
+        let windows = self.split_nonnative_to_4_bit_limbs(n);
+        for i in (0..windows.len()).rev() {
+            result = self.curve_repeated_double(&result, WINDOW_SIZE);
+            let window = windows[i];
+
+            let to_add = self.random_access_curve_points(window, precomputation.clone());
+            let is_zero = self.is_equal(window, zero);
+            let should_add = self.not(is_zero);
+            result = self.curve_conditional_add(&result, &to_add, should_add);
+        }
+
+        let to_subtract = self.constant_affine_point(starting_point_multiplied.to_affine());
+        let to_add = self.curve_neg(&to_subtract);
+        result = self.curve_add(&result, &to_add);
+
+        result
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use core::ops::Neg;
+
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+    use rand::rngs::OsRng;
+    use rand::Rng;
+
+    use super::*;
+    use crate::curve::secp256k1::Secp256K1;
+
+    #[test]
+    fn test_random_access_curve_points() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let num_points = 16;
+        let points: Vec<_> = (0..num_points)
+            .map(|_| {
+                let g = (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE)
+                    .to_affine();
+                builder.constant_affine_point(g)
+            })
+            .collect();
+
+        let mut rng = OsRng;
+        let access_index = rng.gen::<usize>() % num_points;
+
+        let access_index_target = builder.constant(F::from_canonical_usize(access_index));
+        let selected = builder.random_access_curve_points(access_index_target, points.clone());
+        let expected = points[access_index].clone();
+        builder.connect_affine_point(&selected, &expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+
+    #[test]
+    #[ignore]
+    fn test_curve_windowed_mul() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let g =
+            (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE).to_affine();
+        let five = Secp256K1Scalar::from_canonical_usize(5);
+        let neg_five = five.neg();
+        let neg_five_scalar = CurveScalar::<Secp256K1>(neg_five);
+        let neg_five_g = (neg_five_scalar * g.to_projective()).to_affine();
+        let neg_five_g_expected = builder.constant_affine_point(neg_five_g);
+        builder.curve_assert_valid(&neg_five_g_expected);
+
+        let g_target = builder.constant_affine_point(g);
+        let neg_five_target = builder.constant_nonnative(neg_five);
+        let neg_five_g_actual = builder.curve_scalar_mul_windowed(&g_target, &neg_five_target);
+        builder.curve_assert_valid(&neg_five_g_actual);
+
+        builder.connect_affine_point(&neg_five_g_expected, &neg_five_g_actual);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+}

src/gadgets/ecdsa.rs

@@ -0,0 +1,111 @@
+use core::marker::PhantomData;
+
+use plonky2::field::extension::Extendable;
+use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+use plonky2::hash::hash_types::RichField;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+
+use crate::curve::curve_types::Curve;
+use crate::curve::secp256k1::Secp256K1;
+use crate::gadgets::curve::{AffinePointTarget, CircuitBuilderCurve};
+use crate::gadgets::curve_fixed_base::fixed_base_curve_mul_circuit;
+use crate::gadgets::glv::CircuitBuilderGlv;
+use crate::gadgets::nonnative::{CircuitBuilderNonNative, NonNativeTarget};
+
+#[derive(Clone, Debug)]
+pub struct ECDSASecretKeyTarget<C: Curve>(pub NonNativeTarget<C::ScalarField>);
+
+#[derive(Clone, Debug)]
+pub struct ECDSAPublicKeyTarget<C: Curve>(pub AffinePointTarget<C>);
+
+#[derive(Clone, Debug)]
+pub struct ECDSASignatureTarget<C: Curve> {
+    pub r: NonNativeTarget<C::ScalarField>,
+    pub s: NonNativeTarget<C::ScalarField>,
+}
+
+pub fn verify_message_circuit<F: RichField + Extendable<D>, const D: usize>(
+    builder: &mut CircuitBuilder<F, D>,
+    msg: NonNativeTarget<Secp256K1Scalar>,
+    sig: ECDSASignatureTarget<Secp256K1>,
+    pk: ECDSAPublicKeyTarget<Secp256K1>,
+) {
+    let ECDSASignatureTarget { r, s } = sig;
+
+    builder.curve_assert_valid(&pk.0);
+
+    let c = builder.inv_nonnative(&s);
+    let u1 = builder.mul_nonnative(&msg, &c);
+    let u2 = builder.mul_nonnative(&r, &c);
+
+    let point1 = fixed_base_curve_mul_circuit(builder, Secp256K1::GENERATOR_AFFINE, &u1);
+    let point2 = builder.glv_mul(&pk.0, &u2);
+    let point = builder.curve_add(&point1, &point2);
+
+    let x = NonNativeTarget::<Secp256K1Scalar> {
+        value: point.x.value,
+        _phantom: PhantomData,
+    };
+    builder.connect_nonnative(&r, &x);
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::types::Sample;
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use super::*;
+    use crate::curve::curve_types::CurveScalar;
+    use crate::curve::ecdsa::{sign_message, ECDSAPublicKey, ECDSASecretKey, ECDSASignature};
+
+    fn test_ecdsa_circuit_with_config(config: CircuitConfig) -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        type Curve = Secp256K1;
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let msg = Secp256K1Scalar::rand();
+        let msg_target = builder.constant_nonnative(msg);
+
+        let sk = ECDSASecretKey::<Curve>(Secp256K1Scalar::rand());
+        let pk = ECDSAPublicKey((CurveScalar(sk.0) * Curve::GENERATOR_PROJECTIVE).to_affine());
+
+        let pk_target = ECDSAPublicKeyTarget(builder.constant_affine_point(pk.0));
+
+        let sig = sign_message(msg, sk);
+
+        let ECDSASignature { r, s } = sig;
+        let r_target = builder.constant_nonnative(r);
+        let s_target = builder.constant_nonnative(s);
+        let sig_target = ECDSASignatureTarget {
+            r: r_target,
+            s: s_target,
+        };
+
+        verify_message_circuit(&mut builder, msg_target, sig_target, pk_target);
+
+        dbg!(builder.num_gates());
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    #[ignore]
+    fn test_ecdsa_circuit_narrow() -> Result<()> {
+        test_ecdsa_circuit_with_config(CircuitConfig::standard_ecc_config())
+    }
+
+    #[test]
+    #[ignore]
+    fn test_ecdsa_circuit_wide() -> Result<()> {
+        test_ecdsa_circuit_with_config(CircuitConfig::wide_ecc_config())
+    }
+}

src/gadgets/glv.rs

@@ -0,0 +1,180 @@
+use alloc::vec::Vec;
+use core::marker::PhantomData;
+
+use plonky2::field::extension::Extendable;
+use plonky2::field::secp256k1_base::Secp256K1Base;
+use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+use plonky2::field::types::{Field, PrimeField};
+use plonky2::hash::hash_types::RichField;
+use plonky2::iop::generator::{GeneratedValues, SimpleGenerator};
+use plonky2::iop::target::{BoolTarget, Target};
+use plonky2::iop::witness::{PartitionWitness, WitnessWrite};
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+
+use crate::curve::glv::{decompose_secp256k1_scalar, GLV_BETA, GLV_S};
+use crate::curve::secp256k1::Secp256K1;
+use crate::gadgets::biguint::{GeneratedValuesBigUint, WitnessBigUint};
+use crate::gadgets::curve::{AffinePointTarget, CircuitBuilderCurve};
+use crate::gadgets::curve_msm::curve_msm_circuit;
+use crate::gadgets::nonnative::{CircuitBuilderNonNative, NonNativeTarget};
+
+pub trait CircuitBuilderGlv<F: RichField + Extendable<D>, const D: usize> {
+    fn secp256k1_glv_beta(&mut self) -> NonNativeTarget<Secp256K1Base>;
+
+    fn decompose_secp256k1_scalar(
+        &mut self,
+        k: &NonNativeTarget<Secp256K1Scalar>,
+    ) -> (
+        NonNativeTarget<Secp256K1Scalar>,
+        NonNativeTarget<Secp256K1Scalar>,
+        BoolTarget,
+        BoolTarget,
+    );
+
+    fn glv_mul(
+        &mut self,
+        p: &AffinePointTarget<Secp256K1>,
+        k: &NonNativeTarget<Secp256K1Scalar>,
+    ) -> AffinePointTarget<Secp256K1>;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderGlv<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn secp256k1_glv_beta(&mut self) -> NonNativeTarget<Secp256K1Base> {
+        self.constant_nonnative(GLV_BETA)
+    }
+
+    fn decompose_secp256k1_scalar(
+        &mut self,
+        k: &NonNativeTarget<Secp256K1Scalar>,
+    ) -> (
+        NonNativeTarget<Secp256K1Scalar>,
+        NonNativeTarget<Secp256K1Scalar>,
+        BoolTarget,
+        BoolTarget,
+    ) {
+        let k1 = self.add_virtual_nonnative_target_sized::<Secp256K1Scalar>(4);
+        let k2 = self.add_virtual_nonnative_target_sized::<Secp256K1Scalar>(4);
+        let k1_neg = self.add_virtual_bool_target_unsafe();
+        let k2_neg = self.add_virtual_bool_target_unsafe();
+
+        self.add_simple_generator(GLVDecompositionGenerator::<F, D> {
+            k: k.clone(),
+            k1: k1.clone(),
+            k2: k2.clone(),
+            k1_neg,
+            k2_neg,
+            _phantom: PhantomData,
+        });
+
+        // Check that `k1_raw + GLV_S * k2_raw == k`.
+        let k1_raw = self.nonnative_conditional_neg(&k1, k1_neg);
+        let k2_raw = self.nonnative_conditional_neg(&k2, k2_neg);
+        let s = self.constant_nonnative(GLV_S);
+        let mut should_be_k = self.mul_nonnative(&s, &k2_raw);
+        should_be_k = self.add_nonnative(&should_be_k, &k1_raw);
+        self.connect_nonnative(&should_be_k, k);
+
+        (k1, k2, k1_neg, k2_neg)
+    }
+
+    fn glv_mul(
+        &mut self,
+        p: &AffinePointTarget<Secp256K1>,
+        k: &NonNativeTarget<Secp256K1Scalar>,
+    ) -> AffinePointTarget<Secp256K1> {
+        let (k1, k2, k1_neg, k2_neg) = self.decompose_secp256k1_scalar(k);
+
+        let beta = self.secp256k1_glv_beta();
+        let beta_px = self.mul_nonnative(&beta, &p.x);
+        let sp = AffinePointTarget::<Secp256K1> {
+            x: beta_px,
+            y: p.y.clone(),
+        };
+
+        let p_neg = self.curve_conditional_neg(p, k1_neg);
+        let sp_neg = self.curve_conditional_neg(&sp, k2_neg);
+        curve_msm_circuit(self, &p_neg, &sp_neg, &k1, &k2)
+    }
+}
+
+#[derive(Debug)]
+struct GLVDecompositionGenerator<F: RichField + Extendable<D>, const D: usize> {
+    k: NonNativeTarget<Secp256K1Scalar>,
+    k1: NonNativeTarget<Secp256K1Scalar>,
+    k2: NonNativeTarget<Secp256K1Scalar>,
+    k1_neg: BoolTarget,
+    k2_neg: BoolTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
+    for GLVDecompositionGenerator<F, D>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.k.value.limbs.iter().map(|l| l.0).collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let k = Secp256K1Scalar::from_noncanonical_biguint(
+            witness.get_biguint_target(self.k.value.clone()),
+        );
+
+        let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
+
+        out_buffer.set_biguint_target(&self.k1.value, &k1.to_canonical_biguint());
+        out_buffer.set_biguint_target(&self.k2.value, &k2.to_canonical_biguint());
+        out_buffer.set_bool_target(self.k1_neg, k1_neg);
+        out_buffer.set_bool_target(self.k2_neg, k2_neg);
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::Sample;
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use crate::curve::curve_types::{Curve, CurveScalar};
+    use crate::curve::glv::glv_mul;
+    use crate::curve::secp256k1::Secp256K1;
+    use crate::gadgets::curve::CircuitBuilderCurve;
+    use crate::gadgets::glv::CircuitBuilderGlv;
+    use crate::gadgets::nonnative::CircuitBuilderNonNative;
+
+    #[test]
+    #[ignore]
+    fn test_glv_gadget() -> Result<()> {
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let rando =
+            (CurveScalar(Secp256K1Scalar::rand()) * Secp256K1::GENERATOR_PROJECTIVE).to_affine();
+        let randot = builder.constant_affine_point(rando);
+
+        let scalar = Secp256K1Scalar::rand();
+        let scalar_target = builder.constant_nonnative(scalar);
+
+        let rando_glv_scalar = glv_mul(rando.to_projective(), scalar);
+        let expected = builder.constant_affine_point(rando_glv_scalar.to_affine());
+        let actual = builder.glv_mul(&randot, &scalar_target);
+        builder.connect_affine_point(&expected, &actual);
+
+        dbg!(builder.num_gates());
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+
+        data.verify(proof)
+    }
+}

src/gadgets/mod.rs

@@ -0,0 +1,9 @@
+pub mod biguint;
+pub mod curve;
+pub mod curve_fixed_base;
+pub mod curve_msm;
+pub mod curve_windowed_mul;
+pub mod ecdsa;
+pub mod glv;
+pub mod nonnative;
+pub mod split_nonnative;

src/gadgets/nonnative.rs

@@ -0,0 +1,826 @@
+use alloc::vec;
+use alloc::vec::Vec;
+use core::marker::PhantomData;
+
+use num::{BigUint, Integer, One, Zero};
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::{Field, PrimeField};
+use plonky2::hash::hash_types::RichField;
+use plonky2::iop::generator::{GeneratedValues, SimpleGenerator};
+use plonky2::iop::target::{BoolTarget, Target};
+use plonky2::iop::witness::{PartitionWitness, WitnessWrite};
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2::util::ceil_div_usize;
+use plonky2_u32::gadgets::arithmetic_u32::{CircuitBuilderU32, U32Target};
+use plonky2_u32::gadgets::range_check::range_check_u32_circuit;
+use plonky2_u32::witness::GeneratedValuesU32;
+
+use crate::gadgets::biguint::{
+    BigUintTarget, CircuitBuilderBiguint, GeneratedValuesBigUint, WitnessBigUint,
+};
+
+#[derive(Clone, Debug)]
+pub struct NonNativeTarget<FF: Field> {
+    pub(crate) value: BigUintTarget,
+    pub(crate) _phantom: PhantomData<FF>,
+}
+
+pub trait CircuitBuilderNonNative<F: RichField + Extendable<D>, const D: usize> {
+    fn num_nonnative_limbs<FF: Field>() -> usize {
+        ceil_div_usize(FF::BITS, 32)
+    }
+
+    fn biguint_to_nonnative<FF: Field>(&mut self, x: &BigUintTarget) -> NonNativeTarget<FF>;
+
+    fn nonnative_to_canonical_biguint<FF: Field>(
+        &mut self,
+        x: &NonNativeTarget<FF>,
+    ) -> BigUintTarget;
+
+    fn constant_nonnative<FF: PrimeField>(&mut self, x: FF) -> NonNativeTarget<FF>;
+
+    fn zero_nonnative<FF: PrimeField>(&mut self) -> NonNativeTarget<FF>;
+
+    // Assert that two NonNativeTarget's, both assumed to be in reduced form, are equal.
+    fn connect_nonnative<FF: Field>(
+        &mut self,
+        lhs: &NonNativeTarget<FF>,
+        rhs: &NonNativeTarget<FF>,
+    );
+
+    fn add_virtual_nonnative_target<FF: Field>(&mut self) -> NonNativeTarget<FF>;
+
+    fn add_virtual_nonnative_target_sized<FF: Field>(
+        &mut self,
+        num_limbs: usize,
+    ) -> NonNativeTarget<FF>;
+
+    fn add_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF>;
+
+    fn mul_nonnative_by_bool<FF: Field>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: BoolTarget,
+    ) -> NonNativeTarget<FF>;
+
+    fn if_nonnative<FF: PrimeField>(
+        &mut self,
+        b: BoolTarget,
+        x: &NonNativeTarget<FF>,
+        y: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF>;
+
+    fn add_many_nonnative<FF: PrimeField>(
+        &mut self,
+        to_add: &[NonNativeTarget<FF>],
+    ) -> NonNativeTarget<FF>;
+
+    // Subtract two `NonNativeTarget`s.
+    fn sub_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF>;
+
+    fn mul_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF>;
+
+    fn mul_many_nonnative<FF: PrimeField>(
+        &mut self,
+        to_mul: &[NonNativeTarget<FF>],
+    ) -> NonNativeTarget<FF>;
+
+    fn neg_nonnative<FF: PrimeField>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF>;
+
+    fn inv_nonnative<FF: PrimeField>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF>;
+
+    /// Returns `x % |FF|` as a `NonNativeTarget`.
+    fn reduce<FF: Field>(&mut self, x: &BigUintTarget) -> NonNativeTarget<FF>;
+
+    fn reduce_nonnative<FF: Field>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF>;
+
+    fn bool_to_nonnative<FF: Field>(&mut self, b: &BoolTarget) -> NonNativeTarget<FF>;
+
+    // Split a nonnative field element to bits.
+    fn split_nonnative_to_bits<FF: Field>(&mut self, x: &NonNativeTarget<FF>) -> Vec<BoolTarget>;
+
+    fn nonnative_conditional_neg<FF: PrimeField>(
+        &mut self,
+        x: &NonNativeTarget<FF>,
+        b: BoolTarget,
+    ) -> NonNativeTarget<FF>;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderNonNative<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn num_nonnative_limbs<FF: Field>() -> usize {
+        ceil_div_usize(FF::BITS, 32)
+    }
+
+    fn biguint_to_nonnative<FF: Field>(&mut self, x: &BigUintTarget) -> NonNativeTarget<FF> {
+        NonNativeTarget {
+            value: x.clone(),
+            _phantom: PhantomData,
+        }
+    }
+
+    fn nonnative_to_canonical_biguint<FF: Field>(
+        &mut self,
+        x: &NonNativeTarget<FF>,
+    ) -> BigUintTarget {
+        x.value.clone()
+    }
+
+    fn constant_nonnative<FF: PrimeField>(&mut self, x: FF) -> NonNativeTarget<FF> {
+        let x_biguint = self.constant_biguint(&x.to_canonical_biguint());
+        self.biguint_to_nonnative(&x_biguint)
+    }
+
+    fn zero_nonnative<FF: PrimeField>(&mut self) -> NonNativeTarget<FF> {
+        self.constant_nonnative(FF::ZERO)
+    }
+
+    // Assert that two NonNativeTarget's, both assumed to be in reduced form, are equal.
+    fn connect_nonnative<FF: Field>(
+        &mut self,
+        lhs: &NonNativeTarget<FF>,
+        rhs: &NonNativeTarget<FF>,
+    ) {
+        self.connect_biguint(&lhs.value, &rhs.value);
+    }
+
+    fn add_virtual_nonnative_target<FF: Field>(&mut self) -> NonNativeTarget<FF> {
+        let num_limbs = Self::num_nonnative_limbs::<FF>();
+        let value = self.add_virtual_biguint_target(num_limbs);
+
+        NonNativeTarget {
+            value,
+            _phantom: PhantomData,
+        }
+    }
+
+    fn add_virtual_nonnative_target_sized<FF: Field>(
+        &mut self,
+        num_limbs: usize,
+    ) -> NonNativeTarget<FF> {
+        let value = self.add_virtual_biguint_target(num_limbs);
+
+        NonNativeTarget {
+            value,
+            _phantom: PhantomData,
+        }
+    }
+
+    fn add_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF> {
+        let sum = self.add_virtual_nonnative_target::<FF>();
+        let overflow = self.add_virtual_bool_target_unsafe();
+
+        self.add_simple_generator(NonNativeAdditionGenerator::<F, D, FF> {
+            a: a.clone(),
+            b: b.clone(),
+            sum: sum.clone(),
+            overflow,
+            _phantom: PhantomData,
+        });
+
+        let sum_expected = self.add_biguint(&a.value, &b.value);
+
+        let modulus = self.constant_biguint(&FF::order());
+        let mod_times_overflow = self.mul_biguint_by_bool(&modulus, overflow);
+        let sum_actual = self.add_biguint(&sum.value, &mod_times_overflow);
+        self.connect_biguint(&sum_expected, &sum_actual);
+
+        // Range-check result.
+        // TODO: can potentially leave unreduced until necessary (e.g. when connecting values).
+        let cmp = self.cmp_biguint(&sum.value, &modulus);
+        let one = self.one();
+        self.connect(cmp.target, one);
+
+        sum
+    }
+
+    fn mul_nonnative_by_bool<FF: Field>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: BoolTarget,
+    ) -> NonNativeTarget<FF> {
+        NonNativeTarget {
+            value: self.mul_biguint_by_bool(&a.value, b),
+            _phantom: PhantomData,
+        }
+    }
+
+    fn if_nonnative<FF: PrimeField>(
+        &mut self,
+        b: BoolTarget,
+        x: &NonNativeTarget<FF>,
+        y: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF> {
+        let not_b = self.not(b);
+        let maybe_x = self.mul_nonnative_by_bool(x, b);
+        let maybe_y = self.mul_nonnative_by_bool(y, not_b);
+        self.add_nonnative(&maybe_x, &maybe_y)
+    }
+
+    fn add_many_nonnative<FF: PrimeField>(
+        &mut self,
+        to_add: &[NonNativeTarget<FF>],
+    ) -> NonNativeTarget<FF> {
+        if to_add.len() == 1 {
+            return to_add[0].clone();
+        }
+
+        let sum = self.add_virtual_nonnative_target::<FF>();
+        let overflow = self.add_virtual_u32_target();
+        let summands = to_add.to_vec();
+
+        self.add_simple_generator(NonNativeMultipleAddsGenerator::<F, D, FF> {
+            summands: summands.clone(),
+            sum: sum.clone(),
+            overflow,
+            _phantom: PhantomData,
+        });
+
+        range_check_u32_circuit(self, sum.value.limbs.clone());
+        range_check_u32_circuit(self, vec![overflow]);
+
+        let sum_expected = summands
+            .iter()
+            .fold(self.zero_biguint(), |a, b| self.add_biguint(&a, &b.value));
+
+        let modulus = self.constant_biguint(&FF::order());
+        let overflow_biguint = BigUintTarget {
+            limbs: vec![overflow],
+        };
+        let mod_times_overflow = self.mul_biguint(&modulus, &overflow_biguint);
+        let sum_actual = self.add_biguint(&sum.value, &mod_times_overflow);
+        self.connect_biguint(&sum_expected, &sum_actual);
+
+        // Range-check result.
+        // TODO: can potentially leave unreduced until necessary (e.g. when connecting values).
+        let cmp = self.cmp_biguint(&sum.value, &modulus);
+        let one = self.one();
+        self.connect(cmp.target, one);
+
+        sum
+    }
+
+    // Subtract two `NonNativeTarget`s.
+    fn sub_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF> {
+        let diff = self.add_virtual_nonnative_target::<FF>();
+        let overflow = self.add_virtual_bool_target_unsafe();
+
+        self.add_simple_generator(NonNativeSubtractionGenerator::<F, D, FF> {
+            a: a.clone(),
+            b: b.clone(),
+            diff: diff.clone(),
+            overflow,
+            _phantom: PhantomData,
+        });
+
+        range_check_u32_circuit(self, diff.value.limbs.clone());
+        self.assert_bool(overflow);
+
+        let diff_plus_b = self.add_biguint(&diff.value, &b.value);
+        let modulus = self.constant_biguint(&FF::order());
+        let mod_times_overflow = self.mul_biguint_by_bool(&modulus, overflow);
+        let diff_plus_b_reduced = self.sub_biguint(&diff_plus_b, &mod_times_overflow);
+        self.connect_biguint(&a.value, &diff_plus_b_reduced);
+
+        diff
+    }
+
+    fn mul_nonnative<FF: PrimeField>(
+        &mut self,
+        a: &NonNativeTarget<FF>,
+        b: &NonNativeTarget<FF>,
+    ) -> NonNativeTarget<FF> {
+        let prod = self.add_virtual_nonnative_target::<FF>();
+        let modulus = self.constant_biguint(&FF::order());
+        let overflow = self.add_virtual_biguint_target(
+            a.value.num_limbs() + b.value.num_limbs() - modulus.num_limbs(),
+        );
+
+        self.add_simple_generator(NonNativeMultiplicationGenerator::<F, D, FF> {
+            a: a.clone(),
+            b: b.clone(),
+            prod: prod.clone(),
+            overflow: overflow.clone(),
+            _phantom: PhantomData,
+        });
+
+        range_check_u32_circuit(self, prod.value.limbs.clone());
+        range_check_u32_circuit(self, overflow.limbs.clone());
+
+        let prod_expected = self.mul_biguint(&a.value, &b.value);
+
+        let mod_times_overflow = self.mul_biguint(&modulus, &overflow);
+        let prod_actual = self.add_biguint(&prod.value, &mod_times_overflow);
+        self.connect_biguint(&prod_expected, &prod_actual);
+
+        prod
+    }
+
+    fn mul_many_nonnative<FF: PrimeField>(
+        &mut self,
+        to_mul: &[NonNativeTarget<FF>],
+    ) -> NonNativeTarget<FF> {
+        if to_mul.len() == 1 {
+            return to_mul[0].clone();
+        }
+
+        let mut accumulator = self.mul_nonnative(&to_mul[0], &to_mul[1]);
+        for t in to_mul.iter().skip(2) {
+            accumulator = self.mul_nonnative(&accumulator, t);
+        }
+        accumulator
+    }
+
+    fn neg_nonnative<FF: PrimeField>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF> {
+        let zero_target = self.constant_biguint(&BigUint::zero());
+        let zero_ff = self.biguint_to_nonnative(&zero_target);
+
+        self.sub_nonnative(&zero_ff, x)
+    }
+
+    fn inv_nonnative<FF: PrimeField>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF> {
+        let num_limbs = x.value.num_limbs();
+        let inv_biguint = self.add_virtual_biguint_target(num_limbs);
+        let div = self.add_virtual_biguint_target(num_limbs);
+
+        self.add_simple_generator(NonNativeInverseGenerator::<F, D, FF> {
+            x: x.clone(),
+            inv: inv_biguint.clone(),
+            div: div.clone(),
+            _phantom: PhantomData,
+        });
+
+        let product = self.mul_biguint(&x.value, &inv_biguint);
+
+        let modulus = self.constant_biguint(&FF::order());
+        let mod_times_div = self.mul_biguint(&modulus, &div);
+        let one = self.constant_biguint(&BigUint::one());
+        let expected_product = self.add_biguint(&mod_times_div, &one);
+        self.connect_biguint(&product, &expected_product);
+
+        NonNativeTarget::<FF> {
+            value: inv_biguint,
+            _phantom: PhantomData,
+        }
+    }
+
+    /// Returns `x % |FF|` as a `NonNativeTarget`.
+    fn reduce<FF: Field>(&mut self, x: &BigUintTarget) -> NonNativeTarget<FF> {
+        let modulus = FF::order();
+        let order_target = self.constant_biguint(&modulus);
+        let value = self.rem_biguint(x, &order_target);
+
+        NonNativeTarget {
+            value,
+            _phantom: PhantomData,
+        }
+    }
+
+    fn reduce_nonnative<FF: Field>(&mut self, x: &NonNativeTarget<FF>) -> NonNativeTarget<FF> {
+        let x_biguint = self.nonnative_to_canonical_biguint(x);
+        self.reduce(&x_biguint)
+    }
+
+    fn bool_to_nonnative<FF: Field>(&mut self, b: &BoolTarget) -> NonNativeTarget<FF> {
+        let limbs = vec![U32Target(b.target)];
+        let value = BigUintTarget { limbs };
+
+        NonNativeTarget {
+            value,
+            _phantom: PhantomData,
+        }
+    }
+
+    // Split a nonnative field element to bits.
+    fn split_nonnative_to_bits<FF: Field>(&mut self, x: &NonNativeTarget<FF>) -> Vec<BoolTarget> {
+        let num_limbs = x.value.num_limbs();
+        let mut result = Vec::with_capacity(num_limbs * 32);
+
+        for i in 0..num_limbs {
+            let limb = x.value.get_limb(i);
+            let bit_targets = self.split_le_base::<2>(limb.0, 32);
+            let mut bits: Vec<_> = bit_targets
+                .iter()
+                .map(|&t| BoolTarget::new_unsafe(t))
+                .collect();
+
+            result.append(&mut bits);
+        }
+
+        result
+    }
+
+    fn nonnative_conditional_neg<FF: PrimeField>(
+        &mut self,
+        x: &NonNativeTarget<FF>,
+        b: BoolTarget,
+    ) -> NonNativeTarget<FF> {
+        let not_b = self.not(b);
+        let neg = self.neg_nonnative(x);
+        let x_if_true = self.mul_nonnative_by_bool(&neg, b);
+        let x_if_false = self.mul_nonnative_by_bool(x, not_b);
+
+        self.add_nonnative(&x_if_true, &x_if_false)
+    }
+}
+
+#[derive(Debug)]
+struct NonNativeAdditionGenerator<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> {
+    a: NonNativeTarget<FF>,
+    b: NonNativeTarget<FF>,
+    sum: NonNativeTarget<FF>,
+    overflow: BoolTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> SimpleGenerator<F>
+    for NonNativeAdditionGenerator<F, D, FF>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.a
+            .value
+            .limbs
+            .iter()
+            .cloned()
+            .chain(self.b.value.limbs.clone())
+            .map(|l| l.0)
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let a = FF::from_noncanonical_biguint(witness.get_biguint_target(self.a.value.clone()));
+        let b = FF::from_noncanonical_biguint(witness.get_biguint_target(self.b.value.clone()));
+        let a_biguint = a.to_canonical_biguint();
+        let b_biguint = b.to_canonical_biguint();
+        let sum_biguint = a_biguint + b_biguint;
+        let modulus = FF::order();
+        let (overflow, sum_reduced) = if sum_biguint > modulus {
+            (true, sum_biguint - modulus)
+        } else {
+            (false, sum_biguint)
+        };
+
+        out_buffer.set_biguint_target(&self.sum.value, &sum_reduced);
+        out_buffer.set_bool_target(self.overflow, overflow);
+    }
+}
+
+#[derive(Debug)]
+struct NonNativeMultipleAddsGenerator<F: RichField + Extendable<D>, const D: usize, FF: PrimeField>
+{
+    summands: Vec<NonNativeTarget<FF>>,
+    sum: NonNativeTarget<FF>,
+    overflow: U32Target,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> SimpleGenerator<F>
+    for NonNativeMultipleAddsGenerator<F, D, FF>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.summands
+            .iter()
+            .flat_map(|summand| summand.value.limbs.iter().map(|limb| limb.0))
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let summands: Vec<_> = self
+            .summands
+            .iter()
+            .map(|summand| {
+                FF::from_noncanonical_biguint(witness.get_biguint_target(summand.value.clone()))
+            })
+            .collect();
+        let summand_biguints: Vec<_> = summands
+            .iter()
+            .map(|summand| summand.to_canonical_biguint())
+            .collect();
+
+        let sum_biguint = summand_biguints
+            .iter()
+            .fold(BigUint::zero(), |a, b| a + b.clone());
+
+        let modulus = FF::order();
+        let (overflow_biguint, sum_reduced) = sum_biguint.div_rem(&modulus);
+        let overflow = overflow_biguint.to_u64_digits()[0] as u32;
+
+        out_buffer.set_biguint_target(&self.sum.value, &sum_reduced);
+        out_buffer.set_u32_target(self.overflow, overflow);
+    }
+}
+
+#[derive(Debug)]
+struct NonNativeSubtractionGenerator<F: RichField + Extendable<D>, const D: usize, FF: Field> {
+    a: NonNativeTarget<FF>,
+    b: NonNativeTarget<FF>,
+    diff: NonNativeTarget<FF>,
+    overflow: BoolTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> SimpleGenerator<F>
+    for NonNativeSubtractionGenerator<F, D, FF>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.a
+            .value
+            .limbs
+            .iter()
+            .cloned()
+            .chain(self.b.value.limbs.clone())
+            .map(|l| l.0)
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let a = FF::from_noncanonical_biguint(witness.get_biguint_target(self.a.value.clone()));
+        let b = FF::from_noncanonical_biguint(witness.get_biguint_target(self.b.value.clone()));
+        let a_biguint = a.to_canonical_biguint();
+        let b_biguint = b.to_canonical_biguint();
+
+        let modulus = FF::order();
+        let (diff_biguint, overflow) = if a_biguint >= b_biguint {
+            (a_biguint - b_biguint, false)
+        } else {
+            (modulus + a_biguint - b_biguint, true)
+        };
+
+        out_buffer.set_biguint_target(&self.diff.value, &diff_biguint);
+        out_buffer.set_bool_target(self.overflow, overflow);
+    }
+}
+
+#[derive(Debug)]
+struct NonNativeMultiplicationGenerator<F: RichField + Extendable<D>, const D: usize, FF: Field> {
+    a: NonNativeTarget<FF>,
+    b: NonNativeTarget<FF>,
+    prod: NonNativeTarget<FF>,
+    overflow: BigUintTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> SimpleGenerator<F>
+    for NonNativeMultiplicationGenerator<F, D, FF>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.a
+            .value
+            .limbs
+            .iter()
+            .cloned()
+            .chain(self.b.value.limbs.clone())
+            .map(|l| l.0)
+            .collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let a = FF::from_noncanonical_biguint(witness.get_biguint_target(self.a.value.clone()));
+        let b = FF::from_noncanonical_biguint(witness.get_biguint_target(self.b.value.clone()));
+        let a_biguint = a.to_canonical_biguint();
+        let b_biguint = b.to_canonical_biguint();
+
+        let prod_biguint = a_biguint * b_biguint;
+
+        let modulus = FF::order();
+        let (overflow_biguint, prod_reduced) = prod_biguint.div_rem(&modulus);
+
+        out_buffer.set_biguint_target(&self.prod.value, &prod_reduced);
+        out_buffer.set_biguint_target(&self.overflow, &overflow_biguint);
+    }
+}
+
+#[derive(Debug)]
+struct NonNativeInverseGenerator<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> {
+    x: NonNativeTarget<FF>,
+    inv: BigUintTarget,
+    div: BigUintTarget,
+    _phantom: PhantomData<F>,
+}
+
+impl<F: RichField + Extendable<D>, const D: usize, FF: PrimeField> SimpleGenerator<F>
+    for NonNativeInverseGenerator<F, D, FF>
+{
+    fn dependencies(&self) -> Vec<Target> {
+        self.x.value.limbs.iter().map(|&l| l.0).collect()
+    }
+
+    fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
+        let x = FF::from_noncanonical_biguint(witness.get_biguint_target(self.x.value.clone()));
+        let inv = x.inverse();
+
+        let x_biguint = x.to_canonical_biguint();
+        let inv_biguint = inv.to_canonical_biguint();
+        let prod = x_biguint * &inv_biguint;
+        let modulus = FF::order();
+        let (div, _rem) = prod.div_rem(&modulus);
+
+        out_buffer.set_biguint_target(&self.div, &div);
+        out_buffer.set_biguint_target(&self.inv, &inv_biguint);
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_base::Secp256K1Base;
+    use plonky2::field::types::{Field, PrimeField, Sample};
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_builder::CircuitBuilder;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use crate::gadgets::nonnative::CircuitBuilderNonNative;
+
+    #[test]
+    fn test_nonnative_add() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let x_ff = FF::rand();
+        let y_ff = FF::rand();
+        let sum_ff = x_ff + y_ff;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_nonnative(x_ff);
+        let y = builder.constant_nonnative(y_ff);
+        let sum = builder.add_nonnative(&x, &y);
+
+        let sum_expected = builder.constant_nonnative(sum_ff);
+        builder.connect_nonnative(&sum, &sum_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_nonnative_many_adds() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let a_ff = FF::rand();
+        let b_ff = FF::rand();
+        let c_ff = FF::rand();
+        let d_ff = FF::rand();
+        let e_ff = FF::rand();
+        let f_ff = FF::rand();
+        let g_ff = FF::rand();
+        let h_ff = FF::rand();
+        let sum_ff = a_ff + b_ff + c_ff + d_ff + e_ff + f_ff + g_ff + h_ff;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let a = builder.constant_nonnative(a_ff);
+        let b = builder.constant_nonnative(b_ff);
+        let c = builder.constant_nonnative(c_ff);
+        let d = builder.constant_nonnative(d_ff);
+        let e = builder.constant_nonnative(e_ff);
+        let f = builder.constant_nonnative(f_ff);
+        let g = builder.constant_nonnative(g_ff);
+        let h = builder.constant_nonnative(h_ff);
+        let all = [a, b, c, d, e, f, g, h];
+        let sum = builder.add_many_nonnative(&all);
+
+        let sum_expected = builder.constant_nonnative(sum_ff);
+        builder.connect_nonnative(&sum, &sum_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_nonnative_sub() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let x_ff = FF::rand();
+        let mut y_ff = FF::rand();
+        while y_ff.to_canonical_biguint() > x_ff.to_canonical_biguint() {
+            y_ff = FF::rand();
+        }
+        let diff_ff = x_ff - y_ff;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_nonnative(x_ff);
+        let y = builder.constant_nonnative(y_ff);
+        let diff = builder.sub_nonnative(&x, &y);
+
+        let diff_expected = builder.constant_nonnative(diff_ff);
+        builder.connect_nonnative(&diff, &diff_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_nonnative_mul() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let x_ff = FF::rand();
+        let y_ff = FF::rand();
+        let product_ff = x_ff * y_ff;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_nonnative(x_ff);
+        let y = builder.constant_nonnative(y_ff);
+        let product = builder.mul_nonnative(&x, &y);
+
+        let product_expected = builder.constant_nonnative(product_ff);
+        builder.connect_nonnative(&product, &product_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_nonnative_neg() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let x_ff = FF::rand();
+        let neg_x_ff = -x_ff;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_nonnative(x_ff);
+        let neg_x = builder.neg_nonnative(&x);
+
+        let neg_x_expected = builder.constant_nonnative(neg_x_ff);
+        builder.connect_nonnative(&neg_x, &neg_x_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+
+    #[test]
+    fn test_nonnative_inv() -> Result<()> {
+        type FF = Secp256K1Base;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+        let x_ff = FF::rand();
+        let inv_x_ff = x_ff.inverse();
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = builder.constant_nonnative(x_ff);
+        let inv_x = builder.inv_nonnative(&x);
+
+        let inv_x_expected = builder.constant_nonnative(inv_x_ff);
+        builder.connect_nonnative(&inv_x, &inv_x_expected);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+}

src/gadgets/split_nonnative.rs

@@ -0,0 +1,131 @@
+use alloc::vec::Vec;
+use core::marker::PhantomData;
+
+use itertools::Itertools;
+use plonky2::field::extension::Extendable;
+use plonky2::field::types::Field;
+use plonky2::hash::hash_types::RichField;
+use plonky2::iop::target::Target;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+use plonky2_u32::gadgets::arithmetic_u32::{CircuitBuilderU32, U32Target};
+
+use crate::gadgets::biguint::BigUintTarget;
+use crate::gadgets::nonnative::NonNativeTarget;
+
+pub trait CircuitBuilderSplit<F: RichField + Extendable<D>, const D: usize> {
+    fn split_u32_to_4_bit_limbs(&mut self, val: U32Target) -> Vec<Target>;
+
+    fn split_nonnative_to_4_bit_limbs<FF: Field>(
+        &mut self,
+        val: &NonNativeTarget<FF>,
+    ) -> Vec<Target>;
+
+    fn split_nonnative_to_2_bit_limbs<FF: Field>(
+        &mut self,
+        val: &NonNativeTarget<FF>,
+    ) -> Vec<Target>;
+
+    // Note: assumes its inputs are 4-bit limbs, and does not range-check.
+    fn recombine_nonnative_4_bit_limbs<FF: Field>(
+        &mut self,
+        limbs: Vec<Target>,
+    ) -> NonNativeTarget<FF>;
+}
+
+impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilderSplit<F, D>
+    for CircuitBuilder<F, D>
+{
+    fn split_u32_to_4_bit_limbs(&mut self, val: U32Target) -> Vec<Target> {
+        let two_bit_limbs = self.split_le_base::<4>(val.0, 16);
+        let four = self.constant(F::from_canonical_usize(4));
+        let combined_limbs = two_bit_limbs
+            .iter()
+            .tuples()
+            .map(|(&a, &b)| self.mul_add(b, four, a))
+            .collect();
+
+        combined_limbs
+    }
+
+    fn split_nonnative_to_4_bit_limbs<FF: Field>(
+        &mut self,
+        val: &NonNativeTarget<FF>,
+    ) -> Vec<Target> {
+        val.value
+            .limbs
+            .iter()
+            .flat_map(|&l| self.split_u32_to_4_bit_limbs(l))
+            .collect()
+    }
+
+    fn split_nonnative_to_2_bit_limbs<FF: Field>(
+        &mut self,
+        val: &NonNativeTarget<FF>,
+    ) -> Vec<Target> {
+        val.value
+            .limbs
+            .iter()
+            .flat_map(|&l| self.split_le_base::<4>(l.0, 16))
+            .collect()
+    }
+
+    // Note: assumes its inputs are 4-bit limbs, and does not range-check.
+    fn recombine_nonnative_4_bit_limbs<FF: Field>(
+        &mut self,
+        limbs: Vec<Target>,
+    ) -> NonNativeTarget<FF> {
+        let base = self.constant_u32(1 << 4);
+        let u32_limbs = limbs
+            .chunks(8)
+            .map(|chunk| {
+                let mut combined_chunk = self.zero_u32();
+                for i in (0..8).rev() {
+                    let (low, _high) = self.mul_add_u32(combined_chunk, base, U32Target(chunk[i]));
+                    combined_chunk = low;
+                }
+                combined_chunk
+            })
+            .collect();
+
+        NonNativeTarget {
+            value: BigUintTarget { limbs: u32_limbs },
+            _phantom: PhantomData,
+        }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use anyhow::Result;
+    use plonky2::field::secp256k1_scalar::Secp256K1Scalar;
+    use plonky2::field::types::Sample;
+    use plonky2::iop::witness::PartialWitness;
+    use plonky2::plonk::circuit_data::CircuitConfig;
+    use plonky2::plonk::config::{GenericConfig, PoseidonGoldilocksConfig};
+
+    use super::*;
+    use crate::gadgets::nonnative::{CircuitBuilderNonNative, NonNativeTarget};
+
+    #[test]
+    fn test_split_nonnative() -> Result<()> {
+        type FF = Secp256K1Scalar;
+        const D: usize = 2;
+        type C = PoseidonGoldilocksConfig;
+        type F = <C as GenericConfig<D>>::F;
+
+        let config = CircuitConfig::standard_ecc_config();
+        let pw = PartialWitness::new();
+        let mut builder = CircuitBuilder::<F, D>::new(config);
+
+        let x = FF::rand();
+        let x_target = builder.constant_nonnative(x);
+        let split = builder.split_nonnative_to_4_bit_limbs(&x_target);
+        let combined: NonNativeTarget<Secp256K1Scalar> =
+            builder.recombine_nonnative_4_bit_limbs(split);
+        builder.connect_nonnative(&x_target, &combined);
+
+        let data = builder.build::<C>();
+        let proof = data.prove(pw).unwrap();
+        data.verify(proof)
+    }
+}

src/lib.rs

@@ -0,0 +1,7 @@
+#![allow(clippy::needless_range_loop)]
+#![cfg_attr(not(test), no_std)]
+
+extern crate alloc;
+
+pub mod curve;
+pub mod gadgets;