mirror of https://github.com/zcash/halo2.git
283 lines
11 KiB
Rust
283 lines
11 KiB
Rust
//! Constants required for the ECC chip.
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use arrayvec::ArrayVec;
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use group::{
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ff::{Field, PrimeField},
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Curve,
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};
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use halo2_proofs::arithmetic::lagrange_interpolate;
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use pasta_curves::{
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arithmetic::{CurveAffine, FieldExt},
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pallas,
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};
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/// Window size for fixed-base scalar multiplication
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pub const FIXED_BASE_WINDOW_SIZE: usize = 3;
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/// $2^{`FIXED_BASE_WINDOW_SIZE`}$
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pub const H: usize = 1 << FIXED_BASE_WINDOW_SIZE;
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/// Number of windows for a full-width scalar
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pub const NUM_WINDOWS: usize =
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(pallas::Scalar::NUM_BITS as usize + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE;
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/// Number of windows for a short signed scalar
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pub const NUM_WINDOWS_SHORT: usize =
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(L_SCALAR_SHORT + FIXED_BASE_WINDOW_SIZE - 1) / FIXED_BASE_WINDOW_SIZE;
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/// $\ell_\mathsf{value}$
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/// Number of bits in an unsigned short scalar.
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pub(crate) const L_SCALAR_SHORT: usize = 64;
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/// The Pallas scalar field modulus is $q = 2^{254} + \mathsf{t_q}$.
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/// <https://github.com/zcash/pasta>
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pub(crate) const T_Q: u128 = 45560315531506369815346746415080538113;
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/// The Pallas base field modulus is $p = 2^{254} + \mathsf{t_p}$.
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/// <https://github.com/zcash/pasta>
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pub(crate) const T_P: u128 = 45560315531419706090280762371685220353;
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/// For each fixed base, we calculate its scalar multiples in three-bit windows.
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/// Each window will have $2^3 = 8$ points. The tables are computed as described in
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/// [the Halo 2 book](https://zcash.github.io/halo2/design/gadgets/ecc/fixed-base-scalar-mul.html#load-fixed-base).
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fn compute_window_table<C: CurveAffine>(base: C, num_windows: usize) -> Vec<[C; H]> {
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let mut window_table: Vec<[C; H]> = Vec::with_capacity(num_windows);
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// Generate window table entries for all windows but the last.
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// For these first `num_windows - 1` windows, we compute the multiple [(k+2)*(2^3)^w]B.
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// Here, w ranges from [0..`num_windows - 1`)
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for w in 0..(num_windows - 1) {
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window_table.push(
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(0..H)
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.map(|k| {
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// scalar = (k+2)*(8^w)
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let scalar = C::Scalar::from(k as u64 + 2)
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* C::Scalar::from(H as u64).pow(&[w as u64, 0, 0, 0]);
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(base * scalar).to_affine()
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})
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.collect::<ArrayVec<C, H>>()
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.into_inner()
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.unwrap(),
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);
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}
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// Generate window table entries for the last window, w = `num_windows - 1`.
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// For the last window, we compute [k * (2^3)^w - sum]B, where sum is defined
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// as sum = \sum_{j = 0}^{`num_windows - 2`} 2^{3j+1}
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let sum = (0..(num_windows - 1)).fold(C::Scalar::zero(), |acc, j| {
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acc + C::Scalar::from(2).pow(&[FIXED_BASE_WINDOW_SIZE as u64 * j as u64 + 1, 0, 0, 0])
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});
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window_table.push(
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(0..H)
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.map(|k| {
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// scalar = k * (2^3)^w - sum, where w = `num_windows - 1`
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let scalar = C::Scalar::from(k as u64)
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* C::Scalar::from(H as u64).pow(&[(num_windows - 1) as u64, 0, 0, 0])
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- sum;
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(base * scalar).to_affine()
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})
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.collect::<ArrayVec<C, H>>()
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.into_inner()
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.unwrap(),
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);
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window_table
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}
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/// For each window, we interpolate the $x$-coordinate.
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/// Here, we pre-compute and store the coefficients of the interpolation polynomial.
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pub fn compute_lagrange_coeffs<C: CurveAffine>(base: C, num_windows: usize) -> Vec<[C::Base; H]> {
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// We are interpolating over the 3-bit window, k \in [0..8)
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let points: Vec<_> = (0..H).map(|i| C::Base::from(i as u64)).collect();
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let window_table = compute_window_table(base, num_windows);
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window_table
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.iter()
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.map(|window_points| {
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let x_window_points: Vec<_> = window_points
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.iter()
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.map(|point| *point.coordinates().unwrap().x())
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.collect();
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lagrange_interpolate(&points, &x_window_points)
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.into_iter()
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.collect::<ArrayVec<C::Base, H>>()
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.into_inner()
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.unwrap()
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})
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.collect()
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}
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/// For each window, $z$ is a field element such that for each point $(x, y)$ in the window:
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/// - $z + y = u^2$ (some square in the field); and
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/// - $z - y$ is not a square.
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/// If successful, return a vector of `(z: u64, us: [C::Base; H])` for each window.
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///
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/// This function was used to generate the `z`s and `u`s for the Orchard fixed
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/// bases. The outputs of this function have been stored as constants, and it
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/// is not called anywhere in this codebase. However, we keep this function here
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/// as a utility for those who wish to use it with different parameters.
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pub fn find_zs_and_us<C: CurveAffine>(
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base: C,
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num_windows: usize,
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) -> Option<Vec<(u64, [C::Base; H])>> {
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// Closure to find z and u's for one window
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let find_z_and_us = |window_points: &[C]| {
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assert_eq!(H, window_points.len());
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let ys: Vec<_> = window_points
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.iter()
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.map(|point| *point.coordinates().unwrap().y())
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.collect();
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(0..(1000 * (1 << (2 * H)))).find_map(|z| {
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ys.iter()
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.map(|&y| {
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if (-y + C::Base::from(z)).sqrt().is_none().into() {
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(y + C::Base::from(z)).sqrt().into()
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} else {
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None
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}
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})
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.collect::<Option<ArrayVec<C::Base, H>>>()
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.map(|us| (z, us.into_inner().unwrap()))
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})
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};
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let window_table = compute_window_table(base, num_windows);
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window_table
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.iter()
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.map(|window_points| find_z_and_us(window_points))
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.collect()
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}
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/// Test that the z-values and u-values satisfy the conditions:
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/// 1. z + y = u^2,
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/// 2. z - y is not a square
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/// for the y-coordinate of each fixed-base multiple in each window.
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#[cfg(any(test, feature = "test-dependencies"))]
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#[cfg_attr(docsrs, doc(cfg(feature = "test-dependencies")))]
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pub fn test_zs_and_us<C: CurveAffine>(base: C, z: &[u64], u: &[[[u8; 32]; H]], num_windows: usize) {
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let window_table = compute_window_table(base, num_windows);
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for ((u, z), window_points) in u.iter().zip(z.iter()).zip(window_table) {
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for (u, point) in u.iter().zip(window_points.iter()) {
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let y = *point.coordinates().unwrap().y();
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let mut u_repr = <C::Base as PrimeField>::Repr::default();
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u_repr.as_mut().copy_from_slice(u);
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let u = C::Base::from_repr(u_repr).unwrap();
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assert_eq!(C::Base::from(*z) + y, u * u); // allow either square root
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assert!(bool::from((C::Base::from(*z) - y).sqrt().is_none()));
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}
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}
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}
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/// Test that Lagrange interpolation coefficients reproduce the correct x-coordinate
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/// for each fixed-base multiple in each window.
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#[cfg(any(test, feature = "test-dependencies"))]
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#[cfg_attr(docsrs, doc(cfg(feature = "test-dependencies")))]
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pub fn test_lagrange_coeffs<C: CurveAffine>(base: C, num_windows: usize) {
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/// Evaluate y = f(x) given the coefficients of f(x)
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fn evaluate<C: CurveAffine>(x: u8, coeffs: &[C::Base]) -> C::Base {
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let x = C::Base::from(x as u64);
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coeffs
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.iter()
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.rev()
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.cloned()
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.reduce(|acc, coeff| acc * x + coeff)
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.unwrap_or_else(C::Base::zero)
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}
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let lagrange_coeffs = compute_lagrange_coeffs(base, num_windows);
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// Check first 84 windows, i.e. `k_0, k_1, ..., k_83`
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for (idx, coeffs) in lagrange_coeffs[0..(num_windows - 1)].iter().enumerate() {
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// Test each three-bit chunk in this window.
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for bits in 0..(H as u8) {
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{
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// Interpolate the x-coordinate using this window's coefficients
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let interpolated_x = evaluate::<C>(bits, coeffs);
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// Compute the actual x-coordinate of the multiple [(k+2)*(8^w)]B.
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let point = base
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* C::Scalar::from(bits as u64 + 2)
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* C::Scalar::from(H as u64).pow(&[idx as u64, 0, 0, 0]);
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let x = *point.to_affine().coordinates().unwrap().x();
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// Check that the interpolated x-coordinate matches the actual one.
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assert_eq!(x, interpolated_x);
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}
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}
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}
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// Check last window.
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for bits in 0..(H as u8) {
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// Interpolate the x-coordinate using the last window's coefficients
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let interpolated_x = evaluate::<C>(bits, &lagrange_coeffs[num_windows - 1]);
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// Compute the actual x-coordinate of the multiple [k * (8^84) - offset]B,
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// where offset = \sum_{j = 0}^{83} 2^{3j+1}
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let offset = (0..(num_windows - 1)).fold(C::Scalar::zero(), |acc, w| {
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acc + C::Scalar::from(2).pow(&[FIXED_BASE_WINDOW_SIZE as u64 * w as u64 + 1, 0, 0, 0])
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});
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let scalar = C::Scalar::from(bits as u64)
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* C::Scalar::from(H as u64).pow(&[(num_windows - 1) as u64, 0, 0, 0])
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- offset;
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let point = base * scalar;
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let x = *point.to_affine().coordinates().unwrap().x();
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// Check that the interpolated x-coordinate matches the actual one.
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assert_eq!(x, interpolated_x);
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}
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}
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#[cfg(test)]
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mod tests {
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use group::{ff::Field, Curve, Group};
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use pasta_curves::{
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arithmetic::{CurveAffine, FieldExt},
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pallas,
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};
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use proptest::prelude::*;
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use super::{compute_window_table, find_zs_and_us, test_lagrange_coeffs, H, NUM_WINDOWS};
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prop_compose! {
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/// Generate an arbitrary Pallas point.
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pub fn arb_point()(bytes in prop::array::uniform32(0u8..)) -> pallas::Point {
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// Instead of rejecting out-of-range bytes, let's reduce them.
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let mut buf = [0; 64];
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buf[..32].copy_from_slice(&bytes);
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let scalar = pallas::Scalar::from_bytes_wide(&buf);
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pallas::Point::generator() * scalar
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}
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}
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proptest! {
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#[test]
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fn lagrange_coeffs(
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base in arb_point(),
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) {
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test_lagrange_coeffs(base.to_affine(), NUM_WINDOWS);
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}
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}
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#[test]
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fn zs_and_us() {
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let base = pallas::Point::random(rand::rngs::OsRng);
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let (z, u): (Vec<u64>, Vec<[pallas::Base; H]>) =
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find_zs_and_us(base.to_affine(), NUM_WINDOWS)
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.unwrap()
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.into_iter()
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.unzip();
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let window_table = compute_window_table(base.to_affine(), NUM_WINDOWS);
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for ((u, z), window_points) in u.iter().zip(z.iter()).zip(window_table) {
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for (u, point) in u.iter().zip(window_points.iter()) {
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let y = *point.coordinates().unwrap().y();
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assert_eq!(pallas::Base::from(*z) + y, u * u); // allow either square root
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assert!(bool::from((pallas::Base::from(*z) - y).sqrt().is_none()));
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}
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}
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}
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}
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