2021-03-05 15:25:45 -08:00
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//! Constants used in the Orchard protocol.
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2021-03-18 08:38:31 -07:00
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use ff::{Field, PrimeField};
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use group::Curve;
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use halo2::{
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arithmetic::{lagrange_interpolate, CurveAffine, FieldExt},
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pasta::pallas,
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};
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pub mod commit_ivk_r;
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pub mod note_commit_r;
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pub mod nullifier_k;
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pub mod value_commit_r;
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pub mod value_commit_v;
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pub mod util;
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2021-03-05 15:25:45 -08:00
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2021-03-15 13:33:07 -07:00
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/// $\ell^\mathsf{Orchard}_\mathsf{base}$
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pub(crate) const L_ORCHARD_BASE: usize = 255;
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2021-03-18 08:38:18 -07:00
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// SWU hash-to-curve personalizations
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/// SWU hash-to-curve personalization
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/// This is used for the spending key base point and the nullifier base point K^Orchard
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pub const ORCHARD_PERSONALIZATION: &str = "z.cash:Orchard";
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/// SWU hash-to-curve personalization for the group hash for key diversification
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pub const KEY_DIVERSIFICATION_PERSONALIZATION: &str = "z.cash:Orchard-gd";
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/// SWU hash-to-curve personalization for the value commitment generator
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pub const VALUE_COMMITMENT_PERSONALIZATION: &str = "z.cash:Orchard-cv";
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/// SWU hash-to-curve personalization for the note commitment generator
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pub const NOTE_COMMITMENT_PERSONALIZATION: &str = "z.cash:Orchard-NoteCommit";
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/// SWU hash-to-curve personalization for the IVK commitment generator
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pub const COMMIT_IVK_PERSONALIZATION: &str = "z.cash:Orchard-CommitIvk";
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/// SWU hash-to-curve personalization for the Merkle CRH generator
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pub const MERKLE_CRH_PERSONALIZATION: &str = "z.cash:Orchard-MerkleCRH";
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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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2021-03-18 08:38:31 -07:00
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/// Number of windows
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pub const NUM_WINDOWS: usize = pallas::Base::NUM_BITS as usize / FIXED_BASE_WINDOW_SIZE;
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/// Number of bits used in complete addition (for variable-base scalar mul)
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pub const NUM_COMPLETE_BITS: usize = 3;
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#[derive(Copy, Clone, Debug, Eq, PartialEq)]
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pub enum OrchardFixedBases<C: CurveAffine> {
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CommitIvkR(OrchardFixedBase<C>),
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NoteCommitR(OrchardFixedBase<C>),
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NullifierK(OrchardFixedBase<C>),
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ValueCommitR(OrchardFixedBase<C>),
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ValueCommitV(OrchardFixedBase<C>),
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}
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#[derive(Copy, Clone, Debug, Eq, PartialEq)]
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pub struct OrchardFixedBase<C: CurveAffine>(C);
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impl<C: CurveAffine> OrchardFixedBase<C> {
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pub fn new(generator: C) -> Self {
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OrchardFixedBase(generator)
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}
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pub fn value(&self) -> C {
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self.0
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}
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}
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pub trait FixedBase<C: CurveAffine> {
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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.
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fn compute_window_table(&self) -> Vec<Vec<C>>;
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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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fn compute_lagrange_coeffs(&self) -> Vec<Vec<C::Base>>;
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/// For each window, z is a field element
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/// 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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fn find_zs(&self) -> Option<Vec<u64>>;
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}
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impl<C: CurveAffine> FixedBase<C> for OrchardFixedBase<C> {
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fn compute_window_table(&self) -> Vec<Vec<C>> {
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let h: usize = 1 << FIXED_BASE_WINDOW_SIZE;
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let mut window_table: Vec<Vec<C>> = 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 84 windows, we compute the multiple [(k+1)*(8^w)]B.
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// Here, w ranges from [0..84)
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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+1)*(8^w)
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let scalar = C::ScalarExt::from_u64(k as u64 + 1)
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* C::ScalarExt::from_u64(h as u64).pow(&[w as u64, 0, 0, 0]);
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(self.0 * scalar).to_affine()
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})
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.collect(),
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);
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}
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// Generate window table entries for the last window, w = 84.
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// For the last window, we compute [k * (8^w) - sum]B, where sum is defined
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// as sum = \sum_{j = 0}^{83} 8^j
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let sum = (0..(NUM_WINDOWS - 1)).fold(C::ScalarExt::zero(), |acc, w| {
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acc + C::ScalarExt::from_u64(h as u64).pow(&[w as u64, 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 * (8^w) - sum, where w = 84
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let scalar = C::ScalarExt::from_u64(k as u64)
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* C::ScalarExt::from_u64(h as u64).pow(&[
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(NUM_WINDOWS - 1) as u64,
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0,
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0,
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0,
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])
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- sum;
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(self.0 * scalar).to_affine()
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})
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.collect(),
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);
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window_table
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}
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fn compute_lagrange_coeffs(&self) -> Vec<Vec<C::Base>> {
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let h: usize = 1 << FIXED_BASE_WINDOW_SIZE;
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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_u64(i as u64)).collect();
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let window_table = self.compute_window_table();
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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.get_xy().unwrap().0)
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.collect();
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let coeffs = lagrange_interpolate(&points, &x_window_points);
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coeffs
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})
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.collect::<Vec<Vec<_>>>()
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}
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/// For each window, z is a field element
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/// 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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fn find_zs(&self) -> Option<Vec<u64>> {
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// Closure to find z for one window
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let find_z = |window_points: &[C]| {
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let h: usize = 1 << FIXED_BASE_WINDOW_SIZE;
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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.get_xy().unwrap().1)
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.collect();
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let z_for_single_y = |y: C::Base, z: u64| {
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let sum_y_is_square: bool = (y + C::Base::from_u64(z)).sqrt().is_some().into();
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let sum_neg_y_is_square: bool = (-y + C::Base::from_u64(z)).sqrt().is_some().into();
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(sum_y_is_square && !sum_neg_y_is_square) as usize
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};
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for z in 0..(1000 * (1 << (2 * h))) {
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if ys.iter().map(|y| z_for_single_y(*y, z)).sum::<usize>() == h {
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return Some(z);
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}
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}
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None
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};
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let window_table = self.compute_window_table();
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window_table
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.iter()
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.map(|window_points| find_z(window_points))
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.collect()
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}
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}
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pub trait TestFixedBase<C: CurveAffine> {
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fn test_lagrange_coeffs(&self);
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fn test_z(&self, z: &[u64]);
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}
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impl<C: CurveAffine> TestFixedBase<C> for OrchardFixedBase<C> {
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fn test_lagrange_coeffs(&self) {
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let h = 1 << FIXED_BASE_WINDOW_SIZE;
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let lagrange_coeffs = self.compute_lagrange_coeffs();
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let mut points = Vec::<C::CurveExt>::with_capacity(NUM_WINDOWS);
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let scalar = C::Scalar::rand();
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let bits = util::decompose_scalar_fixed::<C>(
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scalar,
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C::Scalar::NUM_BITS as usize,
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FIXED_BASE_WINDOW_SIZE,
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);
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// Check first 84 windows, i.e. `k_0, k_1, ..., k_83`
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for ((idx, bits), coeffs) in bits[0..(NUM_WINDOWS - 1)]
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.iter()
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.enumerate()
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.zip(lagrange_coeffs[0..(NUM_WINDOWS - 1)].iter())
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{
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let interpolated_x = util::evaluate::<C>(*bits, coeffs);
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// [(k+1)*(8^w)]B
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let point = self.0
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* C::Scalar::from_u64(*bits as u64 + 1)
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* C::Scalar::from_u64(h as u64).pow(&[idx as u64, 0, 0, 0]);
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let x = point.to_affine().get_xy().unwrap().0;
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assert_eq!(x, interpolated_x);
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points.push(point);
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}
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// Check last window
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{
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let last_bits = bits[NUM_WINDOWS - 1];
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let interpolated_x = util::evaluate::<C>(last_bits, &lagrange_coeffs[NUM_WINDOWS - 1]);
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// [k * (8^w) - offset]B, where offset = \sum_{j = 0}^{83} 8^j
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let offset = (0..(NUM_WINDOWS - 1)).fold(C::Scalar::zero(), |acc, w| {
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acc + C::Scalar::from_u64(h as u64).pow(&[w as u64, 0, 0, 0])
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});
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let scalar = C::Scalar::from_u64(last_bits as u64)
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* C::Scalar::from_u64(h as u64).pow(&[(NUM_WINDOWS - 1) as u64, 0, 0, 0])
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- offset;
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let point = self.0 * scalar;
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let x = point.to_affine().get_xy().unwrap().0;
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assert_eq!(x, interpolated_x);
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points.push(point);
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}
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// Check the sum of all the window points
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let window_sum = points
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.iter()
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.fold(C::CurveExt::default(), |acc, point| acc + point);
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let multiple = self.0 * scalar;
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assert_eq!(window_sum, multiple);
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}
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fn test_z(&self, z: &[u64]) {
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let window_table = self.compute_window_table();
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for (z, window_points) in z.iter().zip(window_table) {
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for point in window_points.iter() {
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let y = point.get_xy().unwrap().1;
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assert_eq!((C::Base::from_u64(*z) + y).sqrt().is_some().unwrap_u8(), 1);
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assert_eq!((C::Base::from_u64(*z) - y).sqrt().is_some().unwrap_u8(), 0);
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}
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}
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}
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}
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