mirror of https://github.com/zcash/halo2.git
375 lines
14 KiB
Rust
375 lines
14 KiB
Rust
use super::super::NonIdentityEccPoint;
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use super::{X, Y, Z};
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use crate::utilities::bool_check;
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use group::ff::PrimeField;
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use halo2_proofs::{
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circuit::{Region, Value},
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plonk::{
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Advice, Column, ConstraintSystem, Constraints, Error, Expression, Selector, VirtualCells,
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},
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poly::Rotation,
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};
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use pasta_curves::pallas;
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/// A helper struct for implementing single-row double-and-add using incomplete addition.
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#[derive(Copy, Clone, Debug, Eq, PartialEq)]
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pub(crate) struct DoubleAndAdd {
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// x-coordinate of the accumulator in each double-and-add iteration.
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pub(crate) x_a: Column<Advice>,
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// x-coordinate of the point being added in each double-and-add iteration.
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pub(crate) x_p: Column<Advice>,
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// lambda1 in each double-and-add iteration.
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pub(crate) lambda_1: Column<Advice>,
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// lambda2 in each double-and-add iteration.
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pub(crate) lambda_2: Column<Advice>,
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}
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impl DoubleAndAdd {
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/// Derives the expression `x_r = lambda_1^2 - x_a - x_p`.
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pub(crate) fn x_r(
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&self,
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meta: &mut VirtualCells<pallas::Base>,
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rotation: Rotation,
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) -> Expression<pallas::Base> {
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let x_a = meta.query_advice(self.x_a, rotation);
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let x_p = meta.query_advice(self.x_p, rotation);
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let lambda_1 = meta.query_advice(self.lambda_1, rotation);
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lambda_1.square() - x_a - x_p
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}
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/// Derives the expression `Y_A = (lambda_1 + lambda_2) * (x_a - x_r)`.
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///
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/// Note that this is missing the factor of `1/2`; the Sinsemilla constraints factor
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/// it out, so we leave it up to the caller to handle it.
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#[allow(non_snake_case)]
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pub(crate) fn Y_A(
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&self,
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meta: &mut VirtualCells<pallas::Base>,
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rotation: Rotation,
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) -> Expression<pallas::Base> {
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let x_a = meta.query_advice(self.x_a, rotation);
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let lambda_1 = meta.query_advice(self.lambda_1, rotation);
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let lambda_2 = meta.query_advice(self.lambda_2, rotation);
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(lambda_1 + lambda_2) * (x_a - self.x_r(meta, rotation))
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}
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}
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#[derive(Copy, Clone, Debug, Eq, PartialEq)]
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pub(crate) struct Config<const NUM_BITS: usize> {
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// Selector constraining the first row of incomplete addition.
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pub(super) q_mul_1: Selector,
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// Selector constraining the main loop of incomplete addition.
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pub(super) q_mul_2: Selector,
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// Selector constraining the last row of incomplete addition.
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pub(super) q_mul_3: Selector,
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// Cumulative sum used to decompose the scalar.
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pub(super) z: Column<Advice>,
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// Logic specific to merged double-and-add.
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pub(super) double_and_add: DoubleAndAdd,
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// y-coordinate of the point being added in each double-and-add iteration.
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pub(super) y_p: Column<Advice>,
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}
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impl<const NUM_BITS: usize> Config<NUM_BITS> {
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pub(super) fn configure(
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meta: &mut ConstraintSystem<pallas::Base>,
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z: Column<Advice>,
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x_a: Column<Advice>,
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x_p: Column<Advice>,
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y_p: Column<Advice>,
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lambda_1: Column<Advice>,
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lambda_2: Column<Advice>,
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) -> Self {
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meta.enable_equality(z);
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meta.enable_equality(lambda_1);
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let config = Self {
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q_mul_1: meta.selector(),
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q_mul_2: meta.selector(),
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q_mul_3: meta.selector(),
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z,
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double_and_add: DoubleAndAdd {
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x_a,
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x_p,
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lambda_1,
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lambda_2,
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},
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y_p,
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};
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config.create_gate(meta);
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config
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}
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// Gate for incomplete addition part of variable-base scalar multiplication.
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fn create_gate(&self, meta: &mut ConstraintSystem<pallas::Base>) {
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// Closure to compute x_{R,i} = λ_{1,i}^2 - x_{A,i} - x_{P,i}
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let x_r = |meta: &mut VirtualCells<pallas::Base>, rotation: Rotation| {
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self.double_and_add.x_r(meta, rotation)
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};
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// Closure to compute y_{A,i} = (λ_{1,i} + λ_{2,i}) * (x_{A,i} - x_{R,i}) / 2
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let y_a = |meta: &mut VirtualCells<pallas::Base>, rotation: Rotation| {
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self.double_and_add.Y_A(meta, rotation) * pallas::Base::TWO_INV
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};
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// Constraints used for q_mul_{2, 3} == 1
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// https://p.z.cash/halo2-0.1:ecc-var-mul-incomplete-main-loop?partial
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// https://p.z.cash/halo2-0.1:ecc-var-mul-incomplete-last-row?partial
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let for_loop = |meta: &mut VirtualCells<pallas::Base>,
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y_a_next: Expression<pallas::Base>| {
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let one = Expression::Constant(pallas::Base::one());
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// z_i
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let z_cur = meta.query_advice(self.z, Rotation::cur());
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// z_{i+1}
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let z_prev = meta.query_advice(self.z, Rotation::prev());
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// x_{A,i}
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let x_a_cur = meta.query_advice(self.double_and_add.x_a, Rotation::cur());
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// x_{A,i-1}
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let x_a_next = meta.query_advice(self.double_and_add.x_a, Rotation::next());
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// x_{P,i}
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let x_p_cur = meta.query_advice(self.double_and_add.x_p, Rotation::cur());
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// y_{P,i}
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let y_p_cur = meta.query_advice(self.y_p, Rotation::cur());
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// λ_{1,i}
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let lambda1_cur = meta.query_advice(self.double_and_add.lambda_1, Rotation::cur());
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// λ_{2,i}
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let lambda2_cur = meta.query_advice(self.double_and_add.lambda_2, Rotation::cur());
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let y_a_cur = y_a(meta, Rotation::cur());
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// The current bit in the scalar decomposition, k_i = z_i - 2⋅z_{i+1}.
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// Recall that we assigned the cumulative variable `z_i` in descending order,
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// i from n down to 0. So z_{i+1} corresponds to the `z_prev` query.
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let k = z_cur - z_prev * pallas::Base::from(2);
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// Check booleanity of decomposition.
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let bool_check = bool_check(k.clone());
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// λ_{1,i}⋅(x_{A,i} − x_{P,i}) − y_{A,i} + (2k_i - 1) y_{P,i} = 0
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let gradient_1 = lambda1_cur * (x_a_cur.clone() - x_p_cur) - y_a_cur.clone()
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+ (k * pallas::Base::from(2) - one) * y_p_cur;
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// λ_{2,i}^2 − x_{A,i-1} − x_{R,i} − x_{A,i} = 0
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let secant_line = lambda2_cur.clone().square()
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- x_a_next.clone()
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- x_r(meta, Rotation::cur())
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- x_a_cur.clone();
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// λ_{2,i}⋅(x_{A,i} − x_{A,i-1}) − y_{A,i} − y_{A,i-1} = 0
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let gradient_2 = lambda2_cur * (x_a_cur - x_a_next) - y_a_cur - y_a_next;
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std::iter::empty()
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.chain(Some(("bool_check", bool_check)))
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.chain(Some(("gradient_1", gradient_1)))
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.chain(Some(("secant_line", secant_line)))
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.chain(Some(("gradient_2", gradient_2)))
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};
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// q_mul_1 == 1 checks
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// https://p.z.cash/halo2-0.1:ecc-var-mul-incomplete-first-row
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meta.create_gate("q_mul_1 == 1 checks", |meta| {
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let q_mul_1 = meta.query_selector(self.q_mul_1);
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let y_a_next = y_a(meta, Rotation::next());
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let y_a_witnessed = meta.query_advice(self.double_and_add.lambda_1, Rotation::cur());
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Constraints::with_selector(q_mul_1, Some(("init y_a", y_a_witnessed - y_a_next)))
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});
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// q_mul_2 == 1 checks
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// https://p.z.cash/halo2-0.1:ecc-var-mul-incomplete-main-loop?partial
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meta.create_gate("q_mul_2 == 1 checks", |meta| {
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let q_mul_2 = meta.query_selector(self.q_mul_2);
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let y_a_next = y_a(meta, Rotation::next());
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// x_{P,i}
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let x_p_cur = meta.query_advice(self.double_and_add.x_p, Rotation::cur());
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// x_{P,i-1}
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let x_p_next = meta.query_advice(self.double_and_add.x_p, Rotation::next());
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// y_{P,i}
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let y_p_cur = meta.query_advice(self.y_p, Rotation::cur());
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// y_{P,i-1}
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let y_p_next = meta.query_advice(self.y_p, Rotation::next());
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// The base used in double-and-add remains constant. We check that its
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// x- and y- coordinates are the same throughout.
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let x_p_check = x_p_cur - x_p_next;
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let y_p_check = y_p_cur - y_p_next;
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Constraints::with_selector(
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q_mul_2,
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std::iter::empty()
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.chain(Some(("x_p_check", x_p_check)))
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.chain(Some(("y_p_check", y_p_check)))
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.chain(for_loop(meta, y_a_next)),
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)
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});
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// q_mul_3 == 1 checks
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// https://p.z.cash/halo2-0.1:ecc-var-mul-incomplete-last-row?partial
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meta.create_gate("q_mul_3 == 1 checks", |meta| {
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let q_mul_3 = meta.query_selector(self.q_mul_3);
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let y_a_final = meta.query_advice(self.double_and_add.lambda_1, Rotation::next());
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Constraints::with_selector(q_mul_3, for_loop(meta, y_a_final))
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});
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}
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/// We perform incomplete addition on all but the last three bits of the
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/// decomposed scalar.
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/// We split the bits in the incomplete addition range into "hi" and "lo"
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/// halves and process them side by side, using the same rows but with
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/// non-overlapping columns. The base is never the identity point even at
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/// the boundary between halves.
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/// Returns (x, y, z).
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#[allow(clippy::type_complexity)]
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pub(super) fn double_and_add(
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&self,
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region: &mut Region<'_, pallas::Base>,
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offset: usize,
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base: &NonIdentityEccPoint,
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bits: &[Value<bool>],
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acc: (X<pallas::Base>, Y<pallas::Base>, Z<pallas::Base>),
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) -> Result<(X<pallas::Base>, Y<pallas::Base>, Vec<Z<pallas::Base>>), Error> {
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// Check that we have the correct number of bits for this double-and-add.
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assert_eq!(bits.len(), NUM_BITS);
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// Handle exceptional cases
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let (x_p, y_p) = (base.x.value().cloned(), base.y.value().cloned());
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let (x_a, y_a) = (acc.0.value().cloned(), acc.1.value().cloned());
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x_a.zip(y_a)
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.zip(x_p.zip(y_p))
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.error_if_known_and(|((x_a, y_a), (x_p, y_p))| {
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// A is point at infinity
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(x_p.is_zero_vartime() && y_p.is_zero_vartime())
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// Q is point at infinity
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|| (x_a.is_zero_vartime() && y_a.is_zero_vartime())
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// x_p = x_a
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|| (x_p == x_a)
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})?;
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// Set q_mul values
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{
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// q_mul_1 = 1 on offset 0
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self.q_mul_1.enable(region, offset)?;
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let offset = offset + 1;
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// q_mul_2 = 1 on all rows after offset 0, excluding the last row.
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for idx in 0..(NUM_BITS - 1) {
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self.q_mul_2.enable(region, offset + idx)?;
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}
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// q_mul_3 = 1 on the last row.
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self.q_mul_3.enable(region, offset + NUM_BITS - 1)?;
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}
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// Initialise double-and-add
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let (mut x_a, mut y_a, mut z) = {
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// Initialise the running `z` sum for the scalar bits.
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let z = acc.2.copy_advice(|| "starting z", region, self.z, offset)?;
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// Initialise acc
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let x_a = acc.0.copy_advice(
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|| "starting x_a",
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region,
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self.double_and_add.x_a,
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offset + 1,
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)?;
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let y_a = acc.1.copy_advice(
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|| "starting y_a",
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region,
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self.double_and_add.lambda_1,
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offset,
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)?;
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(x_a, y_a.value().cloned(), z)
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};
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// Increase offset by 1; we used row 0 for initializing `z`.
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let offset = offset + 1;
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// Initialise vector to store all interstitial `z` running sum values.
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let mut zs: Vec<Z<pallas::Base>> = Vec::with_capacity(bits.len());
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// Incomplete addition
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for (row, k) in bits.iter().enumerate() {
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// z_{i} = 2 * z_{i+1} + k_i
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// https://p.z.cash/halo2-0.1:ecc-var-mul-witness-scalar?partial
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let z_val = z
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.value()
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.zip(k.as_ref())
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.map(|(z_val, k)| pallas::Base::from(2) * z_val + pallas::Base::from(*k as u64));
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z = region.assign_advice(|| "z", self.z, row + offset, || z_val)?;
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zs.push(Z(z.clone()));
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// Assign `x_p`, `y_p`
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region.assign_advice(|| "x_p", self.double_and_add.x_p, row + offset, || x_p)?;
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region.assign_advice(|| "y_p", self.y_p, row + offset, || y_p)?;
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// If the bit is set, use `y`; if the bit is not set, use `-y`
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let y_p = y_p
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.zip(k.as_ref())
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.map(|(y_p, k)| if !k { -y_p } else { y_p });
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// Compute and assign λ1⋅(x_A − x_P) = y_A − y_P
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let lambda1 = y_a
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.zip(y_p)
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.zip(x_a.value())
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.zip(x_p)
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.map(|(((y_a, y_p), x_a), x_p)| (y_a - y_p) * (x_a - x_p).invert());
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region.assign_advice(
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|| "lambda1",
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self.double_and_add.lambda_1,
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row + offset,
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|| lambda1,
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)?;
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// x_R = λ1^2 - x_A - x_P
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let x_r = lambda1
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.zip(x_a.value())
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.zip(x_p)
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.map(|((lambda1, x_a), x_p)| lambda1.square() - x_a - x_p);
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// λ2 = (2(y_A) / (x_A - x_R)) - λ1
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let lambda2 =
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lambda1
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.zip(y_a)
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.zip(x_a.value())
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.zip(x_r)
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.map(|(((lambda1, y_a), x_a), x_r)| {
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y_a * pallas::Base::from(2) * (x_a - x_r).invert() - lambda1
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});
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region.assign_advice(
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|| "lambda2",
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self.double_and_add.lambda_2,
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row + offset,
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|| lambda2,
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)?;
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// Compute and assign `x_a` for the next row
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let x_a_new = lambda2.square() - x_a.value() - x_r;
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y_a = lambda2 * (x_a.value() - x_a_new) - y_a;
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let x_a_val = x_a_new;
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x_a = region.assign_advice(
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|| "x_a",
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self.double_and_add.x_a,
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row + offset + 1,
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|| x_a_val,
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)?;
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}
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// Witness final y_a
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let y_a = region.assign_advice(
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|| "y_a",
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self.double_and_add.lambda_1,
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offset + NUM_BITS,
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|| y_a,
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)?;
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Ok((X(x_a), Y(y_a), zs))
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}
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}
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