mirror of https://github.com/zcash/orchard.git
542 lines
19 KiB
Rust
542 lines
19 KiB
Rust
use super::{add, CellValue, EccConfig, EccPoint, NonIdentityEccPoint, Var, T_Q};
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use std::ops::{Deref, Range};
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use utilities::copy;
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use bigint::U256;
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use ff::PrimeField;
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use halo2::{
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arithmetic::FieldExt,
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circuit::{Layouter, Region},
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plonk::{ConstraintSystem, Error, Expression, Selector},
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poly::Rotation,
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};
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use pasta_curves::pallas;
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mod complete;
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mod incomplete;
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mod overflow;
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/// Number of bits for which complete addition needs to be used in variable-base
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/// scalar multiplication
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const NUM_COMPLETE_BITS: usize = 3;
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// Bits used in incomplete addition. k_{254} to k_{4} inclusive
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const INCOMPLETE_LEN: usize = pallas::Scalar::NUM_BITS as usize - 1 - NUM_COMPLETE_BITS;
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// Bits k_{254} to k_{4} inclusive are used in incomplete addition.
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// The `hi` half is k_{254} to k_{130} inclusive (length 125 bits).
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// (It is a coincidence that k_{130} matches the boundary of the
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// overflow check described in [the book](https://zcash.github.io/halo2/design/gadgets/ecc/var-base-scalar-mul.html#overflow-check).)
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const INCOMPLETE_HI_RANGE: Range<usize> = 0..(INCOMPLETE_LEN / 2);
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// Bits k_{254} to k_{4} inclusive are used in incomplete addition.
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// The `lo` half is k_{129} to k_{4} inclusive (length 126 bits).
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const INCOMPLETE_LO_RANGE: Range<usize> = (INCOMPLETE_LEN / 2)..INCOMPLETE_LEN;
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// Bits k_{3} to k_{1} inclusive are used in complete addition.
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// Bit k_{0} is handled separately.
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const COMPLETE_RANGE: Range<usize> = INCOMPLETE_LEN..(INCOMPLETE_LEN + NUM_COMPLETE_BITS);
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pub struct Config {
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// Selector used to check switching logic on LSB
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q_mul_lsb: Selector,
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// Configuration used in complete addition
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add_config: add::Config,
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// Configuration used for `hi` bits of the scalar
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hi_config: incomplete::HiConfig,
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// Configuration used for `lo` bits of the scalar
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lo_config: incomplete::LoConfig,
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// Configuration used for complete addition part of double-and-add algorithm
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complete_config: complete::Config,
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// Configuration used to check for overflow
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overflow_config: overflow::Config,
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}
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impl From<&EccConfig> for Config {
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fn from(ecc_config: &EccConfig) -> Self {
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let config = Self {
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q_mul_lsb: ecc_config.q_mul_lsb,
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add_config: ecc_config.into(),
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hi_config: ecc_config.into(),
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lo_config: ecc_config.into(),
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complete_config: ecc_config.into(),
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overflow_config: ecc_config.into(),
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};
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assert_eq!(
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config.hi_config.x_p, config.lo_config.x_p,
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"x_p is shared across hi and lo halves."
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);
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assert_eq!(
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config.hi_config.y_p, config.lo_config.y_p,
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"y_p is shared across hi and lo halves."
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);
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// For both hi_config and lo_config:
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// z and lambda1 are assigned on the same row as the add_config output.
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// Therefore, z and lambda1 must not overlap with add_config.x_qr, add_config.y_qr.
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let add_config_outputs = config.add_config.output_columns();
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for config in [&(*config.hi_config), &(*config.lo_config)].iter() {
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assert!(
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!add_config_outputs.contains(&config.z),
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"incomplete config z cannot overlap with complete addition columns."
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);
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assert!(
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!add_config_outputs.contains(&config.lambda1),
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"incomplete config lambda1 cannot overlap with complete addition columns."
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);
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}
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config
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}
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}
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impl Config {
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pub(super) fn create_gate(&self, meta: &mut ConstraintSystem<pallas::Base>) {
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// If `lsb` is 0, (x, y) = (x_p, -y_p). If `lsb` is 1, (x, y) = (0,0).
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meta.create_gate("LSB check", |meta| {
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let q_mul_lsb = meta.query_selector(self.q_mul_lsb);
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let z_1 = meta.query_advice(self.complete_config.z_complete, Rotation::cur());
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let z_0 = meta.query_advice(self.complete_config.z_complete, Rotation::next());
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let x_p = meta.query_advice(self.add_config.x_p, Rotation::cur());
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let y_p = meta.query_advice(self.add_config.y_p, Rotation::cur());
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let base_x = meta.query_advice(self.add_config.x_p, Rotation::next());
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let base_y = meta.query_advice(self.add_config.y_p, Rotation::next());
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// z_0 = 2 * z_1 + k_0
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// => k_0 = z_0 - 2 * z_1
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let lsb = z_0 - z_1 * pallas::Base::from_u64(2);
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let one_minus_lsb = Expression::Constant(pallas::Base::one()) - lsb.clone();
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let bool_check = lsb.clone() * one_minus_lsb.clone();
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// `lsb` = 0 => (x_p, y_p) = (x, -y)
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// `lsb` = 1 => (x_p, y_p) = (0,0)
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let lsb_x = (lsb.clone() * x_p.clone()) + one_minus_lsb.clone() * (x_p - base_x);
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let lsb_y = (lsb * y_p.clone()) + one_minus_lsb * (y_p + base_y);
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std::array::IntoIter::new([
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("bool_check", bool_check),
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("lsb_x", lsb_x),
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("lsb_y", lsb_y),
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])
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.map(move |(name, poly)| (name, q_mul_lsb.clone() * poly))
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});
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self.hi_config.create_gate(meta);
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self.lo_config.create_gate(meta);
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self.complete_config.create_gate(meta);
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self.overflow_config.create_gate(meta);
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}
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pub(super) fn assign(
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&self,
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mut layouter: impl Layouter<pallas::Base>,
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alpha: CellValue<pallas::Base>,
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base: &NonIdentityEccPoint,
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) -> Result<(EccPoint, CellValue<pallas::Base>), Error> {
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let (result, zs): (EccPoint, Vec<Z<pallas::Base>>) = layouter.assign_region(
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|| "variable-base scalar mul",
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|mut region| {
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let offset = 0;
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// Case `base` into an `EccPoint` for later use.
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let base_point: EccPoint = (*base).into();
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// Decompose `k = alpha + t_q` bitwise (big-endian bit order).
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let bits = decompose_for_scalar_mul(alpha.value());
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// Define ranges for each part of the algorithm.
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let bits_incomplete_hi = &bits[INCOMPLETE_HI_RANGE];
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let bits_incomplete_lo = &bits[INCOMPLETE_LO_RANGE];
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let lsb = bits[pallas::Scalar::NUM_BITS as usize - 1];
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// Initialize the accumulator `acc = [2]base`
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let acc =
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self.add_config
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.assign_region(&base_point, &base_point, offset, &mut region)?;
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// Increase the offset by 1 after complete addition.
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let offset = offset + 1;
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// Initialize the running sum for scalar decomposition to zero
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let z_init = {
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let z_init_cell = region.assign_advice_from_constant(
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|| "z_init = 0",
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self.hi_config.z,
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offset,
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pallas::Base::zero(),
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)?;
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Z(CellValue::new(z_init_cell, Some(pallas::Base::zero())))
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};
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// Double-and-add (incomplete addition) for the `hi` half of the scalar decomposition
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let (x_a, y_a, zs_incomplete_hi) = self.hi_config.double_and_add(
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&mut region,
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offset,
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base,
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bits_incomplete_hi,
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(X(acc.x), Y(acc.y), z_init),
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)?;
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// Double-and-add (incomplete addition) for the `lo` half of the scalar decomposition
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let z = zs_incomplete_hi.last().expect("should not be empty");
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let (x_a, y_a, zs_incomplete_lo) = self.lo_config.double_and_add(
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&mut region,
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offset,
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base,
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bits_incomplete_lo,
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(x_a, y_a, *z),
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)?;
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// Move from incomplete addition to complete addition.
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// Inside incomplete::double_and_add, the offset was increased once after initialization
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// of the running sum.
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// Then, the final assignment of double-and-add was made on row + offset + 1.
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// Outside of incomplete addition, we must account for these offset increases by adding
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// 2 to the incomplete addition length.
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let offset = offset + INCOMPLETE_LO_RANGE.len() + 2;
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// Complete addition
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let (acc, zs_complete) = {
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let z = zs_incomplete_lo.last().expect("should not be empty");
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// Bits used in complete addition. k_{3} to k_{1} inclusive
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// The LSB k_{0} is handled separately.
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let bits_complete = &bits[COMPLETE_RANGE];
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self.complete_config.assign_region(
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&mut region,
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offset,
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bits_complete,
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&base_point,
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x_a,
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y_a,
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*z,
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)?
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};
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// Each iteration of the complete addition uses two rows.
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let offset = offset + COMPLETE_RANGE.len() * 2;
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// Process the least significant bit
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let z_1 = zs_complete.last().unwrap();
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let (result, z_0) = self.process_lsb(&mut region, offset, base, acc, *z_1, lsb)?;
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#[cfg(test)]
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// Check that the correct multiple is obtained.
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{
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use group::Curve;
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let base = base.point();
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let alpha = alpha
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.value()
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.map(|alpha| pallas::Scalar::from_bytes(&alpha.to_bytes()).unwrap());
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let real_mul = base.zip(alpha).map(|(base, alpha)| base * alpha);
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let result = result.point();
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if let (Some(real_mul), Some(result)) = (real_mul, result) {
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assert_eq!(real_mul.to_affine(), result);
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}
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}
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let zs = {
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let mut zs = std::iter::empty()
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.chain(Some(z_init))
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.chain(zs_incomplete_hi.into_iter())
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.chain(zs_incomplete_lo.into_iter())
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.chain(zs_complete.into_iter())
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.chain(Some(z_0))
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.collect::<Vec<_>>();
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assert_eq!(zs.len(), pallas::Scalar::NUM_BITS as usize + 1);
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// This reverses zs to give us [z_0, z_1, ..., z_{254}, z_{255}].
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zs.reverse();
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zs
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};
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Ok((result, zs))
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},
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)?;
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self.overflow_config
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.overflow_check(layouter.namespace(|| "overflow check"), alpha, &zs)?;
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Ok((result, alpha))
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}
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/// Processes the final scalar bit `k_0`.
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///
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/// Assumptions for this sub-region:
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/// - `acc_x` and `acc_y` are assigned in row `offset` by the previous complete
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/// addition. They will be copied into themselves.
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/// - `z_1 is assigned in row `offset` by the mul::complete region assignment. We only
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/// use its value here.
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///
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/// `x_p` and `y_p` are assigned here, and then copied into themselves by the complete
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/// addition subregion.
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///
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/// ```text
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/// | x_p | y_p | acc_x | acc_y | complete addition | z_1 | q_mul_lsb = 1
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/// |base_x|base_y| res_x | res_y | | | | | | z_0 |
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/// ```
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fn process_lsb(
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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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acc: EccPoint,
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z_1: Z<pallas::Base>,
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lsb: Option<bool>,
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) -> Result<(EccPoint, Z<pallas::Base>), Error> {
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// Enforce switching logic on LSB using a custom gate
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self.q_mul_lsb.enable(region, offset)?;
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// z_1 has been assigned at (z_complete, offset).
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// Assign z_0 = 2⋅z_1 + k_0
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let z_0 = {
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let z_0_val = z_1.value().zip(lsb).map(|(z_1, lsb)| {
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let lsb = pallas::Base::from_u64(lsb as u64);
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z_1 * pallas::Base::from_u64(2) + lsb
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});
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let z_0_cell = region.assign_advice(
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|| "z_0",
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self.complete_config.z_complete,
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offset + 1,
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|| z_0_val.ok_or(Error::SynthesisError),
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)?;
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Z(CellValue::new(z_0_cell, z_0_val))
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};
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// Copy in `base_x`, `base_y` to use in the LSB gate
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copy(
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region,
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|| "copy base_x",
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self.add_config.x_p,
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offset + 1,
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&base.x(),
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)?;
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copy(
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region,
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|| "copy base_y",
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self.add_config.y_p,
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offset + 1,
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&base.y(),
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)?;
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// If `lsb` is 0, return `Acc + (-P)`. If `lsb` is 1, simply return `Acc + 0`.
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let x = if let Some(lsb) = lsb {
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if !lsb {
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base.x.value()
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} else {
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Some(pallas::Base::zero())
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}
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} else {
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None
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};
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let y = if let Some(lsb) = lsb {
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if !lsb {
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base.y.value().map(|y_p| -y_p)
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} else {
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Some(pallas::Base::zero())
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}
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} else {
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None
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};
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let x_cell = region.assign_advice(
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|| "x",
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self.add_config.x_p,
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offset,
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|| x.ok_or(Error::SynthesisError),
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)?;
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let y_cell = region.assign_advice(
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|| "y",
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self.add_config.y_p,
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offset,
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|| y.ok_or(Error::SynthesisError),
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)?;
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let p = EccPoint {
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x: CellValue::<pallas::Base>::new(x_cell, x),
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y: CellValue::<pallas::Base>::new(y_cell, y),
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};
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// Return the result of the final complete addition as `[scalar]B`
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let result = self.add_config.assign_region(&p, &acc, offset, region)?;
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Ok((result, z_0))
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}
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}
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#[derive(Clone, Debug)]
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// `x`-coordinate of the accumulator.
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struct X<F: FieldExt>(CellValue<F>);
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impl<F: FieldExt> Deref for X<F> {
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type Target = CellValue<F>;
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fn deref(&self) -> &Self::Target {
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&self.0
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}
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}
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#[derive(Copy, Clone, Debug)]
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// `y`-coordinate of the accumulator.
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struct Y<F: FieldExt>(CellValue<F>);
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impl<F: FieldExt> Deref for Y<F> {
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type Target = CellValue<F>;
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fn deref(&self) -> &Self::Target {
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&self.0
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}
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}
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#[derive(Copy, Clone, Debug)]
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// Cumulative sum `z` used to decompose the scalar.
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struct Z<F: FieldExt>(CellValue<F>);
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impl<F: FieldExt> Deref for Z<F> {
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type Target = CellValue<F>;
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fn deref(&self) -> &Self::Target {
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&self.0
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}
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}
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fn decompose_for_scalar_mul(scalar: Option<pallas::Base>) -> Vec<Option<bool>> {
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let bitstring = scalar.map(|scalar| {
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// We use `k = scalar + t_q` in the double-and-add algorithm, where
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// the scalar field `F_q = 2^254 + t_q`.
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// Note that the addition `scalar + t_q` is not reduced.
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//
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let scalar = U256::from_little_endian(&scalar.to_bytes());
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let t_q = U256::from_little_endian(&T_Q.to_le_bytes());
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let k = scalar + t_q;
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// Big-endian bit representation of `k`.
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let bitstring: Vec<bool> = {
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let mut le_bytes = [0u8; 32];
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k.to_little_endian(&mut le_bytes);
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le_bytes.iter().fold(Vec::new(), |mut bitstring, byte| {
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let bits = (0..8)
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.map(|shift| (byte >> shift) % 2 == 1)
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.collect::<Vec<_>>();
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bitstring.extend_from_slice(&bits);
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bitstring
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})
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};
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// Take the first 255 bits.
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let mut bitstring = bitstring[0..pallas::Scalar::NUM_BITS as usize].to_vec();
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bitstring.reverse();
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bitstring
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});
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if let Some(bitstring) = bitstring {
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bitstring.into_iter().map(Some).collect()
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} else {
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vec![None; pallas::Scalar::NUM_BITS as usize]
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}
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}
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#[cfg(feature = "testing")]
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pub mod tests {
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use group::{Curve, Group};
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use halo2::{
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circuit::{Chip, Layouter},
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plonk::Error,
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};
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use pasta_curves::{arithmetic::FieldExt, pallas};
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use crate::{
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chip::{EccChip, EccPoint},
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gadget::{EccInstructions, FixedPoints, NonIdentityPoint, Point},
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};
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use utilities::UtilitiesInstructions;
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pub fn test_mul<F: FixedPoints<pallas::Affine>>(
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chip: EccChip<F>,
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mut layouter: impl Layouter<pallas::Base>,
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) -> Result<(), Error> {
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// Generate a random point P
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let p_val = pallas::Point::random(rand::rngs::OsRng).to_affine(); // P
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let p = NonIdentityPoint::new(chip.clone(), layouter.namespace(|| "P"), Some(p_val))?;
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let column = chip.config().advices[0];
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fn constrain_equal_non_id<
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EccChip: EccInstructions<pallas::Affine, Point = EccPoint> + Clone + Eq + std::fmt::Debug,
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>(
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chip: EccChip,
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mut layouter: impl Layouter<pallas::Base>,
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base_val: pallas::Affine,
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scalar_val: pallas::Base,
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|
result: Point<pallas::Affine, EccChip>,
|
|
) -> Result<(), Error> {
|
|
// Move scalar from base field into scalar field (which always fits
|
|
// for Pallas).
|
|
let scalar = pallas::Scalar::from_bytes(&scalar_val.to_bytes()).unwrap();
|
|
let expected = NonIdentityPoint::new(
|
|
chip,
|
|
layouter.namespace(|| "expected point"),
|
|
Some((base_val * scalar).to_affine()),
|
|
)?;
|
|
result.constrain_equal(layouter.namespace(|| "constrain result"), &expected)
|
|
}
|
|
|
|
// [a]B
|
|
{
|
|
let scalar_val = pallas::Base::rand();
|
|
let (result, _) = {
|
|
let scalar = chip.load_private(
|
|
layouter.namespace(|| "random scalar"),
|
|
column,
|
|
Some(scalar_val),
|
|
)?;
|
|
p.mul(layouter.namespace(|| "random [a]B"), &scalar)?
|
|
};
|
|
constrain_equal_non_id(
|
|
chip.clone(),
|
|
layouter.namespace(|| "random [a]B"),
|
|
p_val,
|
|
scalar_val,
|
|
result,
|
|
)?;
|
|
}
|
|
|
|
// [0]B should return (0,0) since variable-base scalar multiplication
|
|
// uses complete addition for the final bits of the scalar.
|
|
{
|
|
let scalar_val = pallas::Base::zero();
|
|
let (result, _) = {
|
|
let scalar =
|
|
chip.load_private(layouter.namespace(|| "zero"), column, Some(scalar_val))?;
|
|
p.mul(layouter.namespace(|| "[0]B"), &scalar)?
|
|
};
|
|
assert!(result.inner().is_identity().unwrap());
|
|
}
|
|
|
|
// [-1]B (the largest possible base field element)
|
|
{
|
|
let scalar_val = -pallas::Base::one();
|
|
let (result, _) = {
|
|
let scalar =
|
|
chip.load_private(layouter.namespace(|| "-1"), column, Some(scalar_val))?;
|
|
p.mul(layouter.namespace(|| "[-1]B"), &scalar)?
|
|
};
|
|
constrain_equal_non_id(
|
|
chip,
|
|
layouter.namespace(|| "[-1]B"),
|
|
p_val,
|
|
scalar_val,
|
|
result,
|
|
)?;
|
|
}
|
|
|
|
Ok(())
|
|
}
|
|
}
|