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use alloc::vec::Vec;
use core::iter;
use core::marker::PhantomData;
use core::ops::Mul;
use ff::PrimeField;
use super::Group;
/// Extension trait on a [`Group`] that provides helpers used by [`Wnaf`].
pub trait WnafGroup: Group {
/// Recommends a wNAF window size given the number of scalars you intend to multiply
/// a base by. Always returns a number between 2 and 22, inclusive.
fn recommended_wnaf_for_num_scalars(num_scalars: usize) -> usize;
}
/// Replaces the contents of `table` with a w-NAF window table for the given window size.
pub(crate) fn wnaf_table<G: Group>(table: &mut Vec<G>, mut base: G, window: usize) {
table.truncate(0);
table.reserve(1 << (window - 1));
let dbl = base.double();
for _ in 0..(1 << (window - 1)) {
table.push(base);
base.add_assign(&dbl);
}
}
/// This struct represents a view of a sequence of bytes as a sequence of
/// `u64` limbs in little-endian byte order. It maintains a current index, and
/// allows access to the limb at that index and the one following it. Bytes
/// beyond the end of the original buffer are treated as zero.
struct LimbBuffer<'a> {
buf: &'a [u8],
cur_idx: usize,
cur_limb: u64,
next_limb: u64,
}
impl<'a> LimbBuffer<'a> {
fn new(buf: &'a [u8]) -> Self {
let mut ret = Self {
buf,
cur_idx: 0,
cur_limb: 0,
next_limb: 0,
};
// Initialise the limb buffers.
ret.increment_limb();
ret.increment_limb();
ret.cur_idx = 0usize;
ret
}
fn increment_limb(&mut self) {
self.cur_idx += 1;
self.cur_limb = self.next_limb;
match self.buf.len() {
// There are no more bytes in the buffer; zero-extend.
0 => self.next_limb = 0,
// There are fewer bytes in the buffer than a u64 limb; zero-extend.
x @ 1..=7 => {
let mut next_limb = [0; 8];
next_limb[..x].copy_from_slice(self.buf);
self.next_limb = u64::from_le_bytes(next_limb);
self.buf = &[];
}
// There are at least eight bytes in the buffer; read the next u64 limb.
_ => {
let (next_limb, rest) = self.buf.split_at(8);
self.next_limb = u64::from_le_bytes(next_limb.try_into().unwrap());
self.buf = rest;
}
}
}
fn get(&mut self, idx: usize) -> (u64, u64) {
assert!([self.cur_idx, self.cur_idx + 1].contains(&idx));
if idx > self.cur_idx {
self.increment_limb();
}
(self.cur_limb, self.next_limb)
}
}
/// Replaces the contents of `wnaf` with the w-NAF representation of a little-endian
/// scalar.
pub(crate) fn wnaf_form<S: AsRef<[u8]>>(wnaf: &mut Vec<i64>, c: S, window: usize) {
// Required by the NAF definition
debug_assert!(window >= 2);
// Required so that the NAF digits fit in i64
debug_assert!(window <= 64);
let bit_len = c.as_ref().len() * 8;
wnaf.truncate(0);
wnaf.reserve(bit_len);
// Initialise the current and next limb buffers.
let mut limbs = LimbBuffer::new(c.as_ref());
let width = 1u64 << window;
let window_mask = width - 1;
let mut pos = 0;
let mut carry = 0;
while pos < bit_len {
// Construct a buffer of bits of the scalar, starting at bit `pos`
let u64_idx = pos / 64;
let bit_idx = pos % 64;
let (cur_u64, next_u64) = limbs.get(u64_idx);
let bit_buf = if bit_idx + window < 64 {
// This window's bits are contained in a single u64
cur_u64 >> bit_idx
} else {
// Combine the current u64's bits with the bits from the next u64
(cur_u64 >> bit_idx) | (next_u64 << (64 - bit_idx))
};
// Add the carry into the current window
let window_val = carry + (bit_buf & window_mask);
if window_val & 1 == 0 {
// If the window value is even, preserve the carry and emit 0.
// Why is the carry preserved?
// If carry == 0 and window_val & 1 == 0, then the next carry should be 0
// If carry == 1 and window_val & 1 == 0, then bit_buf & 1 == 1 so the next carry should be 1
wnaf.push(0);
pos += 1;
} else {
wnaf.push(if window_val < width / 2 {
carry = 0;
window_val as i64
} else {
carry = 1;
(window_val as i64).wrapping_sub(width as i64)
});
wnaf.extend(iter::repeat(0).take(window - 1));
pos += window;
}
}
}
/// Performs w-NAF exponentiation with the provided window table and w-NAF form scalar.
///
/// This function must be provided a `table` and `wnaf` that were constructed with
/// the same window size; otherwise, it may panic or produce invalid results.
pub(crate) fn wnaf_exp<G: Group>(table: &[G], wnaf: &[i64]) -> G {
let mut result = G::identity();
let mut found_one = false;
for n in wnaf.iter().rev() {
if found_one {
result = result.double();
}
if *n != 0 {
found_one = true;
if *n > 0 {
result += &table[(n / 2) as usize];
} else {
result -= &table[((-n) / 2) as usize];
}
}
}
result
}
/// A "w-ary non-adjacent form" scalar multiplication (also known as exponentiation)
/// context.
///
/// # Examples
///
/// This struct can be used to implement several patterns:
///
/// ## One base, one scalar
///
/// For this pattern, you can use a transient `Wnaf` context:
///
/// ```ignore
/// use group::Wnaf;
///
/// let result = Wnaf::new().scalar(&scalar).base(base);
/// ```
///
/// ## Many bases, one scalar
///
/// For this pattern, you create a `Wnaf` context, load the scalar into it, and then
/// process each base in turn:
///
/// ```ignore
/// use group::Wnaf;
///
/// let mut wnaf = Wnaf::new();
/// let mut wnaf_scalar = wnaf.scalar(&scalar);
/// let results: Vec<_> = bases
/// .into_iter()
/// .map(|base| wnaf_scalar.base(base))
/// .collect();
/// ```
///
/// ## One base, many scalars
///
/// For this pattern, you create a `Wnaf` context, load the base into it, and then process
/// each scalar in turn:
///
/// ```ignore
/// use group::Wnaf;
///
/// let mut wnaf = Wnaf::new();
/// let mut wnaf_base = wnaf.base(base, scalars.len());
/// let results: Vec<_> = scalars
/// .iter()
/// .map(|scalar| wnaf_base.scalar(scalar))
/// .collect();
/// ```
///
/// ## Many bases, many scalars
///
/// Say you have `n` bases and `m` scalars, and want to produce `n * m` results. For this
/// pattern, you need to cache the w-NAF tables for the bases and then compute the w-NAF
/// form of the scalars on the fly for every base, or vice versa:
///
/// ```ignore
/// use group::Wnaf;
///
/// let mut wnaf_contexts: Vec<_> = (0..bases.len()).map(|_| Wnaf::new()).collect();
/// let mut wnaf_bases: Vec<_> = wnaf_contexts
/// .iter_mut()
/// .zip(bases)
/// .map(|(wnaf, base)| wnaf.base(base, scalars.len()))
/// .collect();
/// let results: Vec<_> = wnaf_bases
/// .iter()
/// .flat_map(|wnaf_base| scalars.iter().map(|scalar| wnaf_base.scalar(scalar)))
/// .collect();
/// ```
///
/// Alternatively, use the [`WnafBase`] and [`WnafScalar`] types, which enable the various
/// tables and w-NAF forms to be cached individually per base and scalar. These types can
/// then be directly multiplied without any additional runtime work, at the cost of fixing
/// a specific window size (rather than choosing the window size dynamically).
#[derive(Debug)]
pub struct Wnaf<W, B, S> {
base: B,
scalar: S,
window_size: W,
}
impl<G: Group> Wnaf<(), Vec<G>, Vec<i64>> {
/// Construct a new wNAF context without allocating.
pub fn new() -> Self {
Wnaf {
base: vec![],
scalar: vec![],
window_size: (),
}
}
}
#[cfg(feature = "wnaf-memuse")]
impl<G: Group + memuse::DynamicUsage> memuse::DynamicUsage for Wnaf<(), Vec<G>, Vec<i64>> {
fn dynamic_usage(&self) -> usize {
self.base.dynamic_usage() + self.scalar.dynamic_usage()
}
fn dynamic_usage_bounds(&self) -> (usize, Option<usize>) {
let (base_lower, base_upper) = self.base.dynamic_usage_bounds();
let (scalar_lower, scalar_upper) = self.scalar.dynamic_usage_bounds();
(
base_lower + scalar_lower,
base_upper.zip(scalar_upper).map(|(a, b)| a + b),
)
}
}
impl<G: WnafGroup> Wnaf<(), Vec<G>, Vec<i64>> {
/// Given a base and a number of scalars, compute a window table and return a `Wnaf` object that
/// can perform exponentiations with `.scalar(..)`.
pub fn base(&mut self, base: G, num_scalars: usize) -> Wnaf<usize, &[G], &mut Vec<i64>> {
// Compute the appropriate window size based on the number of scalars.
let window_size = G::recommended_wnaf_for_num_scalars(num_scalars);
// Compute a wNAF table for the provided base and window size.
wnaf_table(&mut self.base, base, window_size);
// Return a Wnaf object that immutably borrows the computed base storage location,
// but mutably borrows the scalar storage location.
Wnaf {
base: &self.base[..],
scalar: &mut self.scalar,
window_size,
}
}
/// Given a scalar, compute its wNAF representation and return a `Wnaf` object that can perform
/// exponentiations with `.base(..)`.
pub fn scalar(&mut self, scalar: &<G as Group>::Scalar) -> Wnaf<usize, &mut Vec<G>, &[i64]> {
// We hard-code a window size of 4.
let window_size = 4;
// Compute the wNAF form of the scalar.
wnaf_form(&mut self.scalar, scalar.to_repr(), window_size);
// Return a Wnaf object that mutably borrows the base storage location, but
// immutably borrows the computed wNAF form scalar location.
Wnaf {
base: &mut self.base,
scalar: &self.scalar[..],
window_size,
}
}
}
impl<'a, G: Group> Wnaf<usize, &'a [G], &'a mut Vec<i64>> {
/// Constructs new space for the scalar representation while borrowing
/// the computed window table, for sending the window table across threads.
pub fn shared(&self) -> Wnaf<usize, &'a [G], Vec<i64>> {
Wnaf {
base: self.base,
scalar: vec![],
window_size: self.window_size,
}
}
}
#[cfg(feature = "wnaf-memuse")]
impl<'a, G: Group> memuse::DynamicUsage for Wnaf<usize, &'a [G], Vec<i64>> {
fn dynamic_usage(&self) -> usize {
// The heap memory for the window table is counted in the parent `Wnaf`.
self.scalar.dynamic_usage()
}
fn dynamic_usage_bounds(&self) -> (usize, Option<usize>) {
self.scalar.dynamic_usage_bounds()
}
}
impl<'a, G: Group> Wnaf<usize, &'a mut Vec<G>, &'a [i64]> {
/// Constructs new space for the window table while borrowing
/// the computed scalar representation, for sending the scalar representation
/// across threads.
pub fn shared(&self) -> Wnaf<usize, Vec<G>, &'a [i64]> {
Wnaf {
base: vec![],
scalar: self.scalar,
window_size: self.window_size,
}
}
}
#[cfg(feature = "wnaf-memuse")]
impl<'a, G: Group + memuse::DynamicUsage> memuse::DynamicUsage for Wnaf<usize, Vec<G>, &'a [i64]> {
fn dynamic_usage(&self) -> usize {
// The heap memory for the scalar representation is counted in the parent `Wnaf`.
self.base.dynamic_usage()
}
fn dynamic_usage_bounds(&self) -> (usize, Option<usize>) {
self.base.dynamic_usage_bounds()
}
}
impl<B, S: AsRef<[i64]>> Wnaf<usize, B, S> {
/// Performs exponentiation given a base.
pub fn base<G: Group>(&mut self, base: G) -> G
where
B: AsMut<Vec<G>>,
{
wnaf_table(self.base.as_mut(), base, self.window_size);
wnaf_exp(self.base.as_mut(), self.scalar.as_ref())
}
}
impl<B, S: AsMut<Vec<i64>>> Wnaf<usize, B, S> {
/// Performs exponentiation given a scalar.
pub fn scalar<G: Group>(&mut self, scalar: &<G as Group>::Scalar) -> G
where
B: AsRef<[G]>,
{
wnaf_form(self.scalar.as_mut(), scalar.to_repr(), self.window_size);
wnaf_exp(self.base.as_ref(), self.scalar.as_mut())
}
}
/// A "w-ary non-adjacent form" scalar, that uses precomputation to improve the speed of
/// scalar multiplication.
///
/// # Examples
///
/// See [`WnafBase`] for usage examples.
#[derive(Clone, Debug)]
pub struct WnafScalar<F: PrimeField, const WINDOW_SIZE: usize> {
wnaf: Vec<i64>,
field: PhantomData<F>,
}
#[cfg(feature = "wnaf-memuse")]
impl<F: PrimeField, const WINDOW_SIZE: usize> memuse::DynamicUsage for WnafScalar<F, WINDOW_SIZE> {
fn dynamic_usage(&self) -> usize {
self.wnaf.dynamic_usage()
}
fn dynamic_usage_bounds(&self) -> (usize, Option<usize>) {
self.wnaf.dynamic_usage_bounds()
}
}
impl<F: PrimeField, const WINDOW_SIZE: usize> WnafScalar<F, WINDOW_SIZE> {
/// Computes the w-NAF representation of the given scalar with the specified
/// `WINDOW_SIZE`.
pub fn new(scalar: &F) -> Self {
let mut wnaf = vec![];
// Compute the w-NAF form of the scalar.
wnaf_form(&mut wnaf, scalar.to_repr(), WINDOW_SIZE);
WnafScalar {
wnaf,
field: PhantomData::default(),
}
}
}
/// A fixed window table for a group element, precomputed to improve the speed of scalar
/// multiplication.
///
/// This struct is designed for usage patterns that have long-term cached bases and/or
/// scalars, or [Cartesian products] of bases and scalars. The [`Wnaf`] API enables one or
/// the other to be cached, but requires either the base window tables or the scalar w-NAF
/// forms to be computed repeatedly on the fly, which can become a significant performance
/// issue for some use cases.
///
/// `WnafBase` and [`WnafScalar`] enable an alternative trade-off: by fixing the window
/// size at compile time, the precomputations are guaranteed to only occur once per base
/// and once per scalar. Users should select their window size based on how long the bases
/// are expected to live; a larger window size will consume more memory and take longer to
/// precompute, but result in faster scalar multiplications.
///
/// [Cartesian products]: https://en.wikipedia.org/wiki/Cartesian_product
///
/// # Examples
///
/// ```ignore
/// use group::{WnafBase, WnafScalar};
///
/// let wnaf_bases: Vec<_> = bases.into_iter().map(WnafBase::<_, 4>::new).collect();
/// let wnaf_scalars: Vec<_> = scalars.iter().map(WnafScalar::new).collect();
/// let results: Vec<_> = wnaf_bases
/// .iter()
/// .flat_map(|base| wnaf_scalars.iter().map(|scalar| base * scalar))
/// .collect();
/// ```
///
/// Note that this pattern requires specifying a fixed window size (unlike previous
/// patterns that picked a suitable window size internally). This is necessary to ensure
/// in the type system that the base and scalar `Wnaf`s were computed with the same window
/// size, allowing the result to be computed infallibly.
#[derive(Clone, Debug)]
pub struct WnafBase<G: Group, const WINDOW_SIZE: usize> {
table: Vec<G>,
}
#[cfg(feature = "wnaf-memuse")]
impl<G: Group + memuse::DynamicUsage, const WINDOW_SIZE: usize> memuse::DynamicUsage
for WnafBase<G, WINDOW_SIZE>
{
fn dynamic_usage(&self) -> usize {
self.table.dynamic_usage()
}
fn dynamic_usage_bounds(&self) -> (usize, Option<usize>) {
self.table.dynamic_usage_bounds()
}
}
impl<G: Group, const WINDOW_SIZE: usize> WnafBase<G, WINDOW_SIZE> {
/// Computes a window table for the given base with the specified `WINDOW_SIZE`.
pub fn new(base: G) -> Self {
let mut table = vec![];
// Compute a window table for the provided base and window size.
wnaf_table(&mut table, base, WINDOW_SIZE);
WnafBase { table }
}
}
impl<G: Group, const WINDOW_SIZE: usize> Mul<&WnafScalar<G::Scalar, WINDOW_SIZE>>
for &WnafBase<G, WINDOW_SIZE>
{
type Output = G;
fn mul(self, rhs: &WnafScalar<G::Scalar, WINDOW_SIZE>) -> Self::Output {
wnaf_exp(&self.table, &rhs.wnaf)
}
}