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use core::{
cmp::PartialEq,
ops::{Add, Mul, Neg},
};
use subtle::{Choice, ConditionallySelectable, ConstantTimeEq};
use crate::{
constants::COMPRESSED_Y_LENGTH,
field::{FieldElement, FieldImplementation},
montgomery::MontgomeryPoint,
scalar::Scalar,
Error, Result,
};
/// These represent the (X,Y,Z,T) coordinates
#[derive(Clone, Copy, Debug, Default)]
pub struct EdwardsPoint(
// TODO: maybe label them properly
[FieldElement; 4],
);
/// "Compressed" form of a `EdwardsPoint`, whereby
/// the sign of the x-coordinate is stuffed in a
/// spare bit of the y-coordinate
#[derive(Clone, Copy, Debug, Default)]
pub struct CompressedY(pub [u8; 32]);
impl From<&[u8; COMPRESSED_Y_LENGTH]> for CompressedY {
fn from(bytes: &[u8; COMPRESSED_Y_LENGTH]) -> CompressedY {
CompressedY(*bytes)
}
}
impl CompressedY {
/// This is rather tricky: to get the x-coordinate,
/// and not just its sign, need to calculate the square
/// root of `u/v := (y**2 - 1)/(dy**2 + 1)`. Moreover, we want
/// to detect whether our compressed Y actually corresponds
/// to a point on the curve! The original sources are
/// [the Tweet NaCl paper, section 5](tweetnacl) and
/// the [ed25519 paper][ed25519], also section 5.
///
/// [tweetnacl]: http://tweetnacl.cr.yp.to/tweetnacl-20140917.pdf
/// [ed25519]: https://cryptojedi.org/papers/ed25519-20110926.pdf
pub fn decompressed(&self) -> Result<EdwardsPoint> {
#![allow(non_snake_case)]
// point = (X, Y, Z, T)
//
// basic strategy: use exponentiation by `2**252 - 3`,
// which "has all bits set except position 1".
// TODO: actually implement TryFrom
// let Y = FieldElement::try_from(self.as_bytes())?;
let Y = FieldElement::from_bytes_unchecked(self.as_bytes());
let Z = FieldElement::ONE;
let Y_squared = Y.squared();
let u = &Y_squared - &Z; // aka num[erator], y**2 - 1
let v = &(&Y_squared * &FieldElement::D) + &Z; // aka den[ominator], dy**2 + 1
let v2 = v.squared();
let v4 = v2.squared();
let v7 = &(&v4 * &v2) * &v;
let t = &v7 * &u; // term: t = uv**7
let mut X = &(&(&t.pow2523() * &u) * &v2) * &v; // aka `beta`
let chk = &X.squared() * &v;
if chk != u {
X = &X * &FieldElement::I;
}
let chk = &X.squared() * &v;
if chk != u {
return Err(Error::PublicKeyBytesInvalid);
}
// I really don't get it. TweetNaCl checks for equality.
// If we do that, our tests fail. This way, tests pass.
if X.parity() != (self.0[31] >> 7) {
X = -&X;
}
let T = &X * &Y;
Ok(EdwardsPoint([X, Y, Z, T]))
}
// static int unpackneg(gf r[4],const u8 p[32]) {
// // "load curve point"
// gf t, chk, num, den, den2, den4, den6;
// set25519(r[2],gf1); // Z = "one"
// unpack25519(r[1],p); // Y = compressed Y with x's sign bit erased
// S(num,r[1]);
// M(den,num,D);
// Z(num,num,r[2]);
// A(den,r[2],den); // set numerator, denominator as above
// S(den2,den);
// S(den4,den2);
// M(den6,den4,den2);
// M(t,den6,num);
// M(t,t,den); // set t = denominator**7 * numerator
// pow2523(t,t);
// M(t,t,num);
// M(t,t,den);
// M(t,t,den);
// M(r[0],t,den); // X = sqrt(t)*num*den**3
// S(chk,r[0]);
// M(chk,chk,den);
// if (neq25519(chk, num)) M(r[0],r[0],I);
// S(chk,r[0]);
// M(chk,chk,den);
// if (neq25519(chk, num)) return -1;
// if (par25519(r[0]) == (p[31]>>7)) Z(r[0],gf0,r[0]);
// M(r[3],r[0],r[1]);
// return 0;
// }
}
impl EdwardsPoint {
pub fn basepoint() -> EdwardsPoint {
EdwardsPoint([
FieldElement::EDWARDS_BASEPOINT_X,
FieldElement::EDWARDS_BASEPOINT_Y,
FieldElement::ONE,
&FieldElement::EDWARDS_BASEPOINT_X * &FieldElement::EDWARDS_BASEPOINT_Y,
])
}
pub fn neutral_element() -> EdwardsPoint {
EdwardsPoint([
FieldElement::ZERO,
FieldElement::ONE,
FieldElement::ONE,
FieldElement::ZERO,
])
}
pub fn compressed(&self) -> CompressedY {
// normalize X, Y to Z = 1
let z_inverse = &self.0[2].inverse();
let x = &self.0[0] * z_inverse;
let y = &self.0[1] * z_inverse;
// normalized Y coordinate
let mut r = y.to_bytes();
// slot sign of X in the "spare" top bit of last byte
// dalek calls this parity "is_negative"
r[31] ^= x.parity() << 7;
CompressedY(r)
}
/// Convert this `EdwardsPoint` on the Edwards model to the
/// corresponding `MontgomeryPoint` on the Montgomery model.
///
/// This function has one exceptional case; the identity point of
/// the Edwards curve is sent to the 2-torsion point \\((0,0)\\)
/// on the Montgomery curve.
///
/// Note that this is a one-way conversion, since the Montgomery
/// model does not retain sign information.
pub fn to_montgomery(&self) -> MontgomeryPoint {
//// We have u = (1+y)/(1-y) = (Z+Y)/(Z-Y).
////
//// The denominator is zero only when y=1, the identity point of
//// the Edwards curve. Since 0.invert() = 0, in this case we
//// compute the 2-torsion point (0,0).
//let U = &self.Z + &self.Y;
//let W = &self.Z - &self.Y;
//let u = &U * &W.invert();
//MontgomeryPoint(u.to_bytes())
MontgomeryPoint(self.u())
}
/// The x-coordinate of the point
pub fn x(&self) -> FieldElement {
let z_inverse = &self.0[2].inverse();
&self.0[0] * z_inverse
}
/// The y-coordinate of the point
pub fn y(&self) -> FieldElement {
let z_inverse = &self.0[2].inverse();
&self.0[1] * z_inverse
}
/// The u-coordinate of the X25519 point
pub fn u(&self) -> FieldElement {
let y = self.y();
let one = FieldElement::ONE;
&(&y + &one) * &(&one - &y).inverse()
}
}
impl<'a, 'b> Add<&'b EdwardsPoint> for &'a EdwardsPoint {
type Output = EdwardsPoint;
fn add(self, other: &'b EdwardsPoint) -> Self::Output {
let p = &self.0;
let q = &other.0;
let a = &p[1] - &p[0];
let t = &q[1] - &q[0];
let a = &a * &t; // A <- (Y1 - X1)(Y2 - X2)
// let mut b = &p[0] + &p[1];
let b = &p[0] + &p[1];
let t = &q[0] + &q[1];
let b = &b * &t; // B <- (Y1 + X1)*(Y2 + X2)
// b *= &t;
let c = &p[3] * &q[3];
let c = &c * &FieldElement::D2; // C <- k*T1*T2 with k = 2d' =
let d = &p[2] * &q[2];
let d = &d + &d; // D <- 2*Z1*Z2
let e = &b - &a;
let f = &d - &c;
let g = &d + &c;
let h = &b + &a;
let coordinates = [&e * &f, &h * &g, &g * &f, &e * &h];
EdwardsPoint(coordinates)
}
}
impl<'a> Neg for &'a EdwardsPoint {
type Output = EdwardsPoint;
fn neg(self) -> EdwardsPoint {
let p = &self.0;
EdwardsPoint([-&p[0], p[1], p[2], -&p[3]])
}
}
impl<'a, 'b> Mul<&'b EdwardsPoint> for &'a Scalar {
type Output = EdwardsPoint;
fn mul(self, point: &'b EdwardsPoint) -> EdwardsPoint {
let mut p = EdwardsPoint([
FieldElement::ZERO,
FieldElement::ONE,
FieldElement::ONE,
FieldElement::ZERO,
]);
let mut q = *point;
let scalar = self;
for i in (0..=255).rev() {
let b = ((scalar.0[i / 8] >> (i & 7)) & 1).into();
EdwardsPoint::conditional_swap(&mut p, &mut q, b);
q = &q + &p;
p = &p + &p;
EdwardsPoint::conditional_swap(&mut p, &mut q, b);
}
p
}
}
impl ConditionallySelectable for EdwardsPoint {
fn conditional_select(p: &Self, q: &Self, choice: Choice) -> Self {
let mut selection = Self::default();
for (i, (pi, qi)) in p.0.iter().zip(q.0.iter()).enumerate() {
selection.0[i] = FieldElement::conditional_select(pi, qi, choice);
}
selection
}
fn conditional_swap(p: &mut Self, q: &mut Self, choice: Choice) {
for (pi, qi) in p.0.iter_mut().zip(q.0.iter_mut()) {
FieldElement::conditional_swap(pi, qi, choice);
}
}
}
impl CompressedY {
pub fn as_bytes(&self) -> &[u8; 32] {
&self.0
}
pub fn to_bytes(&self) -> [u8; 32] {
self.0
}
}
impl ConstantTimeEq for CompressedY {
fn ct_eq(&self, other: &Self) -> Choice {
self.0.ct_eq(&other.0)
}
}
impl PartialEq for CompressedY {
fn eq(&self, other: &Self) -> bool {
bool::from(self.ct_eq(other))
}
}
impl ConstantTimeEq for EdwardsPoint {
fn ct_eq(&self, other: &Self) -> Choice {
let self_compressed = self.compressed();
let other_compressed = other.compressed();
self_compressed.ct_eq(&other_compressed)
}
}
impl PartialEq for EdwardsPoint {
fn eq(&self, other: &Self) -> bool {
bool::from(self.ct_eq(other))
}
}
#[cfg(test)]
mod tests {
use super::EdwardsPoint;
use crate::Scalar;
#[test]
fn test_neutral_is_neutral() {
let n = 42;
let s = Scalar::from_bytes(&[
n, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0,
]);
let ne = EdwardsPoint::neutral_element();
assert_eq!(ne, &s * &ne);
}
#[test]
fn test_addition_vs_multiplication() {
let p = EdwardsPoint::basepoint();
let p_plus_p = &p + &p;
let two = Scalar::from_bytes(&[
2u8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0,
]);
let two_times_p = &two * &p;
assert_eq!(p_plus_p, two_times_p);
}
#[test]
fn test_some_more() {
let n = 37;
let s = Scalar::from_bytes(&[
n, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0,
]);
let bp = EdwardsPoint::basepoint();
let ne = EdwardsPoint::neutral_element();
let a = (1..=n).fold(ne, |partial_sum, _| &partial_sum + &bp);
let b = &s * &bp;
assert_eq!(a, b);
}
#[test]
fn test_negation() {
let bp = EdwardsPoint::basepoint();
let minus_bp = -&bp;
let maybe_neutral = &bp + &minus_bp;
assert_eq!(maybe_neutral, EdwardsPoint::neutral_element());
}
#[test]
fn to_montgomery() {
let edwards_basepoint = EdwardsPoint::basepoint();
let montgomery_basepoint = crate::montgomery::MontgomeryPoint::basepoint();
assert_eq!(edwards_basepoint.to_montgomery(), montgomery_basepoint);
let scalar = Scalar::from(123456);
assert_eq!(
(&scalar * &edwards_basepoint).to_montgomery(),
&scalar * &montgomery_basepoint
);
}
}