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// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2021 isis lovecruft
// Copyright (c) 2016-2019 Henry de Valence
// See LICENSE for licensing information.
//
// Authors:
// - isis agora lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
//! Scalar multiplication on the Montgomery form of Curve25519.
//!
//! To avoid notational confusion with the Edwards code, we use
//! variables \\( u, v \\) for the Montgomery curve, so that “Montgomery
//! \\(u\\)” here corresponds to “Montgomery \\(x\\)” elsewhere.
//!
//! Montgomery arithmetic works not on the curve itself, but on the
//! \\(u\\)-line, which discards sign information and unifies the curve
//! and its quadratic twist. See [_Montgomery curves and their
//! arithmetic_][costello-smith] by Costello and Smith for more details.
//!
//! The `MontgomeryPoint` struct contains the affine \\(u\\)-coordinate
//! \\(u\_0(P)\\) of a point \\(P\\) on either the curve or the twist.
//! Here the map \\(u\_0 : \mathcal M \rightarrow \mathbb F\_p \\) is
//! defined by \\(u\_0((u,v)) = u\\); \\(u\_0(\mathcal O) = 0\\). See
//! section 5.4 of Costello-Smith for more details.
//!
//! # Scalar Multiplication
//!
//! Scalar multiplication on `MontgomeryPoint`s is provided by the `*`
//! operator, which implements the Montgomery ladder.
//!
//! # Edwards Conversion
//!
//! The \\(2\\)-to-\\(1\\) map from the Edwards model to the Montgomery
//! \\(u\\)-line is provided by `EdwardsPoint::to_montgomery()`.
//!
//! To lift a `MontgomeryPoint` to an `EdwardsPoint`, use
//! `MontgomeryPoint::to_edwards()`, which takes a sign parameter.
//! This function rejects `MontgomeryPoints` which correspond to points
//! on the twist.
//!
//! [costello-smith]: https://eprint.iacr.org/2017/212.pdf
// We allow non snake_case names because coordinates in projective space are
// traditionally denoted by the capitalisation of their respective
// counterparts in affine space. Yeah, you heard me, rustc, I'm gonna have my
// affine and projective cakes and eat both of them too.
#![allow(non_snake_case)]
use core::{
hash::{Hash, Hasher},
ops::{Mul, MulAssign},
};
use crate::constants::{APLUS2_OVER_FOUR, MONTGOMERY_A, MONTGOMERY_A_NEG};
use crate::edwards::{CompressedEdwardsY, EdwardsPoint};
use crate::field::FieldElement;
use crate::scalar::Scalar;
use crate::traits::Identity;
use subtle::Choice;
use subtle::ConstantTimeEq;
use subtle::{ConditionallyNegatable, ConditionallySelectable};
#[cfg(feature = "zeroize")]
use zeroize::Zeroize;
/// Holds the \\(u\\)-coordinate of a point on the Montgomery form of
/// Curve25519 or its twist.
#[derive(Copy, Clone, Debug, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct MontgomeryPoint(pub [u8; 32]);
/// Equality of `MontgomeryPoint`s is defined mod p.
impl ConstantTimeEq for MontgomeryPoint {
fn ct_eq(&self, other: &MontgomeryPoint) -> Choice {
let self_fe = FieldElement::from_bytes(&self.0);
let other_fe = FieldElement::from_bytes(&other.0);
self_fe.ct_eq(&other_fe)
}
}
impl PartialEq for MontgomeryPoint {
fn eq(&self, other: &MontgomeryPoint) -> bool {
self.ct_eq(other).unwrap_u8() == 1u8
}
}
impl Eq for MontgomeryPoint {}
// Equal MontgomeryPoints must hash to the same value. So we have to get them into a canonical
// encoding first
impl Hash for MontgomeryPoint {
fn hash<H: Hasher>(&self, state: &mut H) {
// Do a round trip through a `FieldElement`. `as_bytes` is guaranteed to give a canonical
// 32-byte encoding
let canonical_bytes = FieldElement::from_bytes(&self.0).as_bytes();
canonical_bytes.hash(state);
}
}
impl Identity for MontgomeryPoint {
/// Return the group identity element, which has order 4.
fn identity() -> MontgomeryPoint {
MontgomeryPoint([0u8; 32])
}
}
#[cfg(feature = "zeroize")]
impl Zeroize for MontgomeryPoint {
fn zeroize(&mut self) {
self.0.zeroize();
}
}
impl MontgomeryPoint {
/// View this `MontgomeryPoint` as an array of bytes.
pub const fn as_bytes(&self) -> &[u8; 32] {
&self.0
}
/// Convert this `MontgomeryPoint` to an array of bytes.
pub const fn to_bytes(&self) -> [u8; 32] {
self.0
}
/// Attempt to convert to an `EdwardsPoint`, using the supplied
/// choice of sign for the `EdwardsPoint`.
///
/// # Inputs
///
/// * `sign`: a `u8` donating the desired sign of the resulting
/// `EdwardsPoint`. `0` denotes positive and `1` negative.
///
/// # Return
///
/// * `Some(EdwardsPoint)` if `self` is the \\(u\\)-coordinate of a
/// point on (the Montgomery form of) Curve25519;
///
/// * `None` if `self` is the \\(u\\)-coordinate of a point on the
/// twist of (the Montgomery form of) Curve25519;
///
pub fn to_edwards(&self, sign: u8) -> Option<EdwardsPoint> {
// To decompress the Montgomery u coordinate to an
// `EdwardsPoint`, we apply the birational map to obtain the
// Edwards y coordinate, then do Edwards decompression.
//
// The birational map is y = (u-1)/(u+1).
//
// The exceptional points are the zeros of the denominator,
// i.e., u = -1.
//
// But when u = -1, v^2 = u*(u^2+486662*u+1) = 486660.
//
// Since this is nonsquare mod p, u = -1 corresponds to a point
// on the twist, not the curve, so we can reject it early.
let u = FieldElement::from_bytes(&self.0);
if u == FieldElement::MINUS_ONE {
return None;
}
let one = FieldElement::ONE;
let y = &(&u - &one) * &(&u + &one).invert();
let mut y_bytes = y.as_bytes();
y_bytes[31] ^= sign << 7;
CompressedEdwardsY(y_bytes).decompress()
}
}
/// Perform the Elligator2 mapping to a Montgomery point.
///
/// See <https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-10#section-6.7.1>
//
// TODO Determine how much of the hash-to-group API should be exposed after the CFRG
// draft gets into a more polished/accepted state.
#[allow(unused)]
pub(crate) fn elligator_encode(r_0: &FieldElement) -> MontgomeryPoint {
let one = FieldElement::ONE;
let d_1 = &one + &r_0.square2(); /* 2r^2 */
let d = &MONTGOMERY_A_NEG * &(d_1.invert()); /* A/(1+2r^2) */
let d_sq = &d.square();
let au = &MONTGOMERY_A * &d;
let inner = &(d_sq + &au) + &one;
let eps = &d * &inner; /* eps = d^3 + Ad^2 + d */
let (eps_is_sq, _eps) = FieldElement::sqrt_ratio_i(&eps, &one);
let zero = FieldElement::ZERO;
let Atemp = FieldElement::conditional_select(&MONTGOMERY_A, &zero, eps_is_sq); /* 0, or A if nonsquare*/
let mut u = &d + &Atemp; /* d, or d+A if nonsquare */
u.conditional_negate(!eps_is_sq); /* d, or -d-A if nonsquare */
MontgomeryPoint(u.as_bytes())
}
/// A `ProjectivePoint` holds a point on the projective line
/// \\( \mathbb P(\mathbb F\_p) \\), which we identify with the Kummer
/// line of the Montgomery curve.
#[derive(Copy, Clone, Debug)]
struct ProjectivePoint {
pub U: FieldElement,
pub W: FieldElement,
}
impl Identity for ProjectivePoint {
fn identity() -> ProjectivePoint {
ProjectivePoint {
U: FieldElement::ONE,
W: FieldElement::ZERO,
}
}
}
impl Default for ProjectivePoint {
fn default() -> ProjectivePoint {
ProjectivePoint::identity()
}
}
impl ConditionallySelectable for ProjectivePoint {
fn conditional_select(
a: &ProjectivePoint,
b: &ProjectivePoint,
choice: Choice,
) -> ProjectivePoint {
ProjectivePoint {
U: FieldElement::conditional_select(&a.U, &b.U, choice),
W: FieldElement::conditional_select(&a.W, &b.W, choice),
}
}
}
impl ProjectivePoint {
/// Dehomogenize this point to affine coordinates.
///
/// # Return
///
/// * \\( u = U / W \\) if \\( W \neq 0 \\);
/// * \\( 0 \\) if \\( W \eq 0 \\);
pub fn as_affine(&self) -> MontgomeryPoint {
let u = &self.U * &self.W.invert();
MontgomeryPoint(u.as_bytes())
}
}
/// Perform the double-and-add step of the Montgomery ladder.
///
/// Given projective points
/// \\( (U\_P : W\_P) = u(P) \\),
/// \\( (U\_Q : W\_Q) = u(Q) \\),
/// and the affine difference
/// \\( u\_{P-Q} = u(P-Q) \\), set
/// $$
/// (U\_P : W\_P) \gets u(\[2\]P)
/// $$
/// and
/// $$
/// (U\_Q : W\_Q) \gets u(P + Q).
/// $$
#[rustfmt::skip] // keep alignment of explanatory comments
fn differential_add_and_double(
P: &mut ProjectivePoint,
Q: &mut ProjectivePoint,
affine_PmQ: &FieldElement,
) {
let t0 = &P.U + &P.W;
let t1 = &P.U - &P.W;
let t2 = &Q.U + &Q.W;
let t3 = &Q.U - &Q.W;
let t4 = t0.square(); // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2
let t5 = t1.square(); // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2
let t6 = &t4 - &t5; // 4 U_P W_P
let t7 = &t0 * &t3; // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q
let t8 = &t1 * &t2; // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q
let t9 = &t7 + &t8; // 2 (U_P U_Q - W_P W_Q)
let t10 = &t7 - &t8; // 2 (W_P U_Q - U_P W_Q)
let t11 = t9.square(); // 4 (U_P U_Q - W_P W_Q)^2
let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2
let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q
let t14 = &t4 * &t5; // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2
let t15 = &t13 + &t5; // (U_P - W_P)^2 + (A + 2) U_P W_P
let t16 = &t6 * &t15; // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)
let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2
let t18 = t11; // W_D * 4 (U_P U_Q - W_P W_Q)^2
P.U = t14; // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2
P.W = t16; // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)
Q.U = t18; // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2
Q.W = t17; // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2
}
define_mul_assign_variants!(LHS = MontgomeryPoint, RHS = Scalar);
define_mul_variants!(
LHS = MontgomeryPoint,
RHS = Scalar,
Output = MontgomeryPoint
);
define_mul_variants!(
LHS = Scalar,
RHS = MontgomeryPoint,
Output = MontgomeryPoint
);
/// Multiply this `MontgomeryPoint` by a `Scalar`.
impl<'a, 'b> Mul<&'b Scalar> for &'a MontgomeryPoint {
type Output = MontgomeryPoint;
/// Given `self` \\( = u\_0(P) \\), and a `Scalar` \\(n\\), return \\( u\_0(\[n\]P) \\).
fn mul(self, scalar: &'b Scalar) -> MontgomeryPoint {
// Algorithm 8 of Costello-Smith 2017
let affine_u = FieldElement::from_bytes(&self.0);
let mut x0 = ProjectivePoint::identity();
let mut x1 = ProjectivePoint {
U: affine_u,
W: FieldElement::ONE,
};
// Go through the bits from most to least significant, using a sliding window of 2
let mut bits = scalar.bits_le().rev();
let mut prev_bit = bits.next().unwrap();
for cur_bit in bits {
let choice: u8 = (prev_bit ^ cur_bit) as u8;
debug_assert!(choice == 0 || choice == 1);
ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());
differential_add_and_double(&mut x0, &mut x1, &affine_u);
prev_bit = cur_bit;
}
// The final value of prev_bit above is scalar.bits()[0], i.e., the LSB of scalar
ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(prev_bit as u8));
// Don't leave the bit in the stack
#[cfg(feature = "zeroize")]
prev_bit.zeroize();
x0.as_affine()
}
}
impl<'b> MulAssign<&'b Scalar> for MontgomeryPoint {
fn mul_assign(&mut self, scalar: &'b Scalar) {
*self = (self as &MontgomeryPoint) * scalar;
}
}
impl<'a, 'b> Mul<&'b MontgomeryPoint> for &'a Scalar {
type Output = MontgomeryPoint;
fn mul(self, point: &'b MontgomeryPoint) -> MontgomeryPoint {
point * self
}
}
// ------------------------------------------------------------------------
// Tests
// ------------------------------------------------------------------------
#[cfg(test)]
mod test {
use super::*;
use crate::constants;
#[cfg(feature = "alloc")]
use alloc::vec::Vec;
#[cfg(feature = "rand_core")]
use rand_core::OsRng;
#[test]
fn identity_in_different_coordinates() {
let id_projective = ProjectivePoint::identity();
let id_montgomery = id_projective.as_affine();
assert!(id_montgomery == MontgomeryPoint::identity());
}
#[test]
fn identity_in_different_models() {
assert!(EdwardsPoint::identity().to_montgomery() == MontgomeryPoint::identity());
}
#[test]
#[cfg(feature = "serde")]
fn serde_bincode_basepoint_roundtrip() {
use bincode;
let encoded = bincode::serialize(&constants::X25519_BASEPOINT).unwrap();
let decoded: MontgomeryPoint = bincode::deserialize(&encoded).unwrap();
assert_eq!(encoded.len(), 32);
assert_eq!(decoded, constants::X25519_BASEPOINT);
let raw_bytes = constants::X25519_BASEPOINT.as_bytes();
let bp: MontgomeryPoint = bincode::deserialize(raw_bytes).unwrap();
assert_eq!(bp, constants::X25519_BASEPOINT);
}
/// Test Montgomery -> Edwards on the X/Ed25519 basepoint
#[test]
fn basepoint_montgomery_to_edwards() {
// sign bit = 0 => basepoint
assert_eq!(
constants::ED25519_BASEPOINT_POINT,
constants::X25519_BASEPOINT.to_edwards(0).unwrap()
);
// sign bit = 1 => minus basepoint
assert_eq!(
-constants::ED25519_BASEPOINT_POINT,
constants::X25519_BASEPOINT.to_edwards(1).unwrap()
);
}
/// Test Edwards -> Montgomery on the X/Ed25519 basepoint
#[test]
fn basepoint_edwards_to_montgomery() {
assert_eq!(
constants::ED25519_BASEPOINT_POINT.to_montgomery(),
constants::X25519_BASEPOINT
);
}
/// Check that Montgomery -> Edwards fails for points on the twist.
#[test]
fn montgomery_to_edwards_rejects_twist() {
let one = FieldElement::ONE;
// u = 2 corresponds to a point on the twist.
let two = MontgomeryPoint((&one + &one).as_bytes());
assert!(two.to_edwards(0).is_none());
// u = -1 corresponds to a point on the twist, but should be
// checked explicitly because it's an exceptional point for the
// birational map. For instance, libsignal will accept it.
let minus_one = MontgomeryPoint((-&one).as_bytes());
assert!(minus_one.to_edwards(0).is_none());
}
#[test]
fn eq_defined_mod_p() {
let mut u18_bytes = [0u8; 32];
u18_bytes[0] = 18;
let u18 = MontgomeryPoint(u18_bytes);
let u18_unred = MontgomeryPoint([255; 32]);
assert_eq!(u18, u18_unred);
}
#[test]
#[cfg(feature = "rand_core")]
fn montgomery_ladder_matches_edwards_scalarmult() {
let mut csprng: OsRng = OsRng;
let s: Scalar = Scalar::random(&mut csprng);
let p_edwards = EdwardsPoint::mul_base(&s);
let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();
let expected = s * p_edwards;
let result = s * p_montgomery;
assert_eq!(result, expected.to_montgomery())
}
#[cfg(feature = "alloc")]
const ELLIGATOR_CORRECT_OUTPUT: [u8; 32] = [
0x5f, 0x35, 0x20, 0x00, 0x1c, 0x6c, 0x99, 0x36, 0xa3, 0x12, 0x06, 0xaf, 0xe7, 0xc7, 0xac,
0x22, 0x4e, 0x88, 0x61, 0x61, 0x9b, 0xf9, 0x88, 0x72, 0x44, 0x49, 0x15, 0x89, 0x9d, 0x95,
0xf4, 0x6e,
];
#[test]
#[cfg(feature = "alloc")]
fn montgomery_elligator_correct() {
let bytes: Vec<u8> = (0u8..32u8).collect();
let bits_in: [u8; 32] = (&bytes[..]).try_into().expect("Range invariant broken");
let fe = FieldElement::from_bytes(&bits_in);
let eg = elligator_encode(&fe);
assert_eq!(eg.to_bytes(), ELLIGATOR_CORRECT_OUTPUT);
}
#[test]
fn montgomery_elligator_zero_zero() {
let zero = [0u8; 32];
let fe = FieldElement::from_bytes(&zero);
let eg = elligator_encode(&fe);
assert_eq!(eg.to_bytes(), zero);
}
}