Files
voprf-vx/src/group/p256.rs
T
2021-12-23 15:58:00 -05:00

587 lines
21 KiB
Rust

// Copyright (c) Facebook, Inc. and its affiliates.
//
// This source code is licensed under both the MIT license found in the
// LICENSE-MIT file in the root directory of this source tree and the Apache
// License, Version 2.0 found in the LICENSE-APACHE file in the root directory
// of this source tree.
// Note: This group implementation of p256 is experimental for now, until
// hash-to-curve or crypto-bigint are fully supported.
#![allow(
clippy::borrow_interior_mutable_const,
clippy::declare_interior_mutable_const
)]
use core::ops::{Add, Div, Mul, Neg};
use core::str::FromStr;
use digest::core_api::BlockSizeUser;
use digest::{Digest, FixedOutputReset};
use generic_array::typenum::{Unsigned, U1, U2, U32, U33, U48};
use generic_array::{ArrayLength, GenericArray};
use num_bigint::{BigInt, Sign};
use num_integer::Integer;
use num_traits::{One, ToPrimitive, Zero};
use once_cell::unsync::Lazy;
use p256_::elliptic_curve::group::prime::PrimeCurveAffine;
use p256_::elliptic_curve::group::GroupEncoding;
use p256_::elliptic_curve::sec1::{FromEncodedPoint, ToEncodedPoint};
use p256_::elliptic_curve::Field;
use p256_::{AffinePoint, EncodedPoint, ProjectivePoint};
use rand_core::{CryptoRng, RngCore};
use subtle::{Choice, ConditionallySelectable};
use super::Group;
use crate::errors::InternalError;
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-8.2
// `L: 48`
pub type L = U48;
#[cfg(feature = "p256")]
impl Group for ProjectivePoint {
const SUITE_ID: usize = 0x0003;
// Implements the `hash_to_curve()` function from
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-3
fn hash_to_curve<H: BlockSizeUser + Digest + FixedOutputReset, D: ArrayLength<u8> + Add<U1>>(
msg: &[u8],
dst: GenericArray<u8, D>,
) -> Result<Self, InternalError>
where
<D as Add<U1>>::Output: ArrayLength<u8>,
{
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-8.2
// `p: 2^256 - 2^224 + 2^192 + 2^96 - 1`
const P: Lazy<BigInt> = Lazy::new(|| {
BigInt::from_str(
"115792089210356248762697446949407573530086143415290314195533631308867097853951",
)
.unwrap()
});
// `A: -3`
const A: Lazy<BigInt> = Lazy::new(|| BigInt::from(-3));
// `B: 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b`
const B: Lazy<BigInt> = Lazy::new(|| {
BigInt::parse_bytes(
b"5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b",
16,
)
.unwrap()
});
// `Z: -10`
const Z: Lazy<BigInt> = Lazy::new(|| BigInt::from(-10));
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-3
// `hash_to_curve` calls `hash_to_field` with a `count` of `2`
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-5.3
// `hash_to_field` calls `expand_message` with a `len_in_bytes` of `count * L`
let uniform_bytes =
super::expand::expand_message_xmd::<H, <L as Mul<U2>>::Output, _, _>(Some(msg), dst)?;
// hash to curve
let (q0x, q0y) = hash_to_curve_simple_swu(&uniform_bytes[..L::USIZE], &A, &B, &P, &Z);
let (q1x, q1y) = hash_to_curve_simple_swu(&uniform_bytes[L::USIZE..], &A, &B, &P, &Z);
// convert to `p256` types
let p0 = AffinePoint::from_encoded_point(&EncodedPoint::from_affine_coordinates(
&q0x, &q0y, false,
))
.ok_or(InternalError::PointError)?
.to_curve();
let p1 = AffinePoint::from_encoded_point(&EncodedPoint::from_affine_coordinates(
&q1x, &q1y, false,
))
.ok_or(InternalError::PointError)?;
Ok(p0 + p1)
}
// Implements the `HashToScalar()` function
fn hash_to_scalar<
'a,
H: BlockSizeUser + Digest + FixedOutputReset,
D: ArrayLength<u8> + Add<U1>,
I: IntoIterator<Item = &'a [u8]>,
>(
input: I,
dst: GenericArray<u8, D>,
) -> Result<Self::Scalar, InternalError>
where
<D as Add<U1>>::Output: ArrayLength<u8>,
{
// https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf#[{%22num%22:211,%22gen%22:0},{%22name%22:%22XYZ%22},70,700,0]
// P-256 `n` is defined as
// `115792089210356248762697446949407573529996955224135760342
// 422259061068512044369`
const N: Lazy<BigInt> = Lazy::new(|| {
BigInt::from_str(
"115792089210356248762697446949407573529996955224135760342422259061068512044369",
)
.unwrap()
});
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-5.3
// `HashToScalar` is `hash_to_field`
let uniform_bytes = super::expand::expand_message_xmd::<H, L, _, _>(input, dst)?;
let bytes = BigInt::from_bytes_be(Sign::Plus, &uniform_bytes)
.mod_floor(&N)
.to_bytes_be()
.1;
let mut result = GenericArray::default();
result[..bytes.len()].copy_from_slice(&bytes);
Ok(p256_::Scalar::from_bytes_reduced(&result))
}
type ElemLen = U33;
type Scalar = p256_::Scalar;
type ScalarLen = U32;
fn from_scalar_slice_unchecked(
scalar_bits: &GenericArray<u8, Self::ScalarLen>,
) -> Result<Self::Scalar, InternalError> {
Ok(Self::Scalar::from_bytes_reduced(scalar_bits))
}
fn random_nonzero_scalar<R: RngCore + CryptoRng>(rng: &mut R) -> Self::Scalar {
Self::Scalar::random(rng)
}
fn scalar_as_bytes(scalar: Self::Scalar) -> GenericArray<u8, Self::ScalarLen> {
scalar.into()
}
fn scalar_invert(scalar: &Self::Scalar) -> Self::Scalar {
scalar.invert().unwrap_or(Self::Scalar::zero())
}
fn from_element_slice_unchecked(
element_bits: &GenericArray<u8, Self::ElemLen>,
) -> Result<Self, InternalError> {
Option::from(Self::from_bytes(element_bits)).ok_or(InternalError::PointError)
}
fn to_arr(&self) -> GenericArray<u8, Self::ElemLen> {
let bytes = self.to_affine().to_encoded_point(true);
let bytes = bytes.as_bytes();
let mut result = GenericArray::default();
result[..bytes.len()].copy_from_slice(bytes);
result
}
fn base_point() -> Self {
Self::generator()
}
fn identity() -> Self {
Self::identity()
}
fn scalar_zero() -> Self::Scalar {
Self::Scalar::zero()
}
}
/// Corresponds to the hash_to_curve_simple_swu() function defined in
/// <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#appendix-F.2>
///
/// `cmov`, `mod_floor` and `modpow` needs to be made constant-time, which will
/// be supported after crypto-bigint is no longer experimental. See
/// <https://github.com/novifinancial/voprf/issues/13> for more context.
#[allow(clippy::many_single_char_names)]
fn hash_to_curve_simple_swu<N: ArrayLength<u8>>(
u: &[u8],
a: &BigInt,
b: &BigInt,
p: &BigInt,
z: &BigInt,
) -> (GenericArray<u8, N>, GenericArray<u8, N>) {
#[derive(Clone)]
struct Field<'a>(&'a BigInt);
impl<'a> Field<'a> {
fn new(p: &'a BigInt) -> Self {
Self(p)
}
fn element(&'a self, number: &BigInt) -> FieldElement<'a> {
FieldElement {
number: number.mod_floor(self.0),
f: self,
}
}
fn one(&'a self) -> FieldElement<'a> {
self.element(&BigInt::one())
}
}
/// Finite field arithmetic
#[derive(Clone)]
struct FieldElement<'a> {
number: BigInt,
f: &'a Field<'a>,
}
impl<'a> Add for FieldElement<'a> {
type Output = FieldElement<'a>;
fn add(self, rhs: Self) -> Self::Output {
&self + &rhs
}
}
impl<'a> Add for &FieldElement<'a> {
type Output = FieldElement<'a>;
fn add(self, rhs: Self) -> Self::Output {
self.f.element(&(&self.number + &rhs.number))
}
}
impl<'a> Neg for FieldElement<'a> {
type Output = FieldElement<'a>;
fn neg(self) -> Self::Output {
-&self
}
}
impl<'a> Neg for &FieldElement<'a> {
type Output = FieldElement<'a>;
fn neg(self) -> Self::Output {
self.f.element(&-&self.number)
}
}
impl<'a> Mul for FieldElement<'a> {
type Output = FieldElement<'a>;
fn mul(self, rhs: Self) -> Self::Output {
&self * &rhs
}
}
impl<'a> Mul<&Self> for FieldElement<'a> {
type Output = FieldElement<'a>;
fn mul(self, rhs: &Self) -> Self::Output {
&self * rhs
}
}
impl<'a> Mul<FieldElement<'a>> for &FieldElement<'a> {
type Output = FieldElement<'a>;
fn mul(self, rhs: FieldElement<'a>) -> Self::Output {
self * &rhs
}
}
impl<'a> Mul for &FieldElement<'a> {
type Output = FieldElement<'a>;
fn mul(self, rhs: Self) -> Self::Output {
self.f.element(&(&self.number * &rhs.number))
}
}
impl<'a> Div<&Self> for FieldElement<'a> {
type Output = FieldElement<'a>;
#[allow(clippy::suspicious_arithmetic_impl)]
fn div(self, rhs: &Self) -> Self::Output {
self * rhs.inv0()
}
}
impl<'a> FieldElement<'a> {
fn square(&self) -> Self {
self * self
}
fn pow_internal(&self, exponent: &BigInt) -> Self {
let exponent = exponent.mod_floor(&(self.f.0 - 1));
Self {
number: self.number.modpow(&exponent, self.f.0),
f: self.f,
}
}
/// Corresponds to the sqrt_3mod4() function defined in
/// <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#appendix-I.1>
fn sqrt(&self) -> Self {
// constant
let c1 = (self.f.0 + 1) >> 2;
self.pow_internal(&c1)
}
/// Corresponds to the sgn0_m_eq_1() function defined in
/// <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-4.1>
fn sgn0(&self) -> i32 {
(&self.number % 2_usize).to_i32().unwrap()
}
/// See <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-4>
fn inv0(&self) -> Self {
self.pow_internal(&(self.f.0 - 2))
}
fn is_zero(&self) -> bool {
self.number.is_zero()
}
/// Corresponds to the is_square() function defined in
/// <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-4>
fn is_square(&self) -> bool {
// constant
let exponent = (self.f.0 - 1) >> 1;
let result = self.pow_internal(&exponent);
result.is_zero() || result.number.is_one()
}
fn to_bytes<N: ArrayLength<u8>>(&self) -> GenericArray<u8, N> {
let bytes = self.number.to_bytes_be().1;
let mut result = GenericArray::default();
result[N::USIZE - bytes.len()..].copy_from_slice(&bytes);
result
}
}
fn cmov<'a>(x: &FieldElement<'a>, y: &FieldElement<'a>, b: bool) -> FieldElement<'a> {
let f = x.f;
let x_bytes = x.number.to_bytes_le().1;
let mut x = [0; 32];
x[..x_bytes.len()].copy_from_slice(&x_bytes);
let y_bytes = y.number.to_bytes_le().1;
let mut y = [0; 32];
y[..y_bytes.len()].copy_from_slice(&y_bytes);
let mut bytes = [0; 32];
let choice = Choice::from(u8::from(b));
for ((byte, x), y) in bytes.iter_mut().zip(&x).zip(&y) {
*byte = u8::conditional_select(x, y, choice);
}
FieldElement {
f,
number: BigInt::from_bytes_le(Sign::Plus, &bytes),
}
}
let f = Field::new(p);
let a = f.element(a);
let b = f.element(b);
let z = f.element(z);
let u = f.element(&BigInt::from_bytes_be(Sign::Plus, u));
// Constants:
// 1. c1 = -B / A
let c1 = -&b / &a;
// 2. c2 = -1 / Z
let c2 = -f.one() / &z;
// Steps:
// 1. tv1 = Z * u^2
let tv1 = z * u.square();
// 2. tv2 = tv1^2
let mut tv2 = tv1.square();
// 3. x1 = tv1 + tv2
let mut x1 = &tv1 + &tv2;
// 4. x1 = inv0(x1)
x1 = x1.inv0();
// 5. e1 = x1 == 0
let e1 = x1.is_zero();
// 6. x1 = x1 + 1
x1 = x1 + f.one();
// 7. x1 = CMOV(x1, c2, e1) # If (tv1 + tv2) == 0, set x1 = -1 / Z
x1 = cmov(&x1, &c2, e1);
// 8. x1 = x1 * c1 # x1 = (-B / A) * (1 + (1 / (Z^2 * u^4 + Z * u^2)))
x1 = x1 * c1;
// 9. gx1 = x1^2
let mut gx1 = x1.square();
// 10. gx1 = gx1 + A
gx1 = gx1 + a;
// 11. gx1 = gx1 * x1
gx1 = gx1 * &x1;
// 12. gx1 = gx1 + B # gx1 = g(x1) = x1^3 + A * x1 + B
gx1 = gx1 + b;
// 13. x2 = tv1 * x1 # x2 = Z * u^2 * x1
let x2 = &tv1 * &x1;
// 14. tv2 = tv1 * tv2
tv2 = tv1 * tv2;
// 15. gx2 = gx1 * tv2 # gx2 = (Z * u^2)^3 * gx1
let gx2 = &gx1 * tv2;
// 16. e2 = is_square(gx1)
let e2 = gx1.is_square();
// 17. x = CMOV(x2, x1, e2) # If is_square(gx1), x = x1, else x = x2
let x = cmov(&x2, &x1, e2);
// 18. y2 = CMOV(gx2, gx1, e2) # If is_square(gx1), y2 = gx1, else y2 = gx2
let y2 = cmov(&gx2, &gx1, e2);
// 19. y = sqrt(y2)
let mut y = y2.sqrt();
// 20. e3 = sgn0(u) == sgn0(y) # Fix sign of y
let e3 = u.sgn0() == y.sgn0();
// 21. y = CMOV(-y, y, e3)
y = cmov(&-&y, &y, e3);
// 22. return (x, y)
(x.to_bytes(), y.to_bytes())
}
#[cfg(test)]
mod tests {
use generic_array::typenum::U96;
use super::*;
struct Params {
msg: &'static str,
px: &'static str,
py: &'static str,
u0: &'static str,
u1: &'static str,
q0x: &'static str,
q0y: &'static str,
q1x: &'static str,
q1y: &'static str,
}
#[test]
fn hash_to_curve_simple_swu() {
const P: Lazy<BigInt> = Lazy::new(|| {
BigInt::from_str(
"115792089210356248762697446949407573530086143415290314195533631308867097853951",
)
.unwrap()
});
const A: Lazy<BigInt> = Lazy::new(|| BigInt::from(-3));
const B: Lazy<BigInt> = Lazy::new(|| {
BigInt::parse_bytes(
b"5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b",
16,
)
.unwrap()
});
const Z: Lazy<BigInt> = Lazy::new(|| BigInt::from(-10));
// Test vectors taken from https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#appendix-J.1.1
let test_vectors = alloc::vec![
Params {
msg: "",
px: "2c15230b26dbc6fc9a37051158c95b79656e17a1a920b11394ca91c44247d3e4",
py: "8a7a74985cc5c776cdfe4b1f19884970453912e9d31528c060be9ab5c43e8415",
u0: "ad5342c66a6dd0ff080df1da0ea1c04b96e0330dd89406465eeba11582515009",
u1: "8c0f1d43204bd6f6ea70ae8013070a1518b43873bcd850aafa0a9e220e2eea5a",
q0x: "ab640a12220d3ff283510ff3f4b1953d09fad35795140b1c5d64f313967934d5",
q0y: "dccb558863804a881d4fff3455716c836cef230e5209594ddd33d85c565b19b1",
q1x: "51cce63c50d972a6e51c61334f0f4875c9ac1cd2d3238412f84e31da7d980ef5",
q1y: "b45d1a36d00ad90e5ec7840a60a4de411917fbe7c82c3949a6e699e5a1b66aac",
},
Params {
msg: "abc",
px: "0bb8b87485551aa43ed54f009230450b492fead5f1cc91658775dac4a3388a0f",
py: "5c41b3d0731a27a7b14bc0bf0ccded2d8751f83493404c84a88e71ffd424212e",
u0: "afe47f2ea2b10465cc26ac403194dfb68b7f5ee865cda61e9f3e07a537220af1",
u1: "379a27833b0bfe6f7bdca08e1e83c760bf9a338ab335542704edcd69ce9e46e0",
q0x: "5219ad0ddef3cc49b714145e91b2f7de6ce0a7a7dc7406c7726c7e373c58cb48",
q0y: "7950144e52d30acbec7b624c203b1996c99617d0b61c2442354301b191d93ecf",
q1x: "019b7cb4efcfeaf39f738fe638e31d375ad6837f58a852d032ff60c69ee3875f",
q1y: "589a62d2b22357fed5449bc38065b760095ebe6aeac84b01156ee4252715446e",
},
Params {
msg: "abcdef0123456789",
px: "65038ac8f2b1def042a5df0b33b1f4eca6bff7cb0f9c6c1526811864e544ed80",
py: "cad44d40a656e7aff4002a8de287abc8ae0482b5ae825822bb870d6df9b56ca3",
u0: "0fad9d125a9477d55cf9357105b0eb3a5c4259809bf87180aa01d651f53d312c",
u1: "b68597377392cd3419d8fcc7d7660948c8403b19ea78bbca4b133c9d2196c0fb",
q0x: "a17bdf2965eb88074bc01157e644ed409dac97cfcf0c61c998ed0fa45e79e4a2",
q0y: "4f1bc80c70d411a3cc1d67aeae6e726f0f311639fee560c7f5a664554e3c9c2e",
q1x: "7da48bb67225c1a17d452c983798113f47e438e4202219dd0715f8419b274d66",
q1y: "b765696b2913e36db3016c47edb99e24b1da30e761a8a3215dc0ec4d8f96e6f9",
},
Params {
msg: "q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq\
qqqqqqqqqqqqqqqqqqqqqqqqq",
px: "4be61ee205094282ba8a2042bcb48d88dfbb609301c49aa8b078533dc65a0b5d",
py: "98f8df449a072c4721d241a3b1236d3caccba603f916ca680f4539d2bfb3c29e",
u0: "3bbc30446f39a7befad080f4d5f32ed116b9534626993d2cc5033f6f8d805919",
u1: "76bb02db019ca9d3c1e02f0c17f8baf617bbdae5c393a81d9ce11e3be1bf1d33",
q0x: "c76aaa823aeadeb3f356909cb08f97eee46ecb157c1f56699b5efebddf0e6398",
q0y: "776a6f45f528a0e8d289a4be12c4fab80762386ec644abf2bffb9b627e4352b1",
q1x: "418ac3d85a5ccc4ea8dec14f750a3a9ec8b85176c95a7022f391826794eb5a75",
q1y: "fd6604f69e9d9d2b74b072d14ea13050db72c932815523305cb9e807cc900aff",
},
Params {
msg: "a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
px: "457ae2981f70ca85d8e24c308b14db22f3e3862c5ea0f652ca38b5e49cd64bc5",
py: "ecb9f0eadc9aeed232dabc53235368c1394c78de05dd96893eefa62b0f4757dc",
u0: "4ebc95a6e839b1ae3c63b847798e85cb3c12d3817ec6ebc10af6ee51adb29fec",
u1: "4e21af88e22ea80156aff790750121035b3eefaa96b425a8716e0d20b4e269ee",
q0x: "d88b989ee9d1295df413d4456c5c850b8b2fb0f5402cc5c4c7e815412e926db8",
q0y: "bb4a1edeff506cf16def96afff41b16fc74f6dbd55c2210e5b8f011ba32f4f40",
q1x: "a281e34e628f3a4d2a53fa87ff973537d68ad4fbc28d3be5e8d9f6a2571c5a4b",
q1y: "f6ed88a7aab56a488100e6f1174fa9810b47db13e86be999644922961206e184",
},
];
let dst = GenericArray::from(*b"QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_RO_");
for tv in test_vectors {
let uniform_bytes =
super::super::expand::expand_message_xmd::<sha2::Sha256, U96, _, _>(
Some(tv.msg.as_bytes()),
dst,
)
.unwrap();
let u0 = BigInt::from_bytes_be(Sign::Plus, &uniform_bytes[..48]).mod_floor(&P);
let u1 = BigInt::from_bytes_be(Sign::Plus, &uniform_bytes[48..]).mod_floor(&P);
assert_eq!(BigInt::parse_bytes(tv.u0.as_bytes(), 16).unwrap(), u0);
assert_eq!(BigInt::parse_bytes(tv.u1.as_bytes(), 16).unwrap(), u1);
let (q0x, q0y) = super::hash_to_curve_simple_swu(&u0.to_bytes_be().1, &A, &B, &P, &Z);
let (q1x, q1y) = super::hash_to_curve_simple_swu(&u1.to_bytes_be().1, &A, &B, &P, &Z);
assert_eq!(tv.q0x, hex::encode(q0x));
assert_eq!(tv.q0y, hex::encode(q0y));
assert_eq!(tv.q1x, hex::encode(q1x));
assert_eq!(tv.q1y, hex::encode(q1y));
let p0 = AffinePoint::from_encoded_point(&EncodedPoint::from_affine_coordinates(
&q0x, &q0y, false,
))
.unwrap()
.to_curve();
let p1 = AffinePoint::from_encoded_point(&EncodedPoint::from_affine_coordinates(
&q1x, &q1y, false,
))
.unwrap();
let p = (p0 + p1).to_encoded_point(false);
assert_eq!(tv.px, hex::encode(p.x().unwrap()));
assert_eq!(tv.py, hex::encode(p.y().unwrap()));
}
}
}