Skip to content

/curve25519-dalek forked from dalek-cryptography/curve25519-dalek

Permalink
Branch: master
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
1441 lines (1284 sloc) 47.9 KB
Raw Blame History
// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2018 Isis Lovecruft, Henry de Valence
// Portions Copyright 2017 Brian Smith
// See LICENSE for licensing information.
//
// Authors:
// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
// - Brian Smith <brian@briansmith.org>
//! Arithmetic on scalars (integers mod the group order).
//!
//! Both the Ristretto group and the Ed25519 basepoint have prime order
//! \\( \ell = 2\^{252} + 27742317777372353535851937790883648493 \\).
//!
//! This code is intended to be useful with both the Ristretto group
//! (where everything is done modulo \\( \ell \\)), and the X/Ed25519
//! setting, which mandates specific bit-twiddles that are not
//! well-defined modulo \\( \ell \\).
//!
//! All arithmetic on `Scalars` is done modulo \\( \ell \\).
//!
//! # Constructing a scalar
//!
//! To create a [`Scalar`](struct.Scalar.html) from a supposedly canonical encoding, use
//! [`Scalar::from_canonical_bytes`](struct.Scalar.html#method.from_canonical_bytes).
//!
//! This function does input validation, ensuring that the input bytes
//! are the canonical encoding of a `Scalar`.
//! If they are, we'll get
//! `Some(Scalar)` in return:
//!
//! ```
//! use curve25519_dalek::scalar::Scalar;
//!
//! let one_as_bytes: [u8; 32] = Scalar::one().to_bytes();
//! let a: Option<Scalar> = Scalar::from_canonical_bytes(one_as_bytes);
//!
//! assert!(a.is_some());
//! ```
//!
//! However, if we give it bytes representing a scalar larger than \\( \ell \\)
//! (in this case, \\( \ell + 2 \\)), we'll get `None` back:
//!
//! ```
//! use curve25519_dalek::scalar::Scalar;
//!
//! let l_plus_two_bytes: [u8; 32] = [
//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,
//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,
//! ];
//! let a: Option<Scalar> = Scalar::from_canonical_bytes(l_plus_two_bytes);
//!
//! assert!(a.is_none());
//! ```
//!
//! Another way to create a `Scalar` is by reducing a \\(256\\)-bit integer mod
//! \\( \ell \\), for which one may use the
//! [`Scalar::from_bytes_mod_order`](struct.Scalar.html#method.from_bytes_mod_order)
//! method. In the case of the second example above, this would reduce the
//! resultant scalar \\( \mod \ell \\), producing \\( 2 \\):
//!
//! ```
//! use curve25519_dalek::scalar::Scalar;
//!
//! let l_plus_two_bytes: [u8; 32] = [
//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,
//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,
//! ];
//! let a: Scalar = Scalar::from_bytes_mod_order(l_plus_two_bytes);
//!
//! let two: Scalar = Scalar::one() + Scalar::one();
//!
//! assert!(a == two);
//! ```
//!
//! There is also a constructor that reduces a \\(512\\)-bit integer,
//! [`Scalar::from_bytes_mod_order_wide`](struct.Scalar.html#method.from_bytes_mod_order_wide).
//!
//! To construct a `Scalar` as the hash of some input data, use
//! [`Scalar::hash_from_bytes`](struct.Scalar.html#method.hash_from_bytes),
//! which takes a buffer, or
//! [`Scalar::from_hash`](struct.Scalar.html#method.from_hash),
//! which allows an IUF API.
//!
//! ```
//! # extern crate curve25519_dalek;
//! # extern crate sha2;
//! #
//! # fn main() {
//! use sha2::{Digest, Sha512};
//! use curve25519_dalek::scalar::Scalar;
//!
//! // Hashing a single byte slice
//! let a = Scalar::hash_from_bytes::<Sha512>(b"Abolish ICE");
//!
//! // Streaming data into a hash object
//! let mut hasher = Sha512::default();
//! hasher.input(b"Abolish ");
//! hasher.input(b"ICE");
//! let a2 = Scalar::from_hash(hasher);
//!
//! assert_eq!(a, a2);
//! # }
//! ```
//!
//! Finally, to create a `Scalar` with a specific bit-pattern
//! (e.g., for compatibility with X/Ed25519
//! ["clamping"](https://github.com/isislovecruft/ed25519-dalek/blob/f790bd2ce/src/ed25519.rs#L349)),
//! use [`Scalar::from_bits`](struct.Scalar.html#method.from_bits). This
//! constructs a scalar with exactly the bit pattern given, without any
//! assurances as to reduction modulo the group order:
//!
//! ```
//! use curve25519_dalek::scalar::Scalar;
//!
//! let l_plus_two_bytes: [u8; 32] = [
//! 0xef, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,
//! 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//! 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10,
//! ];
//! let a: Scalar = Scalar::from_bits(l_plus_two_bytes);
//!
//! let two: Scalar = Scalar::one() + Scalar::one();
//!
//! assert!(a != two); // the scalar is not reduced (mod l)…
//! assert!(! a.is_canonical()); // …and therefore is not canonical.
//! assert!(a.reduce() == two); // if we were to reduce it manually, it would be.
//! ```
//!
//! The resulting `Scalar` has exactly the specified bit pattern,
//! **except for the highest bit, which will be set to 0**.
use core::borrow::Borrow;
use core::cmp::{Eq, PartialEq};
use core::fmt::Debug;
use core::iter::{Product, Sum};
use core::ops::Index;
use core::ops::Neg;
use core::ops::{Add, AddAssign};
use core::ops::{Mul, MulAssign};
use core::ops::{Sub, SubAssign};
#[allow(unused_imports)]
use prelude::*;
use rand::{CryptoRng, Rng};
use digest::generic_array::typenum::U64;
use digest::Digest;
use subtle::Choice;
use subtle::ConditionallySelectable;
use subtle::ConstantTimeEq;
use backend;
use constants;
/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u64_backend")]
type UnpackedScalar = backend::u64::scalar::Scalar64;
/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u32_backend")]
type UnpackedScalar = backend::u32::scalar::Scalar32;
/// The `Scalar` struct holds an integer \\(s < 2\^{255} \\) which
/// represents an element of \\(\mathbb Z / \ell\\).
#[derive(Copy, Clone)]
pub struct Scalar {
/// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the
/// group order.
///
/// # Invariant
///
/// The integer representing this scalar must be bounded above by \\(2\^{255}\\), or
/// equivalently the high bit of `bytes[31]` must be zero.
///
/// This ensures that there is room for a carry bit when computing a NAF representation.
//
// XXX This is pub(crate) so we can write literal constants. If const fns were stable, we could
// make the Scalar constructors const fns and use those instead.
pub(crate) bytes: [u8; 32],
}
impl Scalar {
/// Construct a `Scalar` by reducing a 256-bit little-endian integer
/// modulo the group order \\( \ell \\).
pub fn from_bytes_mod_order(bytes: [u8; 32]) -> Scalar {
// Temporarily allow s_unreduced.bytes > 2^255 ...
let s_unreduced = Scalar{bytes: bytes};
// Then reduce mod the group order and return the reduced representative.
let s = s_unreduced.reduce();
debug_assert_eq!(0u8, s[31] >> 7);
s
}
/// Construct a `Scalar` by reducing a 512-bit little-endian integer
/// modulo the group order \\( \ell \\).
pub fn from_bytes_mod_order_wide(input: &[u8; 64]) -> Scalar {
UnpackedScalar::from_bytes_wide(input).pack()
}
/// Attempt to construct a `Scalar` from a canonical byte representation.
///
/// # Return
///
/// - `Some(s)`, where `s` is the `Scalar` corresponding to `bytes`,
/// if `bytes` is a canonical byte representation;
/// - `None` if `bytes` is not a canonical byte representation.
pub fn from_canonical_bytes(bytes: [u8; 32]) -> Option<Scalar> {
// Check that the high bit is not set
if (bytes[31] >> 7) != 0u8 { return None; }
let candidate = Scalar::from_bits(bytes);
if candidate.is_canonical() {
Some(candidate)
} else {
None
}
}
/// Construct a `Scalar` from the low 255 bits of a 256-bit integer.
///
/// This function is intended for applications like X25519 which
/// require specific bit-patterns when performing scalar
/// multiplication.
pub fn from_bits(bytes: [u8; 32]) -> Scalar {
let mut s = Scalar{bytes: bytes};
// Ensure that s < 2^255 by masking the high bit
s.bytes[31] &= 0b0111_1111;
s
}
}
impl Debug for Scalar {
fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
write!(f, "Scalar{{\n\tbytes: {:?},\n}}", &self.bytes)
}
}
impl Eq for Scalar {}
impl PartialEq for Scalar {
fn eq(&self, other: &Self) -> bool {
self.ct_eq(other).unwrap_u8() == 1u8
}
}
impl ConstantTimeEq for Scalar {
fn ct_eq(&self, other: &Self) -> Choice {
self.bytes.ct_eq(&other.bytes)
}
}
impl Index<usize> for Scalar {
type Output = u8;
/// Index the bytes of the representative for this `Scalar`. Mutation is not permitted.
fn index(&self, _index: usize) -> &u8 {
&(self.bytes[_index])
}
}
impl<'b> MulAssign<&'b Scalar> for Scalar {
fn mul_assign(&mut self, _rhs: &'b Scalar) {
*self = UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack();
}
}
define_mul_assign_variants!(LHS = Scalar, RHS = Scalar);
impl<'a, 'b> Mul<&'b Scalar> for &'a Scalar {
type Output = Scalar;
fn mul(self, _rhs: &'b Scalar) -> Scalar {
UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack()
}
}
define_mul_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);
impl<'b> AddAssign<&'b Scalar> for Scalar {
fn add_assign(&mut self, _rhs: &'b Scalar) {
*self = UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack();
}
}
define_add_assign_variants!(LHS = Scalar, RHS = Scalar);
impl<'a, 'b> Add<&'b Scalar> for &'a Scalar {
type Output = Scalar;
fn add(self, _rhs: &'b Scalar) -> Scalar {
UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack()
}
}
define_add_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);
impl<'b> SubAssign<&'b Scalar> for Scalar {
fn sub_assign(&mut self, _rhs: &'b Scalar) {
*self = UnpackedScalar::sub(&self.unpack(), &_rhs.unpack()).pack();
}
}
define_sub_assign_variants!(LHS = Scalar, RHS = Scalar);
impl<'a, 'b> Sub<&'b Scalar> for &'a Scalar {
type Output = Scalar;
fn sub(self, _rhs: &'b Scalar) -> Scalar {
UnpackedScalar::sub(&self.unpack(), &_rhs.unpack()).pack()
}
}
define_sub_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);
impl<'a> Neg for &'a Scalar {
type Output = Scalar;
fn neg(self) -> Scalar {
&Scalar::zero() - self
}
}
impl<'a> Neg for Scalar {
type Output = Scalar;
fn neg(self) -> Scalar {
-&self
}
}
impl ConditionallySelectable for Scalar {
fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
let mut bytes = [0u8; 32];
for i in 0..32 {
bytes[i] = u8::conditional_select(&a.bytes[i], &b.bytes[i], choice);
}
Scalar { bytes }
}
}
#[cfg(feature = "serde")]
use serde::{self, Serialize, Deserialize, Serializer, Deserializer};
#[cfg(feature = "serde")]
use serde::de::Visitor;
#[cfg(feature = "serde")]
impl Serialize for Scalar {
fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
where S: Serializer
{
serializer.serialize_bytes(self.reduce().as_bytes())
}
}
#[cfg(feature = "serde")]
impl<'de> Deserialize<'de> for Scalar {
fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
where D: Deserializer<'de>
{
struct ScalarVisitor;
impl<'de> Visitor<'de> for ScalarVisitor {
type Value = Scalar;
fn expecting(&self, formatter: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
formatter.write_str("a canonically-encoded 32-byte scalar value")
}
fn visit_bytes<E>(self, v: &[u8]) -> Result<Scalar, E>
where E: serde::de::Error
{
if v.len() == 32 {
let mut bytes = [0u8;32];
bytes.copy_from_slice(v);
static ERRMSG: &'static str = "encoding was not canonical";
Scalar::from_canonical_bytes(bytes)
.ok_or(
serde::de::Error::invalid_value(
serde::de::Unexpected::Bytes(v),
&ERRMSG,
)
)
} else {
Err(serde::de::Error::invalid_length(v.len(), &self))
}
}
}
deserializer.deserialize_bytes(ScalarVisitor)
}
}
impl<T> Product<T> for Scalar
where
T: Borrow<Scalar>
{
fn product<I>(iter: I) -> Self
where
I: Iterator<Item = T>
{
iter.fold(Scalar::one(), |acc, item| acc * item.borrow())
}
}
impl<T> Sum<T> for Scalar
where
T: Borrow<Scalar>
{
fn sum<I>(iter: I) -> Self
where
I: Iterator<Item = T>
{
iter.fold(Scalar::zero(), |acc, item| acc + item.borrow())
}
}
impl Default for Scalar {
fn default() -> Scalar {
Scalar::zero()
}
}
impl From<u8> for Scalar {
fn from(x: u8) -> Scalar {
let mut s_bytes = [0u8; 32];
s_bytes[0] = x;
Scalar{ bytes: s_bytes }
}
}
impl From<u16> for Scalar {
fn from(x: u16) -> Scalar {
use byteorder::{ByteOrder, LittleEndian};
let mut s_bytes = [0u8; 32];
LittleEndian::write_u16(&mut s_bytes, x);
Scalar{ bytes: s_bytes }
}
}
impl From<u32> for Scalar {
fn from(x: u32) -> Scalar {
use byteorder::{ByteOrder, LittleEndian};
let mut s_bytes = [0u8; 32];
LittleEndian::write_u32(&mut s_bytes, x);
Scalar{ bytes: s_bytes }
}
}
impl From<u64> for Scalar {
/// Construct a scalar from the given `u64`.
///
/// # Inputs
///
/// An `u64` to convert to a `Scalar`.
///
/// # Returns
///
/// A `Scalar` corresponding to the input `u64`.
///
/// # Example
///
/// ```
/// use curve25519_dalek::scalar::Scalar;
///
/// let fourtytwo = Scalar::from(42u64);
/// let six = Scalar::from(6u64);
/// let seven = Scalar::from(7u64);
///
/// assert!(fourtytwo == six * seven);
/// ```
fn from(x: u64) -> Scalar {
use byteorder::{ByteOrder, LittleEndian};
let mut s_bytes = [0u8; 32];
LittleEndian::write_u64(&mut s_bytes, x);
Scalar{ bytes: s_bytes }
}
}
impl From<u128> for Scalar {
fn from(x: u128) -> Scalar {
use byteorder::{ByteOrder, LittleEndian};
let mut s_bytes = [0u8; 32];
LittleEndian::write_u128(&mut s_bytes, x);
Scalar{ bytes: s_bytes }
}
}
impl Scalar {
/// Return a `Scalar` chosen uniformly at random using a user-provided RNG.
///
/// # Inputs
///
/// * `rng`: any RNG which implements the `rand::CryptoRng` interface.
///
/// # Returns
///
/// A random scalar within ℤ/lℤ.
///
/// # Example
///
/// ```
/// extern crate rand;
/// # extern crate curve25519_dalek;
/// #
/// # fn main() {
/// use curve25519_dalek::scalar::Scalar;
///
/// use rand::OsRng;
///
/// let mut csprng: OsRng = OsRng::new().unwrap();
/// let a: Scalar = Scalar::random(&mut csprng);
/// # }
pub fn random<T: Rng + CryptoRng>(rng: &mut T) -> Self {
let mut scalar_bytes = [0u8; 64];
rng.fill(&mut scalar_bytes);
Scalar::from_bytes_mod_order_wide(&scalar_bytes)
}
/// Hash a slice of bytes into a scalar.
///
/// Takes a type parameter `D`, which is any `Digest` producing 64
/// bytes (512 bits) of output.
///
/// Convenience wrapper around `from_hash`.
///
/// # Example
///
/// ```
/// # extern crate curve25519_dalek;
/// # use curve25519_dalek::scalar::Scalar;
/// extern crate sha2;
///
/// use sha2::Sha512;
///
/// # // Need fn main() here in comment so the doctest compiles
/// # // See https://doc.rust-lang.org/book/documentation.html#documentation-as-tests
/// # fn main() {
/// let msg = "To really appreciate architecture, you may even need to commit a murder";
/// let s = Scalar::hash_from_bytes::<Sha512>(msg.as_bytes());
/// # }
/// ```
pub fn hash_from_bytes<D>(input: &[u8]) -> Scalar
where D: Digest<OutputSize = U64> + Default
{
let mut hash = D::default();
hash.input(input);
Scalar::from_hash(hash)
}
/// Construct a scalar from an existing `Digest` instance.
///
/// Use this instead of `hash_from_bytes` if it is more convenient
/// to stream data into the `Digest` than to pass a single byte
/// slice.
///
/// # Example
///
/// ```
/// # extern crate curve25519_dalek;
/// # use curve25519_dalek::scalar::Scalar;
/// extern crate sha2;
///
/// use sha2::Digest;
/// use sha2::Sha512;
///
/// # fn main() {
/// let mut h = Sha512::new()
/// .chain("To really appreciate architecture, you may even need to commit a murder.")
/// .chain("While the programs used for The Manhattan Transcripts are of the most extreme")
/// .chain("nature, they also parallel the most common formula plot: the archetype of")
/// .chain("murder. Other phantasms were occasionally used to underline the fact that")
/// .chain("perhaps all architecture, rather than being about functional standards, is")
/// .chain("about love and death.");
///
/// let s = Scalar::from_hash(h);
///
/// println!("{:?}", s.to_bytes());
/// assert!(s == Scalar::from_bits([ 21, 88, 208, 252, 63, 122, 210, 152,
/// 154, 38, 15, 23, 16, 167, 80, 150,
/// 192, 221, 77, 226, 62, 25, 224, 148,
/// 239, 48, 176, 10, 185, 69, 168, 11, ]));
/// # }
/// ```
pub fn from_hash<D>(hash: D) -> Scalar
where D: Digest<OutputSize = U64> + Default
{
let mut output = [0u8; 64];
output.copy_from_slice(hash.result().as_slice());
Scalar::from_bytes_mod_order_wide(&output)
}
/// Convert this `Scalar` to its underlying sequence of bytes.
///
/// # Example
///
/// ```
/// use curve25519_dalek::scalar::Scalar;
///
/// let s: Scalar = Scalar::zero();
///
/// assert!(s.to_bytes() == [0u8; 32]);
/// ```
pub fn to_bytes(&self) -> [u8; 32] {
self.bytes
}
/// View the little-endian byte encoding of the integer representing this Scalar.
///
/// # Example
///
/// ```
/// use curve25519_dalek::scalar::Scalar;
///
/// let s: Scalar = Scalar::zero();
///
/// assert!(s.as_bytes() == &[0u8; 32]);
/// ```
pub fn as_bytes(&self) -> &[u8; 32] {
&self.bytes
}
/// Construct the scalar \\( 0 \\).
pub fn zero() -> Self {
Scalar { bytes: [0u8; 32]}
}
/// Construct the scalar \\( 1 \\).
pub fn one() -> Self {
Scalar {
bytes: [
1, 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,
],
}
}
/// Given a nonzero `Scalar`, compute its multiplicative inverse.
///
/// # Warning
///
/// `self` **MUST** be nonzero. If you cannot
/// *prove* that this is the case, you **SHOULD NOT USE THIS
/// FUNCTION**.
///
/// # Returns
///
/// The multiplicative inverse of the this `Scalar`.
///
/// # Example
///
/// ```
/// use curve25519_dalek::scalar::Scalar;
///
/// // x = 2238329342913194256032495932344128051776374960164957527413114840482143558222
/// let X: Scalar = Scalar::from_bytes_mod_order([
/// 0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,
/// 0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,
/// 0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,
/// 0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,
/// ]);
/// // 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244
/// let XINV: Scalar = Scalar::from_bytes_mod_order([
/// 0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb,
/// 0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01, 0x63, 0x47,
/// 0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96,
/// 0xd5, 0x0b, 0xcd, 0x7a, 0x3f, 0x96, 0x2a, 0x0f,
/// ]);
///
/// let inv_X: Scalar = X.invert();
/// assert!(XINV == inv_X);
/// let should_be_one: Scalar = &inv_X * &X;
/// assert!(should_be_one == Scalar::one());
/// ```
pub fn invert(&self) -> Scalar {
self.unpack().invert().pack()
}
/// Given a slice of nonzero (possibly secret) `Scalar`s,
/// compute their inverses in a batch.
///
/// # Return
///
/// Each element of `inputs` is replaced by its inverse.
///
/// The product of all inverses is returned.
///
/// # Warning
///
/// All input `Scalars` **MUST** be nonzero. If you cannot
/// *prove* that this is the case, you **SHOULD NOT USE THIS
/// FUNCTION**.
///
/// # Example
///
/// ```
/// # extern crate curve25519_dalek;
/// # use curve25519_dalek::scalar::Scalar;
/// # fn main() {
/// let mut scalars = [
/// Scalar::from(3u64),
/// Scalar::from(5u64),
/// Scalar::from(7u64),
/// Scalar::from(11u64),
/// ];
///
/// let allinv = Scalar::batch_invert(&mut scalars);
///
/// assert_eq!(allinv, Scalar::from(3*5*7*11u64).invert());
/// assert_eq!(scalars[0], Scalar::from(3u64).invert());
/// assert_eq!(scalars[1], Scalar::from(5u64).invert());
/// assert_eq!(scalars[2], Scalar::from(7u64).invert());
/// assert_eq!(scalars[3], Scalar::from(11u64).invert());
/// # }
/// ```
#[cfg(feature = "alloc")]
pub fn batch_invert(inputs: &mut [Scalar]) -> Scalar {
// This code is essentially identical to the FieldElement
// implementation, and is documented there. Unfortunately,
// it's not easy to write it generically, since here we want
// to use `UnpackedScalar`s internally, and `Scalar`s
// externally, but there's no corresponding distinction for
// field elements.
use clear_on_drop::ClearOnDrop;
use clear_on_drop::clear::ZeroSafe;
// Mark UnpackedScalars as zeroable.
unsafe impl ZeroSafe for UnpackedScalar {}
let n = inputs.len();
let one: UnpackedScalar = Scalar::one().unpack().to_montgomery();
// Wrap the scratch storage in a ClearOnDrop to wipe it when
// we pass out of scope.
let scratch_vec = vec![one; n];
let mut scratch = ClearOnDrop::new(scratch_vec);
// Keep an accumulator of all of the previous products
let mut acc = Scalar::one().unpack().to_montgomery();
// Pass through the input vector, recording the previous
// products in the scratch space
for (input, scratch) in inputs.iter_mut().zip(scratch.iter_mut()) {
*scratch = acc;
// Avoid unnecessary Montgomery multiplication in second pass by
// keeping inputs in Montgomery form
let tmp = input.unpack().to_montgomery();
*input = tmp.pack();
acc = UnpackedScalar::montgomery_mul(&acc, &tmp);
}
// acc is nonzero iff all inputs are nonzero
debug_assert!(acc.pack() != Scalar::zero());
// Compute the inverse of all products
acc = acc.montgomery_invert().from_montgomery();
// We need to return the product of all inverses later
let ret = acc.pack();
// Pass through the vector backwards to compute the inverses
// in place
for (input, scratch) in inputs.iter_mut().rev().zip(scratch.into_iter().rev()) {
let tmp = UnpackedScalar::montgomery_mul(&acc, &input.unpack());
*input = UnpackedScalar::montgomery_mul(&acc, &scratch).pack();
acc = tmp;
}
ret
}
/// Get the bits of the scalar.
pub(crate) fn bits(&self) -> [i8; 256] {
let mut bits = [0i8; 256];
for i in 0..256 {
// As i runs from 0..256, the bottom 3 bits index the bit,
// while the upper bits index the byte.
bits[i] = ((self.bytes[i>>3] >> (i&7)) & 1u8) as i8;
}
bits
}
/// Compute a width-\\(w\\) "Non-Adjacent Form" of this scalar.
///
/// A width-\\(w\\) NAF of a positive integer \\(k\\) is an expression
/// $$
/// k = \sum_{i=0}\^m n\_i 2\^i,
/// $$
/// where each nonzero
/// coefficient \\(n\_i\\) is odd and bounded by \\(|n\_i| < 2\^{w-1}\\),
/// \\(n\_{m-1}\\) is nonzero, and at most one of any \\(w\\) consecutive
/// coefficients is nonzero. (Hankerson, Menezes, Vanstone; def 3.32).
///
/// The length of the NAF is at most one more than the length of
/// the binary representation of \\(k\\). This is why the
/// `Scalar` type maintains an invariant that the top bit is
/// \\(0\\), so that the NAF of a scalar has at most 256 digits.
///
/// Intuitively, this is like a binary expansion, except that we
/// allow some coefficients to grow in magnitude up to
/// \\(2\^{w-1}\\) so that the nonzero coefficients are as sparse
/// as possible.
///
/// When doing scalar multiplication, we can then use a lookup
/// table of precomputed multiples of a point to add the nonzero
/// terms \\( k_i P \\). Using signed digits cuts the table size
/// in half, and using odd digits cuts the table size in half
/// again.
///
/// To compute a \\(w\\)-NAF, we use a modification of Algorithm 3.35 of HMV:
///
/// 1. \\( i \gets 0 \\)
/// 2. While \\( k \ge 1 \\):
/// 1. If \\(k\\) is odd, \\( n_i \gets k \operatorname{mods} 2^w \\), \\( k \gets k - n_i \\).
/// 2. If \\(k\\) is even, \\( n_i \gets 0 \\).
/// 3. \\( k \gets k / 2 \\), \\( i \gets i + 1 \\).
/// 3. Return \\( n_0, n_1, ... , \\)
///
/// Here \\( \bar x = x \operatorname{mods} 2^w \\) means the
/// \\( \bar x \\) with \\( \bar x \equiv x \pmod{2^w} \\) and
/// \\( -2^{w-1} \leq \bar x < 2^w \\).
///
/// We implement this by scanning across the bits of \\(k\\) from
/// least-significant bit to most-significant-bit.
/// Write the bits of \\(k\\) as
/// $$
/// k = \sum\_{i=0}\^m k\_i 2^i,
/// $$
/// and split the sum as
/// $$
/// k = \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \sum\_{i=0} k\_{i+w} 2^i
/// $$
/// where the first part is \\( k \mod 2^w \\).
///
/// If \\( k \mod 2^w\\) is odd, and \\( k \mod 2^w < 2^{w-1} \\), then we emit
/// \\( n_0 = k \mod 2^w \\). Instead of computing
/// \\( k - n_0 \\), we just advance \\(w\\) bits and reindex.
///
/// If \\( k \mod 2^w\\) is odd, and \\( k \mod 2^w \ge 2^{w-1} \\), then
/// \\( n_0 = k \operatorname{mods} 2^w = k \mod 2^w - 2^w \\).
/// The quantity \\( k - n_0 \\) is
/// $$
/// \begin{aligned}
/// k - n_0 &= \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \sum\_{i=0} k\_{i+w} 2^i
/// - \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \\\\
/// &= 2^w + 2^w \sum\_{i=0} k\_{i+w} 2^i
/// \end{aligned}
/// $$
/// so instead of computing the subtraction, we can set a carry
/// bit, advance \\(w\\) bits, and reindex.
///
/// If \\( k \mod 2^w\\) is even, we emit \\(0\\), advance 1 bit
/// and reindex. In fact, by setting all digits to \\(0\\)
/// initially, we don't need to emit anything.
pub(crate) fn non_adjacent_form(&self, w: usize) -> [i8; 256] {
// required by the NAF definition
debug_assert!( w >= 2 );
// required so that the NAF digits fit in i8
debug_assert!( w <= 8 );
use byteorder::{ByteOrder, LittleEndian};
let mut naf = [0i8; 256];
let mut x_u64 = [0u64; 5];
LittleEndian::read_u64_into(&self.bytes, &mut x_u64[0..4]);
let width = 1 << w;
let window_mask = width - 1;
let mut pos = 0;
let mut carry = 0;
while pos < 256 {
// Construct a buffer of bits of the scalar, starting at bit `pos`
let u64_idx = pos / 64;
let bit_idx = pos % 64;
let bit_buf: u64;
if bit_idx < 64 - w {
// This window's bits are contained in a single u64
bit_buf = x_u64[u64_idx] >> bit_idx;
} else {
// Combine the current u64's bits with the bits from the next u64
bit_buf = (x_u64[u64_idx] >> bit_idx) | (x_u64[1+u64_idx] << (64 - bit_idx));
}
// Add the carry into the current window
let window = carry + (bit_buf & window_mask);
if window & 1 == 0 {
// If the window value is even, preserve the carry and continue.
// Why is the carry preserved?
// If carry == 0 and window & 1 == 0, then the next carry should be 0
// If carry == 1 and window & 1 == 0, then bit_buf & 1 == 1 so the next carry should be 1
pos += 1;
continue;
}
if window < width/2 {
carry = 0;
naf[pos] = window as i8;
} else {
carry = 1;
naf[pos] = (window as i8) - (width as i8);
}
pos += w;
}
naf
}
/// Write this scalar in radix 16, with coefficients in \\([-8,8)\\),
/// i.e., compute \\(a\_i\\) such that
/// $$
/// a = a\_0 + a\_1 16\^1 + \cdots + a_{63} 16\^{63},
/// $$
/// with \\(-8 \leq a_i < 8\\) for \\(0 \leq i < 63\\) and \\(-8 \leq a_{63} \leq 8\\).
pub(crate) fn to_radix_16(&self) -> [i8; 64] {
debug_assert!(self[31] <= 127);
let mut output = [0i8; 64];
// Step 1: change radix.
// Convert from radix 256 (bytes) to radix 16 (nibbles)
#[inline(always)]
fn bot_half(x: u8) -> u8 { (x >> 0) & 15 }
#[inline(always)]
fn top_half(x: u8) -> u8 { (x >> 4) & 15 }
for i in 0..32 {
output[2*i ] = bot_half(self[i]) as i8;
output[2*i+1] = top_half(self[i]) as i8;
}
// Precondition note: since self[31] <= 127, output[63] <= 7
// Step 2: recenter coefficients from [0,16) to [-8,8)
for i in 0..63 {
let carry = (output[i] + 8) >> 4;
output[i ] -= carry << 4;
output[i+1] += carry;
}
// Precondition note: output[63] is not recentered. It
// increases by carry <= 1. Thus output[63] <= 8.
output
}
/// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.
pub(crate) fn unpack(&self) -> UnpackedScalar {
UnpackedScalar::from_bytes(&self.bytes)
}
/// Reduce this `Scalar` modulo \\(\ell\\).
#[allow(non_snake_case)]
pub fn reduce(&self) -> Scalar {
let x = self.unpack();
let xR = UnpackedScalar::mul_internal(&x, &constants::R);
let x_mod_l = UnpackedScalar::montgomery_reduce(&xR);
x_mod_l.pack()
}
/// Check whether this `Scalar` is the canonical representative mod \\(\ell\\).
///
/// This is intended for uses like input validation, where variable-time code is acceptable.
///
/// ```
/// # extern crate curve25519_dalek;
/// # extern crate subtle;
/// # use curve25519_dalek::scalar::Scalar;
/// # use subtle::ConditionallySelectable;
/// # fn main() {
/// // 2^255 - 1, since `from_bits` clears the high bit
/// let _2_255_minus_1 = Scalar::from_bits([0xff;32]);
/// assert!(!_2_255_minus_1.is_canonical());
///
/// let reduced = _2_255_minus_1.reduce();
/// assert!(reduced.is_canonical());
/// # }
/// ```
pub fn is_canonical(&self) -> bool {
*self == self.reduce()
}
}
impl UnpackedScalar {
/// Pack the limbs of this `UnpackedScalar` into a `Scalar`.
fn pack(&self) -> Scalar {
Scalar{ bytes: self.to_bytes() }
}
/// Inverts an UnpackedScalar in Montgomery form.
pub fn montgomery_invert(&self) -> UnpackedScalar {
// Uses the addition chain from
// https://briansmith.org/ecc-inversion-addition-chains-01#curve25519_scalar_inversion
let _1 = self;
let _10 = _1.montgomery_square();
let _100 = _10.montgomery_square();
let _11 = UnpackedScalar::montgomery_mul(&_10, &_1);
let _101 = UnpackedScalar::montgomery_mul(&_10, &_11);
let _111 = UnpackedScalar::montgomery_mul(&_10, &_101);
let _1001 = UnpackedScalar::montgomery_mul(&_10, &_111);
let _1011 = UnpackedScalar::montgomery_mul(&_10, &_1001);
let _1111 = UnpackedScalar::montgomery_mul(&_100, &_1011);
// _10000
let mut y = UnpackedScalar::montgomery_mul(&_1111, &_1);
#[inline]
fn square_multiply(y: &mut UnpackedScalar, squarings: usize, x: &UnpackedScalar) {
for _ in 0..squarings {
*y = y.montgomery_square();
}
*y = UnpackedScalar::montgomery_mul(y, x);
}
square_multiply(&mut y, 123 + 3, &_101);
square_multiply(&mut y, 2 + 2, &_11);
square_multiply(&mut y, 1 + 4, &_1111);
square_multiply(&mut y, 1 + 4, &_1111);
square_multiply(&mut y, 4, &_1001);
square_multiply(&mut y, 2, &_11);
square_multiply(&mut y, 1 + 4, &_1111);
square_multiply(&mut y, 1 + 3, &_101);
square_multiply(&mut y, 3 + 3, &_101);
square_multiply(&mut y, 3, &_111);
square_multiply(&mut y, 1 + 4, &_1111);
square_multiply(&mut y, 2 + 3, &_111);
square_multiply(&mut y, 2 + 2, &_11);
square_multiply(&mut y, 1 + 4, &_1011);
square_multiply(&mut y, 2 + 4, &_1011);
square_multiply(&mut y, 6 + 4, &_1001);
square_multiply(&mut y, 2 + 2, &_11);
square_multiply(&mut y, 3 + 2, &_11);
square_multiply(&mut y, 3 + 2, &_11);
square_multiply(&mut y, 1 + 4, &_1001);
square_multiply(&mut y, 1 + 3, &_111);
square_multiply(&mut y, 2 + 4, &_1111);
square_multiply(&mut y, 1 + 4, &_1011);
square_multiply(&mut y, 3, &_101);
square_multiply(&mut y, 2 + 4, &_1111);
square_multiply(&mut y, 3, &_101);
square_multiply(&mut y, 1 + 2, &_11);
y
}
/// Inverts an UnpackedScalar not in Montgomery form.
pub fn invert(&self) -> UnpackedScalar {
self.to_montgomery().montgomery_invert().from_montgomery()
}
}
#[cfg(test)]
mod test {
use super::*;
use constants;
/// x = 2238329342913194256032495932344128051776374960164957527413114840482143558222
pub static X: Scalar = Scalar{
bytes: [
0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,
0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,
0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,
0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,
],
};
/// 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244
pub static XINV: Scalar = Scalar{
bytes: [
0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb,
0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01, 0x63, 0x47,
0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96,
0xd5, 0x0b, 0xcd, 0x7a, 0x3f, 0x96, 0x2a, 0x0f,
],
};
/// y = 2592331292931086675770238855846338635550719849568364935475441891787804997264
pub static Y: Scalar = Scalar{
bytes: [
0x90, 0x76, 0x33, 0xfe, 0x1c, 0x4b, 0x66, 0xa4,
0xa2, 0x8d, 0x2d, 0xd7, 0x67, 0x83, 0x86, 0xc3,
0x53, 0xd0, 0xde, 0x54, 0x55, 0xd4, 0xfc, 0x9d,
0xe8, 0xef, 0x7a, 0xc3, 0x1f, 0x35, 0xbb, 0x05,
],
};
/// x*y = 5690045403673944803228348699031245560686958845067437804563560795922180092780
static X_TIMES_Y: Scalar = Scalar{
bytes: [
0x6c, 0x33, 0x74, 0xa1, 0x89, 0x4f, 0x62, 0x21,
0x0a, 0xaa, 0x2f, 0xe1, 0x86, 0xa6, 0xf9, 0x2c,
0xe0, 0xaa, 0x75, 0xc2, 0x77, 0x95, 0x81, 0xc2,
0x95, 0xfc, 0x08, 0x17, 0x9a, 0x73, 0x94, 0x0c,
],
};
/// sage: l = 2^252 + 27742317777372353535851937790883648493
/// sage: big = 2^256 - 1
/// sage: repr((big % l).digits(256))
static CANONICAL_2_256_MINUS_1: Scalar = Scalar{
bytes: [
28, 149, 152, 141, 116, 49, 236, 214,
112, 207, 125, 115, 244, 91, 239, 198,
254, 255, 255, 255, 255, 255, 255, 255,
255, 255, 255, 255, 255, 255, 255, 15,
],
};
static A_SCALAR: Scalar = Scalar{
bytes: [
0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d,
0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d,
0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1,
0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09,
],
};
static A_NAF: [i8; 256] =
[0,13,0,0,0,0,0,0,0,7,0,0,0,0,0,0,-9,0,0,0,0,-11,0,0,0,0,3,0,0,0,0,1,
0,0,0,0,9,0,0,0,0,-5,0,0,0,0,0,0,3,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,
-9,0,0,0,0,0,-3,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,9,0,
0,0,0,-15,0,0,0,0,-7,0,0,0,0,-9,0,0,0,0,0,5,0,0,0,0,13,0,0,0,0,0,-3,0,
0,0,0,-11,0,0,0,0,-7,0,0,0,0,-13,0,0,0,0,11,0,0,0,0,-9,0,0,0,0,0,1,0,0,
0,0,0,-15,0,0,0,0,1,0,0,0,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,0,13,0,0,0,
0,0,0,11,0,0,0,0,0,15,0,0,0,0,0,-9,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,7,
0,0,0,0,0,-15,0,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0];
#[test]
fn fuzzer_testcase_reduction() {
// LE bytes of 24519928653854221733733552434404946937899825954937634815
let a_bytes = [255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 0, 0, 0, 0, 0, 0, 0, 0, 0];
// LE bytes of 4975441334397345751130612518500927154628011511324180036903450236863266160640
let b_bytes = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 255, 210, 210, 210, 255, 255, 255, 255, 10];
// LE bytes of 6432735165214683820902750800207468552549813371247423777071615116673864412038
let c_bytes = [134, 171, 119, 216, 180, 128, 178, 62, 171, 132, 32, 62, 34, 119, 104, 193, 47, 215, 181, 250, 14, 207, 172, 93, 75, 207, 211, 103, 144, 204, 56, 14];
let a = Scalar::from_bytes_mod_order(a_bytes);
let b = Scalar::from_bytes_mod_order(b_bytes);
let c = Scalar::from_bytes_mod_order(c_bytes);
let mut tmp = [0u8; 64];
// also_a = (a mod l)
tmp[0..32].copy_from_slice(&a_bytes[..]);
let also_a = Scalar::from_bytes_mod_order_wide(&tmp);
// also_b = (b mod l)
tmp[0..32].copy_from_slice(&b_bytes[..]);
let also_b = Scalar::from_bytes_mod_order_wide(&tmp);
let expected_c = &a * &b;
let also_expected_c = &also_a * &also_b;
assert_eq!(c, expected_c);
assert_eq!(c, also_expected_c);
}
#[test]
fn non_adjacent_form() {
let naf = A_SCALAR.non_adjacent_form(5);
for i in 0..256 {
assert_eq!(naf[i], A_NAF[i]);
}
}
#[test]
fn from_u64() {
let val: u64 = 0xdeadbeefdeadbeef;
let s = Scalar::from(val);
assert_eq!(s[7], 0xde);
assert_eq!(s[6], 0xad);
assert_eq!(s[5], 0xbe);
assert_eq!(s[4], 0xef);
assert_eq!(s[3], 0xde);
assert_eq!(s[2], 0xad);
assert_eq!(s[1], 0xbe);
assert_eq!(s[0], 0xef);
}
#[test]
fn scalar_mul_by_one() {
let test_scalar = &X * &Scalar::one();
for i in 0..32 {
assert!(test_scalar[i] == X[i]);
}
}
#[test]
fn impl_add() {
let two = Scalar::from(2u64);
let one = Scalar::one();
let should_be_two = &one + &one;
assert_eq!(should_be_two, two);
}
#[allow(non_snake_case)]
#[test]
fn impl_mul() {
let should_be_X_times_Y = &X * &Y;
assert_eq!(should_be_X_times_Y, X_TIMES_Y);
}
#[allow(non_snake_case)]
#[test]
fn impl_product() {
// Test that product works for non-empty iterators
let X_Y_vector = vec![X, Y];
let should_be_X_times_Y: Scalar = X_Y_vector.iter().product();
assert_eq!(should_be_X_times_Y, X_TIMES_Y);
// Test that product works for the empty iterator
let one = Scalar::one();
let empty_vector = vec![];
let should_be_one: Scalar = empty_vector.iter().product();
assert_eq!(should_be_one, one);
// Test that product works for iterators where Item = Scalar
let xs = [Scalar::from(2u64); 10];
let ys = [Scalar::from(3u64); 10];
// now zs is an iterator with Item = Scalar
let zs = xs.iter().zip(ys.iter()).map(|(x,y)| x * y);
let x_prod: Scalar = xs.iter().product();
let y_prod: Scalar = ys.iter().product();
let z_prod: Scalar = zs.product();
assert_eq!(x_prod, Scalar::from(1024u64));
assert_eq!(y_prod, Scalar::from(59049u64));
assert_eq!(z_prod, Scalar::from(60466176u64));
assert_eq!(x_prod * y_prod, z_prod);
}
#[test]
fn impl_sum() {
// Test that sum works for non-empty iterators
let two = Scalar::from(2u64);
let one_vector = vec![Scalar::one(), Scalar::one()];
let should_be_two: Scalar = one_vector.iter().sum();
assert_eq!(should_be_two, two);
// Test that sum works for the empty iterator
let zero = Scalar::zero();
let empty_vector = vec![];
let should_be_zero: Scalar = empty_vector.iter().sum();
assert_eq!(should_be_zero, zero);
// Test that sum works for owned types
let xs = [Scalar::from(1u64); 10];
let ys = [Scalar::from(2u64); 10];
// now zs is an iterator with Item = Scalar
let zs = xs.iter().zip(ys.iter()).map(|(x,y)| x + y);
let x_sum: Scalar = xs.iter().sum();
let y_sum: Scalar = ys.iter().sum();
let z_sum: Scalar = zs.sum();
assert_eq!(x_sum, Scalar::from(10u64));
assert_eq!(y_sum, Scalar::from(20u64));
assert_eq!(z_sum, Scalar::from(30u64));
assert_eq!(x_sum + y_sum, z_sum);
}
#[test]
fn square() {
let expected = &X * &X;
let actual = X.unpack().square().pack();
for i in 0..32 {
assert!(expected[i] == actual[i]);
}
}
#[test]
fn reduce() {
let biggest = Scalar::from_bytes_mod_order([0xff; 32]);
assert_eq!(biggest, CANONICAL_2_256_MINUS_1);
}
#[test]
fn from_bytes_mod_order_wide() {
let mut bignum = [0u8; 64];
// set bignum = x + 2^256x
for i in 0..32 {
bignum[ i] = X[i];
bignum[32+i] = X[i];
}
// 3958878930004874126169954872055634648693766179881526445624823978500314864344
// = x + 2^256x (mod l)
let reduced = Scalar{
bytes: [
216, 154, 179, 139, 210, 121, 2, 71,
69, 99, 158, 216, 23, 173, 63, 100,
204, 0, 91, 50, 219, 153, 57, 249,
28, 82, 31, 197, 100, 165, 192, 8,
],
};
let test_red = Scalar::from_bytes_mod_order_wide(&bignum);
for i in 0..32 {
assert!(test_red[i] == reduced[i]);
}
}
#[allow(non_snake_case)]
#[test]
fn invert() {
let inv_X = X.invert();
assert_eq!(inv_X, XINV);
let should_be_one = &inv_X * &X;
assert_eq!(should_be_one, Scalar::one());
}
// Negating a scalar twice should result in the original scalar.
#[allow(non_snake_case)]
#[test]
fn neg_twice_is_identity() {
let negative_X = -&X;
let should_be_X = -&negative_X;
assert_eq!(should_be_X, X);
}
#[test]
fn to_bytes_from_bytes_roundtrips() {
let unpacked = X.unpack();
let bytes = unpacked.to_bytes();
let should_be_unpacked = UnpackedScalar::from_bytes(&bytes);
assert_eq!(should_be_unpacked.0, unpacked.0);
}
#[test]
fn montgomery_reduce_matches_from_bytes_mod_order_wide() {
let mut bignum = [0u8; 64];
// set bignum = x + 2^256x
for i in 0..32 {
bignum[ i] = X[i];
bignum[32+i] = X[i];
}
// x + 2^256x (mod l)
// = 3958878930004874126169954872055634648693766179881526445624823978500314864344
let expected = Scalar{
bytes: [
216, 154, 179, 139, 210, 121, 2, 71,
69, 99, 158, 216, 23, 173, 63, 100,
204, 0, 91, 50, 219, 153, 57, 249,
28, 82, 31, 197, 100, 165, 192, 8
],
};
let reduced = Scalar::from_bytes_mod_order_wide(&bignum);
// The reduced scalar should match the expected
assert_eq!(reduced.bytes, expected.bytes);
// (x + 2^256x) * R
let interim = UnpackedScalar::mul_internal(&UnpackedScalar::from_bytes_wide(&bignum),
&constants::R);
// ((x + 2^256x) * R) / R (mod l)
let montgomery_reduced = UnpackedScalar::montgomery_reduce(&interim);
// The Montgomery reduced scalar should match the reduced one, as well as the expected
assert_eq!(montgomery_reduced.0, reduced.unpack().0);
assert_eq!(montgomery_reduced.0, expected.unpack().0)
}
#[test]
fn canonical_decoding() {
// canonical encoding of 1667457891
let canonical_bytes = [99, 99, 99, 99, 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,];
// encoding of
// 7265385991361016183439748078976496179028704920197054998554201349516117938192
// = 28380414028753969466561515933501938171588560817147392552250411230663687203 (mod l)
// non_canonical because unreduced mod l
let non_canonical_bytes_because_unreduced = [16; 32];
// encoding with high bit set, to check that the parser isn't pre-masking the high bit
let non_canonical_bytes_because_highbit = [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, 128];
assert!( Scalar::from_canonical_bytes(canonical_bytes).is_some() );
assert!( Scalar::from_canonical_bytes(non_canonical_bytes_because_unreduced).is_none() );
assert!( Scalar::from_canonical_bytes(non_canonical_bytes_because_highbit).is_none() );
}
#[test]
#[cfg(feature = "serde")]
fn serde_bincode_scalar_roundtrip() {
use bincode;
let output = bincode::serialize(&X).unwrap();
let parsed: Scalar = bincode::deserialize(&output).unwrap();
assert_eq!(parsed, X);
}
#[cfg(debug_assertions)]
#[test]
#[should_panic]
fn batch_invert_with_a_zero_input_panics() {
let mut xs = vec![Scalar::one(); 16];
xs[3] = Scalar::zero();
// This should panic in debug mode.
Scalar::batch_invert(&mut xs);
}
#[test]
fn batch_invert_empty() {
assert_eq!(Scalar::one(), Scalar::batch_invert(&mut []));
}
#[test]
fn batch_invert_consistency() {
let mut x = Scalar::from(1u64);
let mut v1: Vec<_> = (0..16).map(|_| {let tmp = x; x = x + x; tmp}).collect();
let v2 = v1.clone();
let expected: Scalar = v1.iter().product();
let expected = expected.invert();
let ret = Scalar::batch_invert(&mut v1);
assert_eq!(ret, expected);
for (a, b) in v1.iter().zip(v2.iter()) {
assert_eq!(a * b, Scalar::one());
}
}
}
You can’t perform that action at this time.