mirror of
https://github.com/saymrwulf/risc0-curve25519-dalek-source.git
synced 2026-09-07 20:30:39 +00:00
Merge remote-tracking branch 'origin/develop' into develop
This commit is contained in:
commit
9422804e13
14 changed files with 46 additions and 126 deletions
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|
@ -1,6 +1,6 @@
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||||||
[package]
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[package]
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||||||
name = "curve25519-dalek"
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name = "curve25519-dalek"
|
||||||
version = "0.13.2"
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version = "0.14.0"
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||||||
authors = ["Isis Lovecruft <isis@patternsinthevoid.net>",
|
authors = ["Isis Lovecruft <isis@patternsinthevoid.net>",
|
||||||
"Henry de Valence <hdevalence@hdevalence.ca>"]
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"Henry de Valence <hdevalence@hdevalence.ca>"]
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||||||
readme = "README.md"
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readme = "README.md"
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||||||
|
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@ -12,6 +12,7 @@ categories = ["cryptography", "no-std"]
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||||||
keywords = ["cryptography", "curve25519", "elliptic", "curve", "ECC"]
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keywords = ["cryptography", "curve25519", "elliptic", "curve", "ECC"]
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||||||
description = "A low-level cryptographic library for point, group, field, and scalar operations on a curve isomorphic to the twisted Edwards curve defined by -x²+y² = 1 - 121665/121666 x²y² over GF(2²⁵⁵ - 19)."
|
description = "A low-level cryptographic library for point, group, field, and scalar operations on a curve isomorphic to the twisted Edwards curve defined by -x²+y² = 1 - 121665/121666 x²y² over GF(2²⁵⁵ - 19)."
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||||||
exclude = [
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exclude = [
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||||||
|
"**/.gitignore",
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||||||
".gitignore",
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".gitignore",
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||||||
".travis.yml",
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".travis.yml",
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||||||
]
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]
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||||||
|
|
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||||||
20
README.md
20
README.md
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|
@ -44,22 +44,34 @@ Extensive documentation is available [here](https://docs.rs/curve25519-dalek).
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||||||
To install, add the following to the dependencies section of your project's
|
To install, add the following to the dependencies section of your project's
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||||||
`Cargo.toml`:
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`Cargo.toml`:
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||||||
|
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||||||
curve25519-dalek = "^0.13"
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```toml
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||||||
|
curve25519-dalek = "^0.14"
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||||||
|
```
|
||||||
|
|
||||||
Then, in your library or executable source, add:
|
Then, in your library or executable source, add:
|
||||||
|
|
||||||
extern crate curve25519_dalek
|
extern crate curve25519_dalek;
|
||||||
|
|
||||||
|
## Features
|
||||||
|
|
||||||
On nightly Rust, using the `nightly` feature enables a radix-51 field
|
On nightly Rust, using the `nightly` feature enables a radix-51 field
|
||||||
arithmetic implementation using `u128`s, which is approximately twice as
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arithmetic implementation using `u128`s, which is approximately twice as
|
||||||
fast.
|
fast. It will also enable additional developer documentation when
|
||||||
|
compiling via `make doc-internal`.
|
||||||
|
|
||||||
|
By default, the benchmarks are not compiled without the `bench`
|
||||||
|
feature. To run the benchmarks, do:
|
||||||
|
|
||||||
|
```sh
|
||||||
|
cargo bench --features="bench"
|
||||||
|
```
|
||||||
|
|
||||||
## TODO
|
## TODO
|
||||||
|
|
||||||
We intend to stabilise the following before curve25519-dalek-1.0.0:
|
We intend to stabilise the following before curve25519-dalek-1.0.0:
|
||||||
|
|
||||||
* Implement hashing to a point on the curve (Elligator).
|
* Implement hashing to a point on the curve (Elligator).
|
||||||
* Finish Ristretto (Decaf for curve25519) implementation.
|
* Finish Ristretto documentation.
|
||||||
|
|
||||||
## Contributing
|
## Contributing
|
||||||
|
|
||||||
|
|
|
||||||
4
fuzz/.gitignore
vendored
4
fuzz/.gitignore
vendored
|
|
@ -1,4 +0,0 @@
|
||||||
|
|
||||||
target
|
|
||||||
corpus
|
|
||||||
artifacts
|
|
||||||
|
|
@ -1,32 +0,0 @@
|
||||||
|
|
||||||
[package]
|
|
||||||
name = "curve25519-dalek-fuzz"
|
|
||||||
version = "0.0.1"
|
|
||||||
authors = ["Automatically generated"]
|
|
||||||
publish = false
|
|
||||||
|
|
||||||
[package.metadata]
|
|
||||||
cargo-fuzz = true
|
|
||||||
|
|
||||||
[dependencies.curve25519-dalek]
|
|
||||||
path = ".."
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|
||||||
|
|
||||||
[features]
|
|
||||||
yolocrypto = ["curve25519-dalek/yolocrypto"]
|
|
||||||
nightly = ["curve25519-dalek/nightly"]
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|
||||||
radix_51 = ["curve25519-dalek/radix_51"]
|
|
||||||
|
|
||||||
[dependencies.libfuzzer-sys]
|
|
||||||
git = "https://github.com/rust-fuzz/libfuzzer-sys.git"
|
|
||||||
|
|
||||||
# Prevent this from interfering with workspaces
|
|
||||||
[workspace]
|
|
||||||
members = ["."]
|
|
||||||
|
|
||||||
[[bin]]
|
|
||||||
name = "decaf"
|
|
||||||
path = "fuzz_targets/decaf.rs"
|
|
||||||
|
|
||||||
[[bin]]
|
|
||||||
name = "scalar_constructor_accepts_256bit_values"
|
|
||||||
path = "fuzz_targets/scalar_constructor_accepts_256bit_values.rs"
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|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
#![no_main]
|
|
||||||
#[macro_use] extern crate libfuzzer_sys;
|
|
||||||
extern crate curve25519_dalek;
|
|
||||||
|
|
||||||
use curve25519_dalek::curve::ValidityCheck;
|
|
||||||
use curve25519_dalek::decaf::DecafPoint;
|
|
||||||
use curve25519_dalek::field::FieldElement;
|
|
||||||
|
|
||||||
fuzz_target!(|data: &[u8]| {
|
|
||||||
if data.len() != 32 {
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|
||||||
return;
|
|
||||||
}
|
|
||||||
let mut field_bytes = [0u8; 32];
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|
||||||
for (by, data) in field_bytes.iter_mut().zip(data.iter()) {
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|
||||||
*by = *data;
|
|
||||||
}
|
|
||||||
let fe = FieldElement::from_bytes(&field_bytes);
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|
||||||
let p = DecafPoint::elligator_decaf_flavour(&fe);
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|
||||||
assert!(p.0.is_valid());
|
|
||||||
p.compress();
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|
||||||
});
|
|
||||||
|
|
@ -1,36 +0,0 @@
|
||||||
#![no_main]
|
|
||||||
#[macro_use] extern crate libfuzzer_sys;
|
|
||||||
extern crate curve25519_dalek;
|
|
||||||
|
|
||||||
use curve25519_dalek::scalar::Scalar;
|
|
||||||
|
|
||||||
/// Check that the Scalar constructor accepts 256-bit input values and
|
|
||||||
/// behaves correctly on them.
|
|
||||||
///
|
|
||||||
/// Specifically, we take 256-bit values `a` and `b` from the fuzzer
|
|
||||||
/// input data and check that `(a mod l) * (b mod l) == (a * b) mod l`.
|
|
||||||
fuzz_target!(|data: &[u8]| {
|
|
||||||
if data.len() != 64 {
|
|
||||||
return;
|
|
||||||
}
|
|
||||||
let mut a_bytes = [0u8; 32];
|
|
||||||
let mut b_bytes = [0u8; 32];
|
|
||||||
|
|
||||||
// Set a, b to be random 256-bit integers
|
|
||||||
a_bytes.copy_from_slice(&data[ 0..32]);
|
|
||||||
b_bytes.copy_from_slice(&data[32..64]);
|
|
||||||
|
|
||||||
// Compute c = a*b (mod l)
|
|
||||||
let c1 = &Scalar(a_bytes) * &Scalar(b_bytes);
|
|
||||||
|
|
||||||
// Compute c = (a mod l) * (b mod l)
|
|
||||||
let mut tmp = [0u8; 64];
|
|
||||||
tmp[0..32].copy_from_slice(&a_bytes[..]);
|
|
||||||
let a_mod_l = Scalar::reduce(&tmp);
|
|
||||||
tmp[0..32].copy_from_slice(&b_bytes[..]);
|
|
||||||
let b_mod_l = Scalar::reduce(&tmp);
|
|
||||||
|
|
||||||
let c2 = &a_mod_l * &b_mod_l;
|
|
||||||
|
|
||||||
assert_eq!(c1, c2);
|
|
||||||
});
|
|
||||||
|
|
@ -97,7 +97,7 @@ pub const ED25519_BASEPOINT_POINT: ExtendedPoint = ExtendedPoint{
|
||||||
/// array is `i*P`, where `P` is a point of order 8 generating Ɛ[8].
|
/// array is `i*P`, where `P` is a point of order 8 generating Ɛ[8].
|
||||||
///
|
///
|
||||||
/// Thus Ɛ[4] is the points indexed by 0,2,4,6 and Ɛ[2] is the points
|
/// Thus Ɛ[4] is the points indexed by 0,2,4,6 and Ɛ[2] is the points
|
||||||
/// indexed by 0,4.
|
/// indexed by 0,4.
|
||||||
pub const EIGHT_TORSION: [ExtendedPoint; 8] = [
|
pub const EIGHT_TORSION: [ExtendedPoint; 8] = [
|
||||||
ExtendedPoint{
|
ExtendedPoint{
|
||||||
X: FieldElement32([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
|
X: FieldElement32([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]),
|
||||||
|
|
|
||||||
|
|
@ -38,11 +38,11 @@ use subtle::ConditionallyAssignable;
|
||||||
/// to \\(2\^{25+b}\\) or \\(2\^{26+b}\\), where \\(b = 1.75\\).
|
/// to \\(2\^{25+b}\\) or \\(2\^{26+b}\\), where \\(b = 1.75\\).
|
||||||
///
|
///
|
||||||
/// # Note
|
/// # Note
|
||||||
///
|
///
|
||||||
/// The `curve25519_dalek::field` module provides a type alias
|
/// The `curve25519_dalek::field` module provides a type alias
|
||||||
/// `curve25519_dalek::field::FieldElement` to either `FieldElement64`
|
/// `curve25519_dalek::field::FieldElement` to either `FieldElement64`
|
||||||
/// or `FieldElement32`.
|
/// or `FieldElement32`.
|
||||||
///
|
///
|
||||||
/// The backend-specific type `FieldElement32` should not be used
|
/// The backend-specific type `FieldElement32` should not be used
|
||||||
/// outside of the `curve25519_dalek::field` module.
|
/// outside of the `curve25519_dalek::field` module.
|
||||||
#[derive(Copy, Clone)]
|
#[derive(Copy, Clone)]
|
||||||
|
|
@ -133,7 +133,7 @@ impl<'a, 'b> Mul<&'b FieldElement32> for &'a FieldElement32 {
|
||||||
let y1_19 = 19 * y[1]; // This fits in a u32
|
let y1_19 = 19 * y[1]; // This fits in a u32
|
||||||
let y2_19 = 19 * y[2]; // iff 26 + b + lg(19) < 32
|
let y2_19 = 19 * y[2]; // iff 26 + b + lg(19) < 32
|
||||||
let y3_19 = 19 * y[3]; // if b < 32 - 26 - 4.248 = 1.752
|
let y3_19 = 19 * y[3]; // if b < 32 - 26 - 4.248 = 1.752
|
||||||
let y4_19 = 19 * y[4];
|
let y4_19 = 19 * y[4];
|
||||||
let y5_19 = 19 * y[5]; // below, b<2.5: this is a bottleneck,
|
let y5_19 = 19 * y[5]; // below, b<2.5: this is a bottleneck,
|
||||||
let y6_19 = 19 * y[6]; // could be avoided by promoting to
|
let y6_19 = 19 * y[6]; // could be avoided by promoting to
|
||||||
let y7_19 = 19 * y[7]; // u64 here instead of in m()
|
let y7_19 = 19 * y[7]; // u64 here instead of in m()
|
||||||
|
|
@ -181,7 +181,7 @@ impl<'a, 'b> Mul<&'b FieldElement32> for &'a FieldElement32 {
|
||||||
// How big is the contribution to z[i+j] from x[i], y[j]?
|
// How big is the contribution to z[i+j] from x[i], y[j]?
|
||||||
//
|
//
|
||||||
// Using the bounds above, we get:
|
// Using the bounds above, we get:
|
||||||
//
|
//
|
||||||
// i even, j even: x[i]*y[j] < 2^(26+b)*2^(26+b) = 2*2^(51+2*b)
|
// i even, j even: x[i]*y[j] < 2^(26+b)*2^(26+b) = 2*2^(51+2*b)
|
||||||
// i odd, j even: x[i]*y[j] < 2^(25+b)*2^(26+b) = 1*2^(51+2*b)
|
// i odd, j even: x[i]*y[j] < 2^(25+b)*2^(26+b) = 1*2^(51+2*b)
|
||||||
// i even, j odd: x[i]*y[j] < 2^(26+b)*2^(25+b) = 1*2^(51+2*b)
|
// i even, j odd: x[i]*y[j] < 2^(26+b)*2^(25+b) = 1*2^(51+2*b)
|
||||||
|
|
@ -191,7 +191,7 @@ impl<'a, 'b> Mul<&'b FieldElement32> for &'a FieldElement32 {
|
||||||
// (since 2^255 - 19 = 0 mod p). This adds a factor of 19, so
|
// (since 2^255 - 19 = 0 mod p). This adds a factor of 19, so
|
||||||
// we get the bounds (z0 is the biggest one, but calculated for
|
// we get the bounds (z0 is the biggest one, but calculated for
|
||||||
// posterity here in case finer estimation is needed later):
|
// posterity here in case finer estimation is needed later):
|
||||||
//
|
//
|
||||||
// z0 < ( 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 249*2^(51 + 2*b)
|
// z0 < ( 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 249*2^(51 + 2*b)
|
||||||
// z1 < ( 1 + 1 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 154*2^(51 + 2*b)
|
// z1 < ( 1 + 1 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 + 1*19 )*2^(51 + 2b) = 154*2^(51 + 2*b)
|
||||||
// z2 < ( 2 + 1 + 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 195*2^(51 + 2*b)
|
// z2 < ( 2 + 1 + 2 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 + 2*19 + 1*19 )*2^(51 + 2b) = 195*2^(51 + 2*b)
|
||||||
|
|
@ -232,15 +232,15 @@ impl FieldElement32 {
|
||||||
pub fn negate(&mut self) {
|
pub fn negate(&mut self) {
|
||||||
// Compute -b as ((2^4 * p) - b) to avoid underflow.
|
// Compute -b as ((2^4 * p) - b) to avoid underflow.
|
||||||
let neg = FieldElement32::reduce([
|
let neg = FieldElement32::reduce([
|
||||||
((0x3ffffed << 4) - self.0[0]) as u64,
|
((0x3ffffed << 4) - self.0[0]) as u64,
|
||||||
((0x1ffffff << 4) - self.0[1]) as u64,
|
((0x1ffffff << 4) - self.0[1]) as u64,
|
||||||
((0x3ffffff << 4) - self.0[2]) as u64,
|
((0x3ffffff << 4) - self.0[2]) as u64,
|
||||||
((0x1ffffff << 4) - self.0[3]) as u64,
|
((0x1ffffff << 4) - self.0[3]) as u64,
|
||||||
((0x3ffffff << 4) - self.0[4]) as u64,
|
((0x3ffffff << 4) - self.0[4]) as u64,
|
||||||
((0x1ffffff << 4) - self.0[5]) as u64,
|
((0x1ffffff << 4) - self.0[5]) as u64,
|
||||||
((0x3ffffff << 4) - self.0[6]) as u64,
|
((0x3ffffff << 4) - self.0[6]) as u64,
|
||||||
((0x1ffffff << 4) - self.0[7]) as u64,
|
((0x1ffffff << 4) - self.0[7]) as u64,
|
||||||
((0x3ffffff << 4) - self.0[8]) as u64,
|
((0x3ffffff << 4) - self.0[8]) as u64,
|
||||||
((0x1ffffff << 4) - self.0[9]) as u64,
|
((0x1ffffff << 4) - self.0[9]) as u64,
|
||||||
]);
|
]);
|
||||||
self.0 = neg.0;
|
self.0 = neg.0;
|
||||||
|
|
@ -298,7 +298,7 @@ impl FieldElement32 {
|
||||||
// Since z[3] < 2^64, c < 2^(64-25) = 2^39,
|
// Since z[3] < 2^64, c < 2^(64-25) = 2^39,
|
||||||
// so z[4] < 2^26 + 2^39 < 2^39.0002
|
// so z[4] < 2^26 + 2^39 < 2^39.0002
|
||||||
carry(&mut z, 4); carry(&mut z, 8);
|
carry(&mut z, 4); carry(&mut z, 8);
|
||||||
// Now z[4] < 2^26
|
// Now z[4] < 2^26
|
||||||
// and z[5] < 2^25 + 2^13.0002 < 2^25.0004 (good enough)
|
// and z[5] < 2^25 + 2^13.0002 < 2^25.0004 (good enough)
|
||||||
|
|
||||||
// Last carry has a multiplication by 19:
|
// Last carry has a multiplication by 19:
|
||||||
|
|
@ -396,7 +396,7 @@ impl FieldElement32 {
|
||||||
const LOW_26_BITS: u32 = (1 << 26) - 1;
|
const LOW_26_BITS: u32 = (1 << 26) - 1;
|
||||||
|
|
||||||
h[0] += 19*q;
|
h[0] += 19*q;
|
||||||
|
|
||||||
// Now carry the result to compute r + 19q...
|
// Now carry the result to compute r + 19q...
|
||||||
h[1] += h[0] >> 26;
|
h[1] += h[0] >> 26;
|
||||||
h[0] = h[0] & LOW_26_BITS;
|
h[0] = h[0] & LOW_26_BITS;
|
||||||
|
|
@ -416,7 +416,7 @@ impl FieldElement32 {
|
||||||
h[7] = h[7] & LOW_25_BITS;
|
h[7] = h[7] & LOW_25_BITS;
|
||||||
h[9] += h[8] >> 26;
|
h[9] += h[8] >> 26;
|
||||||
h[8] = h[8] & LOW_26_BITS;
|
h[8] = h[8] & LOW_26_BITS;
|
||||||
|
|
||||||
// ... but instead of carrying the value
|
// ... but instead of carrying the value
|
||||||
// (h[9] >> 25) = q*2^255 into another limb,
|
// (h[9] >> 25) = q*2^255 into another limb,
|
||||||
// discard it, subtracting the value from h.
|
// discard it, subtracting the value from h.
|
||||||
|
|
|
||||||
|
|
@ -9,7 +9,7 @@
|
||||||
// - Henry de Valence <hdevalence@hdevalence.ca>
|
// - Henry de Valence <hdevalence@hdevalence.ca>
|
||||||
|
|
||||||
//! The `u32` backend uses `u32`s and a `(u32, u32) -> u64` multiplier.
|
//! The `u32` backend uses `u32`s and a `(u32, u32) -> u64` multiplier.
|
||||||
//!
|
//!
|
||||||
//! This code is intended to be portable, but it requires that
|
//! This code is intended to be portable, but it requires that
|
||||||
//! multiplication of two \\(32\\)-bit values to a \\(64\\)-bit result
|
//! multiplication of two \\(32\\)-bit values to a \\(64\\)-bit result
|
||||||
//! is constant-time on the target platform.
|
//! is constant-time on the target platform.
|
||||||
|
|
|
||||||
|
|
@ -70,7 +70,7 @@ pub const ED25519_BASEPOINT_POINT: ExtendedPoint = ExtendedPoint{
|
||||||
/// array is `i*P`, where `P` is a point of order 8 generating Ɛ[8].
|
/// array is `i*P`, where `P` is a point of order 8 generating Ɛ[8].
|
||||||
///
|
///
|
||||||
/// Thus Ɛ[4] is the points indexed by 0,2,4,6 and Ɛ[2] is the points
|
/// Thus Ɛ[4] is the points indexed by 0,2,4,6 and Ɛ[2] is the points
|
||||||
/// indexed by 0,4.
|
/// indexed by 0,4.
|
||||||
pub const EIGHT_TORSION: [ExtendedPoint; 8] = [
|
pub const EIGHT_TORSION: [ExtendedPoint; 8] = [
|
||||||
ExtendedPoint {
|
ExtendedPoint {
|
||||||
X: FieldElement64([0, 0, 0, 0, 0]),
|
X: FieldElement64([0, 0, 0, 0, 0]),
|
||||||
|
|
|
||||||
|
|
@ -27,11 +27,11 @@ use subtle::ConditionallyAssignable;
|
||||||
/// grow up to \\(2\^{54}\\) between reductions modulo \\(p\\).
|
/// grow up to \\(2\^{54}\\) between reductions modulo \\(p\\).
|
||||||
///
|
///
|
||||||
/// # Note
|
/// # Note
|
||||||
///
|
///
|
||||||
/// The `curve25519_dalek::field` module provides a type alias
|
/// The `curve25519_dalek::field` module provides a type alias
|
||||||
/// `curve25519_dalek::field::FieldElement` to either `FieldElement64`
|
/// `curve25519_dalek::field::FieldElement` to either `FieldElement64`
|
||||||
/// or `FieldElement32`.
|
/// or `FieldElement32`.
|
||||||
///
|
///
|
||||||
/// The backend-specific type `FieldElement64` should not be used
|
/// The backend-specific type `FieldElement64` should not be used
|
||||||
/// outside of the `curve25519_dalek::field` module.
|
/// outside of the `curve25519_dalek::field` module.
|
||||||
#[derive(Copy, Clone)]
|
#[derive(Copy, Clone)]
|
||||||
|
|
|
||||||
|
|
@ -9,11 +9,11 @@
|
||||||
// - Henry de Valence <hdevalence@hdevalence.ca>
|
// - Henry de Valence <hdevalence@hdevalence.ca>
|
||||||
|
|
||||||
//! The `u64` backend uses `u64`s and a `(u64, u64) -> u128` multiplier.
|
//! The `u64` backend uses `u64`s and a `(u64, u64) -> u128` multiplier.
|
||||||
//!
|
//!
|
||||||
//! On x86_64, the idiom `(x as u128) * (y as u128)` lowers to `MUL`
|
//! On x86_64, the idiom `(x as u128) * (y as u128)` lowers to `MUL`
|
||||||
//! instructions taking 64-bit inputs and producing 128-bit outputs. On
|
//! instructions taking 64-bit inputs and producing 128-bit outputs. On
|
||||||
//! other platforms, this implementation is not recommended.
|
//! other platforms, this implementation is not recommended.
|
||||||
//!
|
//!
|
||||||
//! On Haswell and newer, the BMI2 extension provides `MULX`, and on
|
//! On Haswell and newer, the BMI2 extension provides `MULX`, and on
|
||||||
//! Broadwell and newer, the ADX extension provides `ADCX` and `ADOX`
|
//! Broadwell and newer, the ADX extension provides `ADCX` and `ADOX`
|
||||||
//! (allowing the CPU to compute two carry chains in parallel). These
|
//! (allowing the CPU to compute two carry chains in parallel). These
|
||||||
|
|
|
||||||
|
|
@ -16,7 +16,7 @@ use core::ops::{Index, IndexMut};
|
||||||
|
|
||||||
use constants;
|
use constants;
|
||||||
|
|
||||||
/// The `Scalar64` struct represents an element in
|
/// The `Scalar64` struct represents an element in
|
||||||
/// \\(\mathbb Z / \ell \mathbb Z\\) as 5 \\(52\\)-bit limbs.
|
/// \\(\mathbb Z / \ell \mathbb Z\\) as 5 \\(52\\)-bit limbs.
|
||||||
#[derive(Copy,Clone)]
|
#[derive(Copy,Clone)]
|
||||||
pub struct Scalar64(pub [u64; 5]);
|
pub struct Scalar64(pub [u64; 5]);
|
||||||
|
|
|
||||||
|
|
@ -10,7 +10,7 @@
|
||||||
|
|
||||||
//! This module contains internal curve representations which are not part
|
//! This module contains internal curve representations which are not part
|
||||||
//! of the public API.
|
//! of the public API.
|
||||||
//!
|
//!
|
||||||
//! # Curve representations
|
//! # Curve representations
|
||||||
//!
|
//!
|
||||||
//! Internally, we use several different models for the curve. Here
|
//! Internally, we use several different models for the curve. Here
|
||||||
|
|
@ -62,7 +62,7 @@
|
||||||
//! $$
|
//! $$
|
||||||
//! \frac {W\_1} {W\_3} = \frac {XT} {ZT} = \frac X Z = x,
|
//! \frac {W\_1} {W\_3} = \frac {XT} {ZT} = \frac X Z = x,
|
||||||
//! $$
|
//! $$
|
||||||
//! and
|
//! and
|
||||||
//! $$
|
//! $$
|
||||||
//! \frac {W\_2} {W\_3} = \frac {YZ} {ZT} = \frac Y T = y,
|
//! \frac {W\_2} {W\_3} = \frac {YZ} {ZT} = \frac Y T = y,
|
||||||
//! $$
|
//! $$
|
||||||
|
|
|
||||||
Loading…
Reference in a new issue