Struct Edwards 

pub struct Edwards { /* private fields */ }

A twisted Edwards curve a x^2 + y^2 = 1 + d x^2 y^2 over the field of a prime p > 3, with a a square and d not, which makes its addition law complete: one formula for every pair of points, doublings and the neutral element (0, 1) included.

Its points are held in extended coordinates, x = X / Z, y = Y / Z, x y = T / Z, the fastest known form of the group, with a formula of its own for a = -1, as on Ed25519.

Examples

use a::math::ec::{Curve, Edwards};
use a::math::numbers::{ByteOrder, Number};

let n = |x| Number::new_from_u32(x, ByteOrder::BigEndian);
// x^2 + y^2 = 1 + 2 x^2 y^2 over F_13, where 1 is a square and 2 is not.
let curve = Edwards::new(&n(1), &n(2), &n(13)).unwrap();
let o = curve.identity();
assert_eq!(o.coordinates(), Some((n(0), n(1))));

Implementations

impl Edwards

pub fn new(a: &Number, d: &Number, p: &Number) -> Result<Edwards, CurveError>

The curve a x^2 + y^2 = 1 + d x^2 y^2 modulo p, in p’s byte order. a and d are reduced modulo p.

Errors

CurveError::NotPrime when p is not a prime above 3, by is_probable_prime; CurveError::Singular when a, d or a - d is zero modulo p; CurveError::NotComplete when a is not a square or d is.

pub fn a(&self) -> &Number

The coefficient a, below p.

pub fn d(&self) -> &Number

The coefficient d, below p.

pub fn to_montgomery(&self) -> Montgomery

The Montgomery curve this one is birationally equivalent to, with A = 2 (a + d) / (a - d) and B = 4 / (a - d).

pub fn map_to_montgomery<'t>( &self, target: &'t Montgomery, point: &Point<'_, Edwards>, ) -> Point<'t, Montgomery>

The point of target, from to_montgomery, that point of this curve corresponds to: ((1 + y) / (1 - y), (1 + y) / ((1 - y) x)), with the neutral element going to the point at infinity and (0, -1) to (0, 0).

Panics

Panics if point is not on this curve or target is not the curve from to_montgomery.

Trait Implementations

impl Curve for Edwards

fn discriminant(&self) -> Number

a d (a - d)^4 / 16 modulo p, the discriminant of the curve’s Montgomery form.

fn j_invariant(&self) -> Number

16 (a^2 + 14 a d + d^2)^3 / (a d (a - d)^4) modulo p.

fn modulus(&self) -> &Number

The prime of the field.

fn identity(&self) -> Point<'_, Self>

The neutral element of the curve’s group: the point at infinity, or (0, 1) on an Edwards curve.

fn point(&self, x: &Number, y: &Number) -> Result<Point<'_, Self>, NotOnCurve>

The point (x, y), reduced modulo p.

fn lift(&self, coordinate: &Number, odd: bool) -> Option<Point<'_, Self>>

The point with the given coordinate, reduced modulo p, whose other coordinate has the parity odd: x is given and y computed on Weierstrass and Montgomery curves, and the other way round on Edwards curves, or None when there is no such point.

fn point_from_bytes(&self, bytes: &[u8]) -> Option<Point<'_, Self>>

The point in the bytes of Point::compress or Point::to_bytes, or None when they are not the encoding of a point of this curve.

fn multi_mul<'c>( &'c self, terms: &[(&Number, &Point<'c, Self>)], ) -> Point<'c, Self>

The sum of k p over the terms: Straus’s method for a few, Pippenger’s for many, much faster than as many products.