Implementation of fundamental circuitry and primitive Jubjub curve arithmetic.
This commit is contained in:
Sean Bowe
@@ -0,0 +1,547 @@
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use pairing::{
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Engine,
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Field,
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SqrtField,
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PrimeField,
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PrimeFieldRepr,
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BitIterator
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};
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use super::{
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JubjubParams,
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Unknown,
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PrimeOrder,
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Fs,
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FsRepr,
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montgomery
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};
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use rand::{
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Rng
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};
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use std::marker::PhantomData;
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// Represents the affine point (X/Z, Y/Z) via the extended
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// twisted Edwards coordinates.
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pub struct Point<E: Engine, Subgroup> {
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x: E::Fr,
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y: E::Fr,
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t: E::Fr,
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z: E::Fr,
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_marker: PhantomData<Subgroup>
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}
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fn convert_subgroup<E: Engine, S1, S2>(from: &Point<E, S1>) -> Point<E, S2>
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{
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Point {
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x: from.x,
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y: from.y,
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t: from.t,
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z: from.z,
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_marker: PhantomData
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}
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}
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impl<E: Engine> From<Point<E, PrimeOrder>> for Point<E, Unknown>
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{
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fn from(p: Point<E, PrimeOrder>) -> Point<E, Unknown>
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{
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convert_subgroup(&p)
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}
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}
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impl<E: Engine, Subgroup> Clone for Point<E, Subgroup>
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{
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fn clone(&self) -> Self {
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convert_subgroup(self)
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}
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}
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impl<E: Engine, Subgroup> PartialEq for Point<E, Subgroup> {
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fn eq(&self, other: &Point<E, Subgroup>) -> bool {
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// p1 = (x1/z1, y1/z1)
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// p2 = (x2/z2, y2/z2)
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// Deciding that these two points are equal is a matter of
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// determining that x1/z1 = x2/z2, or equivalently that
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// x1*z2 = x2*z1, and similarly for y.
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let mut x1 = self.x;
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x1.mul_assign(&other.z);
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let mut y1 = self.y;
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y1.mul_assign(&other.z);
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let mut x2 = other.x;
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x2.mul_assign(&self.z);
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let mut y2 = other.y;
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y2.mul_assign(&self.z);
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x1 == x2 && y1 == y2
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}
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}
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impl<E: Engine> Point<E, Unknown> {
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/// This guarantees the point is in the prime order subgroup
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pub fn mul_by_cofactor(&self, params: &JubjubParams<E>) -> Point<E, PrimeOrder>
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{
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let tmp = self.double(params)
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.double(params)
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.double(params);
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convert_subgroup(&tmp)
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}
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pub fn rand<R: Rng>(rng: &mut R, params: &JubjubParams<E>) -> Self
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{
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loop {
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// given an x on the curve, y^2 = (1 + x^2) / (1 - dx^2)
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let x: E::Fr = rng.gen();
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let mut x2 = x;
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x2.square();
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let mut num = E::Fr::one();
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num.add_assign(&x2);
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x2.mul_assign(¶ms.edwards_d);
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let mut den = E::Fr::one();
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den.sub_assign(&x2);
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match den.inverse() {
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Some(invden) => {
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num.mul_assign(&invden);
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match num.sqrt() {
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Some(mut y) => {
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if y.into_repr().is_odd() != rng.gen() {
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y.negate();
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}
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let mut t = x;
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t.mul_assign(&y);
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return Point {
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x: x,
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y: y,
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t: t,
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z: E::Fr::one(),
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_marker: PhantomData
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}
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},
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None => {}
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}
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},
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None => {}
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}
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}
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}
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}
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impl<E: Engine, Subgroup> Point<E, Subgroup> {
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/// Convert from a Montgomery point
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pub fn from_montgomery(
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m: &montgomery::Point<E, Subgroup>,
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params: &JubjubParams<E>
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) -> Self
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{
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match m.into_xy() {
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None => {
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// Map the point at infinity to the neutral element.
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Point::zero()
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},
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Some((x, y)) => {
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// The map from a Montgomery curve is defined as:
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// (x, y) -> (u, v) where
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// u = x / y
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// v = (x - 1) / (x + 1)
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//
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// This map is not defined for y = 0 and x = -1.
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//
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// y = 0 is a valid point only for x = 0:
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// y^2 = x^3 + A.x^2 + x
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// 0 = x^3 + A.x^2 + x
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// 0 = x(x^2 + A.x + 1)
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// We have: x = 0 OR x^2 + A.x + 1 = 0
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// x^2 + A.x + 1 = 0
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// (2.x + A)^2 = A^2 - 4 (Complete the square.)
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// The left hand side is a square, and so if A^2 - 4
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// is nonsquare, there is no solution. Indeed, A^2 - 4
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// is nonsquare.
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//
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// (0, 0) is a point of order 2, and so we map it to
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// (0, -1) in the twisted Edwards curve, which is the
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// only point of order 2 that is not the neutral element.
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if y.is_zero() {
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// This must be the point (0, 0) as above.
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let mut neg1 = E::Fr::one();
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neg1.negate();
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Point {
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x: E::Fr::zero(),
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y: neg1,
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t: E::Fr::zero(),
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z: E::Fr::one(),
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_marker: PhantomData
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}
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} else {
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// Otherwise, as stated above, the mapping is still
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// not defined at x = -1. However, x = -1 is not
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// on the curve when A - 2 is nonsquare:
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// y^2 = x^3 + A.x^2 + x
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// y^2 = (-1) + A + (-1)
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// y^2 = A - 2
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// Indeed, A - 2 is nonsquare.
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let mut u = x;
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u.mul_assign(&y.inverse().expect("y is nonzero"));
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let mut v = x;
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v.sub_assign(&E::Fr::one());
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{
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let mut tmp = x;
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tmp.add_assign(&E::Fr::one());
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v.mul_assign(&tmp.inverse().expect("A - 2 is nonsquare"));
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}
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// The resulting x-coordinate needs to be scaled.
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u.mul_assign(¶ms.scale);
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let mut t = u;
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t.mul_assign(&v);
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Point {
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x: u,
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y: v,
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t: t,
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z: E::Fr::one(),
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_marker: PhantomData
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}
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}
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}
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}
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}
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/// Attempts to cast this as a prime order element, failing if it's
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/// not in the prime order subgroup.
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pub fn as_prime_order(&self, params: &JubjubParams<E>) -> Option<Point<E, PrimeOrder>> {
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if self.mul(Fs::char(), params) == Point::zero() {
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Some(convert_subgroup(self))
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} else {
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None
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}
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}
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pub fn zero() -> Self {
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Point {
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x: E::Fr::zero(),
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y: E::Fr::one(),
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t: E::Fr::zero(),
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z: E::Fr::one(),
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_marker: PhantomData
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}
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}
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pub fn into_xy(&self) -> (E::Fr, E::Fr)
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{
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let zinv = self.z.inverse().unwrap();
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let mut x = self.x;
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x.mul_assign(&zinv);
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let mut y = self.y;
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y.mul_assign(&zinv);
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(x, y)
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}
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pub fn negate(&self) -> Self {
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let mut p = self.clone();
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p.x.negate();
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p.t.negate();
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p
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}
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pub fn double(&self, params: &JubjubParams<E>) -> Self {
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self.add(self, params)
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}
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pub fn add(&self, other: &Self, params: &JubjubParams<E>) -> Self
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{
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// A = x1 * x2
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let mut a = self.x;
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a.mul_assign(&other.x);
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// B = y1 * y2
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let mut b = self.y;
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b.mul_assign(&other.y);
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// C = d * t1 * t2
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let mut c = params.edwards_d;
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c.mul_assign(&self.t);
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c.mul_assign(&other.t);
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// D = z1 * z2
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let mut d = self.z;
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d.mul_assign(&other.z);
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// H = B - aA
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// = B + A
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let mut h = b;
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h.add_assign(&a);
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// E = (x1 + y1) * (x2 + y2) - A - B
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// = (x1 + y1) * (x2 + y2) - H
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let mut e = self.x;
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e.add_assign(&self.y);
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{
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let mut tmp = other.x;
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tmp.add_assign(&other.y);
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e.mul_assign(&tmp);
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}
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e.sub_assign(&h);
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// F = D - C
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let mut f = d;
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f.sub_assign(&c);
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// G = D + C
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let mut g = d;
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g.add_assign(&c);
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// x3 = E * F
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let mut x3 = e;
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x3.mul_assign(&f);
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// y3 = G * H
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let mut y3 = g;
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y3.mul_assign(&h);
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// t3 = E * H
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let mut t3 = e;
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t3.mul_assign(&h);
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// z3 = F * G
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let mut z3 = f;
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z3.mul_assign(&g);
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Point {
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x: x3,
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y: y3,
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t: t3,
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z: z3,
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_marker: PhantomData
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}
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}
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pub fn mul<S: Into<FsRepr>>(&self, scalar: S, params: &JubjubParams<E>) -> Self
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{
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let mut res = Self::zero();
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for b in BitIterator::new(scalar.into()) {
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res = res.double(params);
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if b {
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res = res.add(self, params);
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}
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}
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res
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}
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}
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#[cfg(test)]
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mod test {
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use rand::{XorShiftRng, SeedableRng, Rand};
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use super::{JubjubParams, Point, PrimeOrder, Fs};
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use pairing::bls12_381::{Bls12};
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use pairing::{Engine, Field};
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fn is_on_curve<E: Engine>(
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x: E::Fr,
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y: E::Fr,
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params: &JubjubParams<E>
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) -> bool
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{
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let mut x2 = x;
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x2.square();
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let mut y2 = y;
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y2.square();
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// -x^2 + y^2
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let mut lhs = y2;
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lhs.sub_assign(&x2);
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// 1 + d x^2 y^2
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let mut rhs = y2;
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rhs.mul_assign(&x2);
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rhs.mul_assign(¶ms.edwards_d);
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rhs.add_assign(&E::Fr::one());
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lhs == rhs
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}
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#[test]
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fn test_rand() {
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let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = JubjubParams::new();
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for _ in 0..100 {
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let (x, y) = Point::rand(&mut rng, ¶ms).into_xy();
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assert!(is_on_curve(x, y, ¶ms));
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}
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}
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#[test]
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fn test_identities() {
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let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = JubjubParams::new();
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let z = Point::<Bls12, PrimeOrder>::zero();
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assert!(z.double(¶ms) == z);
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assert!(z.negate() == z);
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for _ in 0..100 {
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let r = Point::rand(&mut rng, ¶ms);
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assert!(r.add(&Point::zero(), ¶ms) == r);
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assert!(r.add(&r.negate(), ¶ms) == Point::zero());
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}
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}
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#[test]
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fn test_associativity() {
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let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = JubjubParams::new();
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for _ in 0..1000 {
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let a = Point::rand(&mut rng, ¶ms);
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let b = Point::rand(&mut rng, ¶ms);
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let c = Point::rand(&mut rng, ¶ms);
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assert!(a.add(&b, ¶ms).add(&c, ¶ms) == c.add(&a, ¶ms).add(&b, ¶ms));
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}
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}
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#[test]
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fn test_order() {
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let rng = &mut XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = &JubjubParams::new();
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// The neutral element is in the prime order subgroup.
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assert!(Point::<Bls12, PrimeOrder>::zero().as_prime_order(params).is_some());
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for _ in 0..50 {
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// Pick a random point and multiply it by the cofactor
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let base = Point::rand(rng, params).mul_by_cofactor(params);
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// Any point multiplied by the cofactor will be in the prime
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// order subgroup
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assert!(base.as_prime_order(params).is_some());
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}
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// It's very likely that at least one out of 50 random points on the curve
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// is not in the prime order subgroup.
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let mut at_least_one_not_in_prime_order_subgroup = false;
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for _ in 0..50 {
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// Pick a random point.
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let base = Point::rand(rng, params);
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at_least_one_not_in_prime_order_subgroup |= base.as_prime_order(params).is_none();
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}
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assert!(at_least_one_not_in_prime_order_subgroup);
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}
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#[test]
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fn test_mul_associativity() {
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let rng = &mut XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = &JubjubParams::new();
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for _ in 0..100 {
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// Pick a random point and multiply it by the cofactor
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let base = Point::rand(rng, params).mul_by_cofactor(params);
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let mut a = Fs::rand(rng);
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let b = Fs::rand(rng);
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let c = Fs::rand(rng);
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let res1 = base.mul(a, params).mul(b, params).mul(c, params);
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let res2 = base.mul(b, params).mul(c, params).mul(a, params);
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let res3 = base.mul(c, params).mul(a, params).mul(b, params);
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a.mul_assign(&b);
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a.mul_assign(&c);
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let res4 = base.mul(a, params);
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assert!(res1 == res2);
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assert!(res2 == res3);
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assert!(res3 == res4);
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let (x, y) = res1.into_xy();
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assert!(is_on_curve(x, y, params));
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let (x, y) = res2.into_xy();
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assert!(is_on_curve(x, y, params));
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let (x, y) = res3.into_xy();
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assert!(is_on_curve(x, y, params));
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}
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}
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#[test]
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fn test_montgomery_conversion() {
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use super::montgomery;
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let rng = &mut XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
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let params = &JubjubParams::new();
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for _ in 0..200 {
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// compute base in montgomery
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let base = montgomery::Point::rand(rng, params);
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// sample random exponent
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let exp = Fs::rand(rng);
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// exponentiate in montgomery, convert to edwards
|
||||
let ed_expected = Point::from_montgomery(&base.mul(exp, params), params);
|
||||
|
||||
// convert to edwards and exponentiate
|
||||
let ed_exponentiated = Point::from_montgomery(&base, params).mul(exp, params);
|
||||
|
||||
let (x, y) = ed_expected.into_xy();
|
||||
assert!(is_on_curve(x, y, params));
|
||||
|
||||
assert!(ed_exponentiated == ed_expected);
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_back_and_forth() {
|
||||
use super::montgomery;
|
||||
|
||||
let rng = &mut XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
|
||||
let params = &JubjubParams::new();
|
||||
|
||||
for _ in 0..200 {
|
||||
// compute base in montgomery
|
||||
let base = montgomery::Point::rand(rng, params);
|
||||
|
||||
// convert to edwards
|
||||
let base_ed = Point::from_montgomery(&base, params);
|
||||
|
||||
{
|
||||
let (x, y) = base_ed.into_xy();
|
||||
assert!(is_on_curve(x, y, params));
|
||||
}
|
||||
|
||||
// convert back to montgomery
|
||||
let base_mont = montgomery::Point::from_edwards(&base_ed, params);
|
||||
|
||||
assert!(base == base_mont);
|
||||
}
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user