elliptic-curve: generic impl of complete prime order formulas (#1022)

Adds a generic implementation of the complete addition formulas from
Renes-Costello-Batina 2015[1] adapted from @str4d's original
implementation for the `p256` crate:

RustCrypto/elliptic-curves#15

This implementation has been copied-and-pasted into the `p384` crate,
hence the motivation to make it generic and extract it somewhere that it
can be reused.

The API exposed is fairly low-level, however it's difficult to better
encapsulate it without making breaking changes to the `elliptic-curve`
crate. Thus this PR opts to provide an initial low-level generic
implementation with the goal of exploring removing more duplication with
a higher-level API as followup work to be done at a time when breaking
changes are permitted.

[1]: https://eprint.iacr.org/2015/1060

Author: tarcieri

elliptic-curve/src/lib.rs

@@ -66,13 +66,30 @@ extern crate std;
 #[cfg_attr(docsrs, doc(cfg(feature = "rand_core")))]
 pub use rand_core;
 
+#[macro_use]
+mod macros;
+
 pub mod ops;
 
+#[cfg(feature = "dev")]
+#[cfg_attr(docsrs, doc(cfg(feature = "dev")))]
+pub mod dev;
+
+#[cfg(feature = "ecdh")]
+#[cfg_attr(docsrs, doc(cfg(feature = "ecdh")))]
+pub mod ecdh;
+
+#[cfg(feature = "hash2curve")]
+#[cfg_attr(docsrs, doc(cfg(feature = "hash2curve")))]
+pub mod hash2curve;
+
 #[cfg(feature = "sec1")]
+#[cfg_attr(docsrs, doc(cfg(feature = "sec1")))]
 pub mod sec1;
 
-#[macro_use]
-mod macros;
+#[cfg(feature = "arithmetic")]
+#[cfg_attr(docsrs, doc(cfg(feature = "arithmetic")))]
+pub mod weierstrass;
 
 mod error;
 mod point;
@@ -84,21 +101,9 @@ mod arithmetic;
 #[cfg(feature = "arithmetic")]
 mod public_key;
 
-#[cfg(feature = "dev")]
-#[cfg_attr(docsrs, doc(cfg(feature = "dev")))]
-pub mod dev;
-
-#[cfg(feature = "ecdh")]
-#[cfg_attr(docsrs, doc(cfg(feature = "ecdh")))]
-pub mod ecdh;
-
 #[cfg(feature = "jwk")]
 mod jwk;
 
-#[cfg(feature = "hash2curve")]
-#[cfg_attr(docsrs, doc(cfg(feature = "hash2curve")))]
-pub mod hash2curve;
-
 pub use crate::{
     error::{Error, Result},
     point::{

elliptic-curve/src/weierstrass.rs

@@ -0,0 +1,128 @@
+//! Complete projective formulas for prime order elliptic curves as described
+//! in [Renes-Costello-Batina 2015].
+//!
+//! [Renes-Costello-Batina 2015]: https://eprint.iacr.org/2015/1060
+
+#![allow(clippy::op_ref)]
+
+use ff::Field;
+
+/// Affine point whose coordinates are represented by the given field element.
+pub type AffinePoint<Fe> = (Fe, Fe);
+
+/// Projective point whose coordinates are represented by the given field element.
+pub type ProjectivePoint<Fe> = (Fe, Fe, Fe);
+
+/// Implements the complete addition formula from [Renes-Costello-Batina 2015]
+/// (Algorithm 4).
+///
+/// [Renes-Costello-Batina 2015]: https://eprint.iacr.org/2015/1060
+#[inline(always)]
+pub fn add<Fe>(
+    (ax, ay, az): ProjectivePoint<Fe>,
+    (bx, by, bz): ProjectivePoint<Fe>,
+    curve_equation_b: Fe,
+) -> ProjectivePoint<Fe>
+where
+    Fe: Field,
+{
+    // The comments after each line indicate which algorithm steps are being
+    // performed.
+    let xx = ax * bx; // 1
+    let yy = ay * by; // 2
+    let zz = az * bz; // 3
+    let xy_pairs = ((ax + ay) * &(bx + by)) - &(xx + &yy); // 4, 5, 6, 7, 8
+    let yz_pairs = ((ay + az) * &(by + bz)) - &(yy + &zz); // 9, 10, 11, 12, 13
+    let xz_pairs = ((ax + az) * &(bx + bz)) - &(xx + &zz); // 14, 15, 16, 17, 18
+
+    let bzz_part = xz_pairs - &(curve_equation_b * &zz); // 19, 20
+    let bzz3_part = bzz_part.double() + &bzz_part; // 21, 22
+    let yy_m_bzz3 = yy - &bzz3_part; // 23
+    let yy_p_bzz3 = yy + &bzz3_part; // 24
+
+    let zz3 = zz.double() + &zz; // 26, 27
+    let bxz_part = (curve_equation_b * &xz_pairs) - &(zz3 + &xx); // 25, 28, 29
+    let bxz3_part = bxz_part.double() + &bxz_part; // 30, 31
+    let xx3_m_zz3 = xx.double() + &xx - &zz3; // 32, 33, 34
+
+    (
+        (yy_p_bzz3 * &xy_pairs) - &(yz_pairs * &bxz3_part), // 35, 39, 40
+        (yy_p_bzz3 * &yy_m_bzz3) + &(xx3_m_zz3 * &bxz3_part), // 36, 37, 38
+        (yy_m_bzz3 * &yz_pairs) + &(xy_pairs * &xx3_m_zz3), // 41, 42, 43
+    )
+}
+
+/// Implements the complete mixed addition formula from
+/// [Renes-Costello-Batina 2015] (Algorithm 5).
+///
+/// [Renes-Costello-Batina 2015]: https://eprint.iacr.org/2015/1060
+#[inline(always)]
+pub fn add_mixed<Fe>(
+    (ax, ay, az): ProjectivePoint<Fe>,
+    (bx, by): AffinePoint<Fe>,
+    curve_equation_b: Fe,
+) -> ProjectivePoint<Fe>
+where
+    Fe: Field,
+{
+    // The comments after each line indicate which algorithm steps are being
+    // performed.
+    let xx = ax * &bx; // 1
+    let yy = ay * &by; // 2
+    let xy_pairs = ((ax + &ay) * &(bx + &by)) - &(xx + &yy); // 3, 4, 5, 6, 7
+    let yz_pairs = (by * &az) + &ay; // 8, 9 (t4)
+    let xz_pairs = (bx * &az) + &ax; // 10, 11 (y3)
+
+    let bz_part = xz_pairs - &(curve_equation_b * &az); // 12, 13
+    let bz3_part = bz_part.double() + &bz_part; // 14, 15
+    let yy_m_bzz3 = yy - &bz3_part; // 16
+    let yy_p_bzz3 = yy + &bz3_part; // 17
+
+    let z3 = az.double() + &az; // 19, 20
+    let bxz_part = (curve_equation_b * &xz_pairs) - &(z3 + &xx); // 18, 21, 22
+    let bxz3_part = bxz_part.double() + &bxz_part; // 23, 24
+    let xx3_m_zz3 = xx.double() + &xx - &z3; // 25, 26, 27
+
+    (
+        (yy_p_bzz3 * &xy_pairs) - &(yz_pairs * &bxz3_part), // 28, 32, 33
+        (yy_p_bzz3 * &yy_m_bzz3) + &(xx3_m_zz3 * &bxz3_part), // 29, 30, 31
+        (yy_m_bzz3 * &yz_pairs) + &(xy_pairs * &xx3_m_zz3), // 34, 35, 36
+    )
+}
+
+/// Implements the exception-free point doubling formula from
+/// [Renes-Costello-Batina 2015] (Algorithm 6).
+///
+/// [Renes-Costello-Batina 2015]: https://eprint.iacr.org/2015/1060
+#[inline(always)]
+pub fn double<Fe>((x, y, z): ProjectivePoint<Fe>, curve_equation_b: Fe) -> ProjectivePoint<Fe>
+where
+    Fe: Field,
+{
+    // The comments after each line indicate which algorithm steps are being
+    // performed.
+    let xx = x.square(); // 1
+    let yy = y.square(); // 2
+    let zz = z.square(); // 3
+    let xy2 = (x * &y).double(); // 4, 5
+    let xz2 = (x * &z).double(); // 6, 7
+
+    let bzz_part = (curve_equation_b * &zz) - &xz2; // 8, 9
+    let bzz3_part = bzz_part.double() + &bzz_part; // 10, 11
+    let yy_m_bzz3 = yy - &bzz3_part; // 12
+    let yy_p_bzz3 = yy + &bzz3_part; // 13
+    let y_frag = yy_p_bzz3 * &yy_m_bzz3; // 14
+    let x_frag = yy_m_bzz3 * &xy2; // 15
+
+    let zz3 = zz.double() + &zz; // 16, 17
+    let bxz2_part = (curve_equation_b * &xz2) - &(zz3 + &xx); // 18, 19, 20
+    let bxz6_part = bxz2_part.double() + &bxz2_part; // 21, 22
+    let xx3_m_zz3 = xx.double() + &xx - &zz3; // 23, 24, 25
+
+    let dy = y_frag + &(xx3_m_zz3 * &bxz6_part); // 26, 27
+    let yz2 = (y * &z).double(); // 28, 29
+    let dx = x_frag - &(bxz6_part * &yz2); // 30, 31
+    let dz = (yz2 * &yy).double().double(); // 32, 33, 34
+
+    (dx, dy, dz)
+}