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// Copyright 2016 Brian Smith.
//
// Permission to use, copy, modify, and/or distribute this software for any
// purpose with or without fee is hereby granted, provided that the above
// copyright notice and this permission notice appear in all copies.
//
// THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHORS DISCLAIM ALL WARRANTIES
// WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY
// SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
// WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
// OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
// CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
//! Elliptic curve operations on P-256 & P-384.
use self::ops::*;
use arithmetic::montgomery::*;
use crate::{der, ec, error, pkcs8};
use untrusted;
// NIST SP 800-56A Step 3: "If q is an odd prime p, verify that
// yQ**2 = xQ**3 + axQ + b in GF(p), where the arithmetic is performed modulo
// p."
//
// That is, verify that (x, y) is on the curve, which is true iif:
//
// y**2 == x**3 + a*x + b (mod q)
//
// Or, equivalently, but more efficiently:
//
// y**2 == (x**2 + a)*x + b (mod q)
//
fn verify_affine_point_is_on_the_curve(
ops: &CommonOps, (x, y): (&Elem<R>, &Elem<R>),
) -> Result<(), error::Unspecified> {
verify_affine_point_is_on_the_curve_scaled(ops, (x, y), &ops.a, &ops.b)
}
// Use `verify_affine_point_is_on_the_curve` instead of this function whenever
// the affine coordinates are available or will become available. This function
// should only be used then the affine coordinates are never calculated. See
// the notes for `verify_affine_point_is_on_the_curve_scaled`.
//
// The value `z**2` is returned on success because it is useful for ECDSA
// verification.
//
// This function also verifies that the point is not at infinity.
fn verify_jacobian_point_is_on_the_curve(
ops: &CommonOps, p: &Point,
) -> Result<Elem<R>, error::Unspecified> {
let z = ops.point_z(p);
// Verify that the point is not at infinity.
ops.elem_verify_is_not_zero(&z)?;
let x = ops.point_x(p);
let y = ops.point_y(p);
// We are given Jacobian coordinates (x, y, z). So, we have:
//
// (x/z**2, y/z**3) == (x', y'),
//
// where (x', y') are the affine coordinates. The curve equation is:
//
// y'**2 == x'**3 + a*x' + b == (x'**2 + a)*x' + b
//
// Substituting our Jacobian coordinates, we get:
//
// / y \**2 / / x \**2 \ / x \
// | ---- | == | | ---- | + a | * | ---- | + b
// \ z**3 / \ \ z**2 / / \ z**2 /
//
// Simplify:
//
// y**2 / x**2 \ x
// ---- == | ---- + a | * ---- + b
// z**6 \ z**4 / z**2
//
// Multiply both sides by z**6:
//
// z**6 / x**2 \ z**6
// ---- * y**2 == | ---- + a | * ---- * x + (z**6) * b
// z**6 \ z**4 / z**2
//
// Simplify:
//
// / x**2 \
// y**2 == | ---- + a | * z**4 * x + (z**6) * b
// \ z**4 /
//
// Distribute z**4:
//
// / z**4 \
// y**2 == | ---- * x**2 + z**4 * a | * x + (z**6) * b
// \ z**4 /
//
// Simplify:
//
// y**2 == (x**2 + z**4 * a) * x + (z**6) * b
//
let z2 = ops.elem_squared(&z);
let z4 = ops.elem_squared(&z2);
let z4_a = ops.elem_product(&z4, &ops.a);
let z6 = ops.elem_product(&z4, &z2);
let z6_b = ops.elem_product(&z6, &ops.b);
verify_affine_point_is_on_the_curve_scaled(ops, (&x, &y), &z4_a, &z6_b)?;
Ok(z2)
}
// Handles the common logic of point-is-on-the-curve checks for both affine and
// Jacobian cases.
//
// When doing the check that the point is on the curve after a computation,
// to avoid fault attacks or mitigate potential bugs, it is better for security
// to use `verify_affine_point_is_on_the_curve` on the affine coordinates,
// because it provides some protection against faults that occur in the
// computation of the inverse of `z`. See the paper and presentation "Fault
// Attacks on Projective-to-Affine Coordinates Conversion" by Diana Maimuţ,
// Cédric Murdica, David Naccache, Mehdi Tibouchi. That presentation concluded
// simply "Check the validity of the result after conversion to affine
// coordinates." (It seems like a good idea to verify that
// z_inv * z == 1 mod q too).
//
// In the case of affine coordinates (x, y), `a_scaled` and `b_scaled` are
// `a` and `b`, respectively. In the case of Jacobian coordinates (x, y, z),
// the computation and comparison is the same, except `a_scaled` and `b_scaled`
// are (z**4 * a) and (z**6 * b), respectively. Thus, performance is another
// reason to prefer doing the check on the affine coordinates, as Jacobian
// computation requires 3 extra multiplications and 2 extra squarings.
//
// An example of a fault attack that isn't mitigated by a point-on-the-curve
// check after multiplication is given in "Sign Change Fault Attacks On
// Elliptic Curve Cryptosystems" by Johannes Blömer, Martin Otto, and
// Jean-Pierre Seifert.
fn verify_affine_point_is_on_the_curve_scaled(
ops: &CommonOps, (x, y): (&Elem<R>, &Elem<R>), a_scaled: &Elem<R>, b_scaled: &Elem<R>,
) -> Result<(), error::Unspecified> {
let lhs = ops.elem_squared(y);
let mut rhs = ops.elem_squared(x);
ops.elem_add(&mut rhs, a_scaled);
ops.elem_mul(&mut rhs, x);
ops.elem_add(&mut rhs, b_scaled);
if !ops.elems_are_equal(&lhs, &rhs) {
return Err(error::Unspecified);
}
Ok(())
}
pub(crate) fn key_pair_from_pkcs8(
curve: &ec::Curve, template: &pkcs8::Template, input: untrusted::Input,
) -> Result<ec::KeyPair, error::Unspecified> {
let (ec_private_key, _) = pkcs8::unwrap_key(template, pkcs8::Version::V1Only, input)?;
let (private_key, public_key) = ec_private_key.read_all(error::Unspecified, |input| {
// https://tools.ietf.org/html/rfc5915#section-3
der::nested(input, der::Tag::Sequence, error::Unspecified, |input| {
let version = der::small_nonnegative_integer(input)?;
if version != 1 {
return Err(error::Unspecified);
}
let private_key = der::expect_tag_and_get_value(input, der::Tag::OctetString)?;
// [0] parameters (optional).
if input.peek(der::Tag::ContextSpecificConstructed0 as u8) {
let actual_alg_id =
der::expect_tag_and_get_value(input, der::Tag::ContextSpecificConstructed0)?;
if actual_alg_id != template.curve_oid() {
return Err(error::Unspecified);
}
}
// [1] publicKey. The RFC says it is optional, but we require it
// to be present.
let public_key = der::nested(
input,
der::Tag::ContextSpecificConstructed1,
error::Unspecified,
der::bit_string_with_no_unused_bits,
)?;
Ok((private_key, public_key))
})
})?;
key_pair_from_bytes(curve, private_key, public_key)
}
pub fn key_pair_from_bytes(
curve: &ec::Curve, private_key_bytes: untrusted::Input, public_key_bytes: untrusted::Input,
) -> Result<ec::KeyPair, error::Unspecified> {
let private_key = ec::PrivateKey::from_bytes(curve, private_key_bytes)?;
let mut public_key_check = [0; ec::PUBLIC_KEY_MAX_LEN];
{
// Borrow `public_key_check`.
let public_key_check = &mut public_key_check[..curve.public_key_len];
(curve.public_from_private)(public_key_check, &private_key)?;
if public_key_bytes != &*public_key_check {
return Err(error::Unspecified);
}
}
Ok(ec::KeyPair {
private_key,
public_key: public_key_check,
})
}
pub mod curve;
pub mod ecdh;
pub mod ecdsa;
#[macro_use]
mod ops;
mod private_key;
mod public_key;