blob: b203d265d240950a2f3dfc061fc71619fcf44128 [file] [log] [blame]
// Copyright 2015-2016 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.
//! Multi-precision integers.
//!
//! # Modular Arithmetic.
//!
//! Modular arithmetic is done in finite commutative rings ℤ/mℤ for some
//! modulus *m*. We work in finite commutative rings instead of finite fields
//! because the RSA public modulus *n* is not prime, which means ℤ/nℤ contains
//! nonzero elements that have no multiplicative inverse, so ℤ/nℤ is not a
//! finite field.
//!
//! In some calculations we need to deal with multiple rings at once. For
//! example, RSA private key operations operate in the rings ℤ/nℤ, ℤ/pℤ, and
//! ℤ/qℤ. Types and functions dealing with such rings are all parameterized
//! over a type `M` to ensure that we don't wrongly mix up the math, e.g. by
//! multiplying an element of ℤ/pℤ by an element of ℤ/qℤ modulo q. This follows
//! the "unit" pattern described in [Static checking of units in Servo]; `Elem`,
//! and `Modulus` are analogous to `geom::Length`, and `super::N` and
//! `super::signing::{P, QQ, Q}` are analogous to `Mm` and `Inch`.
//!
//! `Elem` also uses the static unit checking pattern to statically track the
//! Montgomery factors that need to be canceled out in each value using it's
//! `E` parameter.
//!
//! [Static checking of units in Servo]:
//! https://blog.mozilla.org/research/2014/06/23/static-checking-of-units-in-servo/
#![allow(box_pointers)]
use crate::{
arithmetic::montgomery::*,
bits, bssl, c, error,
limb::{self, Limb, LimbMask, LIMB_BITS, LIMB_BYTES},
};
use core::{
self,
marker::PhantomData,
ops::{Deref, DerefMut},
};
use untrusted;
pub unsafe trait Prime {}
struct Width<M> {
num_limbs: usize,
/// The modulus *m* that the width originated from.
m: PhantomData<M>,
}
/// All `BoxedLimbs<M>` are stored in the same number of limbs.
struct BoxedLimbs<M> {
limbs: Box<[Limb]>,
/// The modulus *m* that determines the size of `limbx`.
m: PhantomData<M>,
}
impl<M> Deref for BoxedLimbs<M> {
type Target = [Limb];
#[inline]
fn deref(&self) -> &Self::Target { &self.limbs }
}
impl<M> DerefMut for BoxedLimbs<M> {
#[inline]
fn deref_mut(&mut self) -> &mut Self::Target { &mut self.limbs }
}
// TODO: `derive(Clone)` after https://github.com/rust-lang/rust/issues/26925
// is resolved or restrict `M: Clone`.
impl<M> Clone for BoxedLimbs<M> {
fn clone(&self) -> Self {
Self {
limbs: self.limbs.clone(),
m: self.m.clone(),
}
}
}
impl<M> BoxedLimbs<M> {
fn positive_minimal_width_from_be_bytes(
input: untrusted::Input,
) -> Result<Self, error::KeyRejected> {
// Reject leading zeros. Also reject the value zero ([0]) because zero
// isn't positive.
if untrusted::Reader::new(input).peek(0) {
return Err(error::KeyRejected::invalid_encoding());
}
let num_limbs = (input.len() + LIMB_BYTES - 1) / LIMB_BYTES;
let mut r = Self::zero(Width {
num_limbs,
m: PhantomData,
});
limb::parse_big_endian_and_pad_consttime(input, &mut r)
.map_err(|error::Unspecified| error::KeyRejected::unexpected_error())?;
Ok(r)
}
fn minimal_width_from_unpadded(limbs: &[Limb]) -> Self {
debug_assert_ne!(limbs.last(), Some(&0));
use std::borrow::ToOwned;
Self {
limbs: limbs.to_owned().into_boxed_slice(),
m: PhantomData,
}
}
fn from_be_bytes_padded_less_than(
input: untrusted::Input, m: &Modulus<M>,
) -> Result<Self, error::Unspecified> {
let mut r = Self::zero(m.width());
limb::parse_big_endian_and_pad_consttime(input, &mut r)?;
if limb::limbs_less_than_limbs_consttime(&r, &m.limbs) != LimbMask::True {
return Err(error::Unspecified);
}
Ok(r)
}
#[inline]
fn is_zero(&self) -> bool { limb::limbs_are_zero_constant_time(&self.limbs) == LimbMask::True }
fn zero(width: Width<M>) -> Self {
use std::borrow::ToOwned;
Self {
limbs: vec![0; width.num_limbs].to_owned().into_boxed_slice(),
m: PhantomData,
}
}
fn width(&self) -> Width<M> {
Width {
num_limbs: self.limbs.len(),
m: PhantomData,
}
}
}
/// A modulus *s* that is smaller than another modulus *l* so every element of
/// ℤ/sℤ is also an element of ℤ/lℤ.
pub unsafe trait SmallerModulus<L> {}
/// A modulus *s* where s < l < 2*s for the given larger modulus *l*. This is
/// the precondition for reduction by conditional subtraction,
/// `elem_reduce_once()`.
pub unsafe trait SlightlySmallerModulus<L>: SmallerModulus<L> {}
/// A modulus *s* where √l <= s < l for the given larger modulus *l*. This is
/// the precondition for the more general Montgomery reduction from ℤ/lℤ to
/// ℤ/sℤ.
pub unsafe trait NotMuchSmallerModulus<L>: SmallerModulus<L> {}
/// The x86 implementation of `GFp_bn_mul_mont`, at least, requires at least 4
/// limbs. For a long time we have required 4 limbs for all targets, though
/// this may be unnecessary. TODO: Replace this with
/// `n.len() < 256 / LIMB_BITS` so that 32-bit and 64-bit platforms behave the
/// same.
pub const MODULUS_MIN_LIMBS: usize = 4;
pub const MODULUS_MAX_LIMBS: usize = 8192 / LIMB_BITS;
/// The modulus *m* for a ring ℤ/mℤ, along with the precomputed values needed
/// for efficient Montgomery multiplication modulo *m*. The value must be odd
/// and larger than 2. The larger-than-1 requirement is imposed, at least, by
/// the modular inversion code.
pub struct Modulus<M> {
limbs: BoxedLimbs<M>, // Also `value >= 3`.
// n0 * N == -1 (mod r).
//
// r == 2**(N0_LIMBS_USED * LIMB_BITS) and LG_LITTLE_R == lg(r). This
// ensures that we can do integer division by |r| by simply ignoring
// `N0_LIMBS_USED` limbs. Similarly, we can calculate values modulo `r` by
// just looking at the lowest `N0_LIMBS_USED` limbs. This is what makes
// Montgomery multiplication efficient.
//
// As shown in Algorithm 1 of "Fast Prime Field Elliptic Curve Cryptography
// with 256 Bit Primes" by Shay Gueron and Vlad Krasnov, in the loop of a
// multi-limb Montgomery multiplication of a * b (mod n), given the
// unreduced product t == a * b, we repeatedly calculate:
//
// t1 := t % r |t1| is |t|'s lowest limb (see previous paragraph).
// t2 := t1*n0*n
// t3 := t + t2
// t := t3 / r copy all limbs of |t3| except the lowest to |t|.
//
// In the last step, it would only make sense to ignore the lowest limb of
// |t3| if it were zero. The middle steps ensure that this is the case:
//
// t3 == 0 (mod r)
// t + t2 == 0 (mod r)
// t + t1*n0*n == 0 (mod r)
// t1*n0*n == -t (mod r)
// t*n0*n == -t (mod r)
// n0*n == -1 (mod r)
// n0 == -1/n (mod r)
//
// Thus, in each iteration of the loop, we multiply by the constant factor
// n0, the negative inverse of n (mod r).
//
// TODO(perf): Not all 32-bit platforms actually make use of n0[1]. For the
// ones that don't, we could use a shorter `R` value and use faster `Limb`
// calculations instead of double-precision `u64` calculations.
n0: N0,
oneRR: One<M, RR>,
}
impl core::fmt::Debug for Modulus<super::N> {
fn fmt(&self, fmt: &mut ::core::fmt::Formatter) -> Result<(), ::core::fmt::Error> {
fmt.debug_struct("Modulus")
// TODO: Print modulus value.
.finish()
}
}
impl<M> Modulus<M> {
pub fn from_be_bytes_with_bit_length(
input: untrusted::Input,
) -> Result<(Self, bits::BitLength), error::KeyRejected> {
let limbs = BoxedLimbs::positive_minimal_width_from_be_bytes(input)?;
Self::from_boxed_limbs(limbs)
}
pub fn from_nonnegative_with_bit_length(
n: Nonnegative,
) -> Result<(Self, bits::BitLength), error::KeyRejected> {
let limbs = BoxedLimbs {
limbs: n.limbs.into_boxed_slice(),
m: PhantomData,
};
Self::from_boxed_limbs(limbs)
}
fn from_boxed_limbs(n: BoxedLimbs<M>) -> Result<(Self, bits::BitLength), error::KeyRejected> {
if n.len() > MODULUS_MAX_LIMBS {
return Err(error::KeyRejected::too_large());
}
if n.len() < MODULUS_MIN_LIMBS {
return Err(error::KeyRejected::unexpected_error());
}
if limb::limbs_are_even_constant_time(&n) != LimbMask::False {
return Err(error::KeyRejected::invalid_component());
}
if limb::limbs_less_than_limb_constant_time(&n, 3) != LimbMask::False {
return Err(error::KeyRejected::unexpected_error());
}
// n_mod_r = n % r. As explained in the documentation for `n0`, this is
// done by taking the lowest `N0_LIMBS_USED` limbs of `n`.
let n0 = {
extern "C" {
fn GFp_bn_neg_inv_mod_r_u64(n: u64) -> u64;
}
// XXX: u64::from isn't guaranteed to be constant time.
let mut n_mod_r: u64 = u64::from(n[0]);
if N0_LIMBS_USED == 2 {
// XXX: If we use `<< LIMB_BITS` here then 64-bit builds
// fail to compile because of `deny(exceeding_bitshifts)`.
debug_assert_eq!(LIMB_BITS, 32);
n_mod_r |= u64::from(n[1]) << 32;
}
N0::from(unsafe { GFp_bn_neg_inv_mod_r_u64(n_mod_r) })
};
let bits = limb::limbs_minimal_bits(&n.limbs);
let oneRR = {
let partial = PartialModulus {
limbs: &n.limbs,
n0: n0.clone(),
m: PhantomData,
};
One::newRR(&partial, bits)
};
Ok((
Modulus {
limbs: n,
n0,
oneRR,
},
bits,
))
}
#[inline]
fn width(&self) -> Width<M> { self.limbs.width() }
fn zero<E>(&self) -> Elem<M, E> {
Elem {
limbs: BoxedLimbs::zero(self.width()),
encoding: PhantomData,
}
}
// TODO: Get rid of this
fn one(&self) -> Elem<M, Unencoded> {
let mut r = self.zero();
r.limbs[0] = 1;
r
}
pub fn oneRR(&self) -> &One<M, RR> { &self.oneRR }
pub fn to_elem<L>(&self, l: &Modulus<L>) -> Elem<L, Unencoded>
where
M: SmallerModulus<L>,
{
// TODO: Encode this assertion into the `where` above.
assert_eq!(self.width().num_limbs, l.width().num_limbs);
let limbs = self.limbs.clone();
Elem {
limbs: BoxedLimbs {
limbs: limbs.limbs,
m: PhantomData,
},
encoding: PhantomData,
}
}
fn as_partial(&self) -> PartialModulus<M> {
PartialModulus {
limbs: &self.limbs,
n0: self.n0.clone(),
m: PhantomData,
}
}
}
struct PartialModulus<'a, M> {
limbs: &'a [Limb],
n0: N0,
m: PhantomData<M>,
}
impl<'a, M> PartialModulus<'a, M> {
// TODO: XXX Avoid duplication with `Modulus`.
fn zero(&self) -> Elem<M, R> {
let width = Width {
num_limbs: self.limbs.len(),
m: PhantomData,
};
Elem {
limbs: BoxedLimbs::zero(width),
encoding: PhantomData,
}
}
}
/// Elements of ℤ/mℤ for some modulus *m*.
//
// Defaulting `E` to `Unencoded` is a convenience for callers from outside this
// submodule. However, for maximum clarity, we always explicitly use
// `Unencoded` within the `bigint` submodule.
pub struct Elem<M, E = Unencoded> {
limbs: BoxedLimbs<M>,
/// The number of Montgomery factors that need to be canceled out from
/// `value` to get the actual value.
encoding: PhantomData<E>,
}
// TODO: `derive(Clone)` after https://github.com/rust-lang/rust/issues/26925
// is resolved or restrict `M: Clone` and `E: Clone`.
impl<M, E> Clone for Elem<M, E> {
fn clone(&self) -> Self {
Elem {
limbs: self.limbs.clone(),
encoding: self.encoding.clone(),
}
}
}
impl<M, E> Elem<M, E> {
#[inline]
pub fn is_zero(&self) -> bool { self.limbs.is_zero() }
}
impl<M, E: ReductionEncoding> Elem<M, E> {
fn decode_once(self, m: &Modulus<M>) -> Elem<M, <E as ReductionEncoding>::Output> {
// A multiplication isn't required since we're multiplying by the
// unencoded value one (1); only a Montgomery reduction is needed.
// However the only non-multiplication Montgomery reduction function we
// have requires the input to be large, so we avoid using it here.
let mut limbs = self.limbs;
let num_limbs = m.width().num_limbs;
let mut one = [0; MODULUS_MAX_LIMBS];
one[0] = 1;
let one = &one[..num_limbs]; // assert!(num_limbs <= MODULUS_MAX_LIMBS);
limbs_mont_mul(&mut limbs, &one, &m.limbs, &m.n0);
Elem {
limbs,
encoding: PhantomData,
}
}
}
impl<M> Elem<M, R> {
#[inline]
pub fn into_unencoded(self, m: &Modulus<M>) -> Elem<M, Unencoded> { self.decode_once(m) }
}
impl<M> Elem<M, Unencoded> {
pub fn from_be_bytes_padded(
input: untrusted::Input, m: &Modulus<M>,
) -> Result<Self, error::Unspecified> {
Ok(Elem {
limbs: BoxedLimbs::from_be_bytes_padded_less_than(input, m)?,
encoding: PhantomData,
})
}
#[inline]
pub fn fill_be_bytes(&self, out: &mut [u8]) {
// See Falko Strenzke, "Manger's Attack revisited", ICICS 2010.
limb::big_endian_from_limbs(&self.limbs, out)
}
pub fn into_modulus<MM>(self) -> Result<Modulus<MM>, error::KeyRejected> {
let (m, _bits) =
Modulus::from_boxed_limbs(BoxedLimbs::minimal_width_from_unpadded(&self.limbs))?;
Ok(m)
}
fn is_one(&self) -> bool {
limb::limbs_equal_limb_constant_time(&self.limbs, 1) == LimbMask::True
}
}
pub fn elem_mul<M, AF, BF>(
a: &Elem<M, AF>, b: Elem<M, BF>, m: &Modulus<M>,
) -> Elem<M, <(AF, BF) as ProductEncoding>::Output>
where
(AF, BF): ProductEncoding,
{
elem_mul_(a, b, &m.as_partial())
}
fn elem_mul_<M, AF, BF>(
a: &Elem<M, AF>, mut b: Elem<M, BF>, m: &PartialModulus<M>,
) -> Elem<M, <(AF, BF) as ProductEncoding>::Output>
where
(AF, BF): ProductEncoding,
{
limbs_mont_mul(&mut b.limbs, &a.limbs, &m.limbs, &m.n0);
Elem {
limbs: b.limbs,
encoding: PhantomData,
}
}
fn elem_mul_by_2<M, AF>(a: &mut Elem<M, AF>, m: &PartialModulus<M>) {
extern "C" {
fn LIMBS_shl_mod(r: *mut Limb, a: *const Limb, m: *const Limb, num_limbs: c::size_t);
}
unsafe {
LIMBS_shl_mod(
a.limbs.as_mut_ptr(),
a.limbs.as_ptr(),
m.limbs.as_ptr(),
m.limbs.len(),
);
}
}
pub fn elem_reduced_once<Larger, Smaller: SlightlySmallerModulus<Larger>>(
a: &Elem<Larger, Unencoded>, m: &Modulus<Smaller>,
) -> Elem<Smaller, Unencoded> {
let mut r = a.limbs.clone();
assert!(r.len() <= m.limbs.len());
limb::limbs_reduce_once_constant_time(&mut r, &m.limbs);
Elem {
limbs: BoxedLimbs {
limbs: r.limbs,
m: PhantomData,
},
encoding: PhantomData,
}
}
#[inline]
pub fn elem_reduced<Larger, Smaller: NotMuchSmallerModulus<Larger>>(
a: &Elem<Larger, Unencoded>, m: &Modulus<Smaller>,
) -> Result<Elem<Smaller, RInverse>, error::Unspecified> {
extern "C" {
fn GFp_bn_from_montgomery_in_place(
r: *mut Limb, num_r: c::size_t, a: *mut Limb, num_a: c::size_t, n: *const Limb,
num_n: c::size_t, n0: &N0,
) -> bssl::Result;
}
let mut tmp = [0; MODULUS_MAX_LIMBS];
let tmp = &mut tmp[..a.limbs.len()];
tmp.copy_from_slice(&a.limbs);
let mut r = m.zero();
Result::from(unsafe {
GFp_bn_from_montgomery_in_place(
r.limbs.as_mut_ptr(),
r.limbs.len(),
tmp.as_mut_ptr(),
tmp.len(),
m.limbs.as_ptr(),
m.limbs.len(),
&m.n0,
)
})?;
Ok(r)
}
fn elem_squared<M, E>(
mut a: Elem<M, E>, m: &PartialModulus<M>,
) -> Elem<M, <(E, E) as ProductEncoding>::Output>
where
(E, E): ProductEncoding,
{
limbs_mont_square(&mut a.limbs, &m.limbs, &m.n0);
Elem {
limbs: a.limbs,
encoding: PhantomData,
}
}
pub fn elem_widen<Larger, Smaller: SmallerModulus<Larger>>(
a: Elem<Smaller, Unencoded>, m: &Modulus<Larger>,
) -> Elem<Larger, Unencoded> {
let mut r = m.zero();
r.limbs[..a.limbs.len()].copy_from_slice(&a.limbs);
r
}
// TODO: Document why this works for all Montgomery factors.
pub fn elem_add<M, E>(mut a: Elem<M, E>, b: Elem<M, E>, m: &Modulus<M>) -> Elem<M, E> {
extern "C" {
// `r` and `a` may alias.
fn LIMBS_add_mod(
r: *mut Limb, a: *const Limb, b: *const Limb, m: *const Limb, num_limbs: c::size_t,
);
}
unsafe {
LIMBS_add_mod(
a.limbs.as_mut_ptr(),
a.limbs.as_ptr(),
b.limbs.as_ptr(),
m.limbs.as_ptr(),
m.limbs.len(),
)
}
a
}
// TODO: Document why this works for all Montgomery factors.
pub fn elem_sub<M, E>(mut a: Elem<M, E>, b: &Elem<M, E>, m: &Modulus<M>) -> Elem<M, E> {
extern "C" {
// `r` and `a` may alias.
fn LIMBS_sub_mod(
r: *mut Limb, a: *const Limb, b: *const Limb, m: *const Limb, num_limbs: c::size_t,
);
}
unsafe {
LIMBS_sub_mod(
a.limbs.as_mut_ptr(),
a.limbs.as_ptr(),
b.limbs.as_ptr(),
m.limbs.as_ptr(),
m.limbs.len(),
);
}
a
}
// The value 1, Montgomery-encoded some number of times.
pub struct One<M, E>(Elem<M, E>);
impl<M> One<M, RR> {
// Returns RR = = R**2 (mod n) where R = 2**r is the smallest power of
// 2**LIMB_BITS such that R > m.
//
// Even though the assembly on some 32-bit platforms works with 64-bit
// values, using `LIMB_BITS` here, rather than `N0_LIMBS_USED * LIMB_BITS`,
// is correct because R**2 will still be a multiple of the latter as
// `N0_LIMBS_USED` is either one or two.
fn newRR(m: &PartialModulus<M>, m_bits: bits::BitLength) -> One<M, RR> {
let m_bits = m_bits.as_usize_bits();
let r = (m_bits + (LIMB_BITS - 1)) / LIMB_BITS * LIMB_BITS;
// base = 2**(lg m - 1).
let bit = m_bits - 1;
let mut base = m.zero();
base.limbs[bit / LIMB_BITS] = 1 << (bit % LIMB_BITS);
// Double `base` so that base == R == 2**r (mod m). For normal moduli
// that have the high bit of the highest limb set, this requires one
// doubling. Unusual moduli require more doublings but we are less
// concerned about the performance of those.
//
// Then double `base` again so that base == 2*R (mod n), i.e. `2` in
// Montgomery form (`elem_exp_vartime_()` requires the base to be in
// Montgomery form). Then compute
// RR = R**2 == base**r == R**r == (2**r)**r (mod n).
//
// Take advantage of the fact that `elem_mul_by_2` is faster than
// `elem_squared` by replacing some of the early squarings with shifts.
// TODO: Benchmark shift vs. squaring performance to determine the
// optimal value of `lg_base`.
let lg_base = 2usize; // Shifts vs. squaring trade-off.
debug_assert_eq!(lg_base.count_ones(), 1); // Must 2**n for n >= 0.
let shifts = r - bit + lg_base;
let exponent = (r / lg_base) as u64;
for _ in 0..shifts {
elem_mul_by_2(&mut base, m)
}
let RR = elem_exp_vartime_(base, exponent, m);
One(Elem {
limbs: RR.limbs,
encoding: PhantomData, // PhantomData<RR>
})
}
}
impl<M, E> AsRef<Elem<M, E>> for One<M, E> {
fn as_ref(&self) -> &Elem<M, E> { &self.0 }
}
/// A non-secret odd positive value in the range
/// [3, PUBLIC_EXPONENT_MAX_VALUE].
#[derive(Clone, Copy, Debug)]
pub struct PublicExponent(u64);
impl PublicExponent {
pub fn from_be_bytes(
input: untrusted::Input, min_value: u64,
) -> Result<Self, error::KeyRejected> {
if input.len() > 5 {
return Err(error::KeyRejected::too_large());
}
let value = input.read_all_mut(error::KeyRejected::invalid_encoding(), |input| {
// The exponent can't be zero and it can't be prefixed with
// zero-valued bytes.
if input.peek(0) {
return Err(error::KeyRejected::invalid_encoding());
}
let mut value = 0u64;
loop {
let byte = input
.read_byte()
.map_err(|untrusted::EndOfInput| error::KeyRejected::invalid_encoding())?;
value = (value << 8) | u64::from(byte);
if input.at_end() {
return Ok(value);
}
}
})?;
// Step 2 / Step b. NIST SP800-89 defers to FIPS 186-3, which requires
// `e >= 65537`. We enforce this when signing, but are more flexible in
// verification, for compatibility. Only small public exponents are
// supported.
if value & 1 != 1 {
return Err(error::KeyRejected::invalid_component());
}
debug_assert!(min_value & 1 == 1);
debug_assert!(min_value <= PUBLIC_EXPONENT_MAX_VALUE);
if min_value < 3 {
return Err(error::KeyRejected::invalid_component());
}
if value < min_value {
return Err(error::KeyRejected::too_small());
}
if value > PUBLIC_EXPONENT_MAX_VALUE {
return Err(error::KeyRejected::too_large());
}
Ok(PublicExponent(value))
}
}
// This limit was chosen to bound the performance of the simple
// exponentiation-by-squaring implementation in `elem_exp_vartime`. In
// particular, it helps mitigate theoretical resource exhaustion attacks. 33
// bits was chosen as the limit based on the recommendations in [1] and
// [2]. Windows CryptoAPI (at least older versions) doesn't support values
// larger than 32 bits [3], so it is unlikely that exponents larger than 32
// bits are being used for anything Windows commonly does.
//
// [1] https://www.imperialviolet.org/2012/03/16/rsae.html
// [2] https://www.imperialviolet.org/2012/03/17/rsados.html
// [3] https://msdn.microsoft.com/en-us/library/aa387685(VS.85).aspx
const PUBLIC_EXPONENT_MAX_VALUE: u64 = (1u64 << 33) - 1;
/// Calculates base**exponent (mod m).
// TODO: The test coverage needs to be expanded, e.g. test with the largest
// accepted exponent and with the most common values of 65537 and 3.
pub fn elem_exp_vartime<M>(
base: Elem<M, Unencoded>, PublicExponent(exponent): PublicExponent, m: &Modulus<M>,
) -> Elem<M, R> {
let base = elem_mul(m.oneRR().as_ref(), base, &m);
elem_exp_vartime_(base, exponent, &m.as_partial())
}
/// Calculates base**exponent (mod m).
fn elem_exp_vartime_<M>(base: Elem<M, R>, exponent: u64, m: &PartialModulus<M>) -> Elem<M, R> {
// Use what [Knuth] calls the "S-and-X binary method", i.e. variable-time
// square-and-multiply that scans the exponent from the most significant
// bit to the least significant bit (left-to-right). Left-to-right requires
// less storage compared to right-to-left scanning, at the cost of needing
// to compute `exponent.leading_zeros()`, which we assume to be cheap.
//
// During RSA public key operations the exponent is almost always either 65537
// (0b10000000000000001) or 3 (0b11), both of which have a Hamming weight
// of 2. During Montgomery setup the exponent is almost always a power of two,
// with Hamming weight 1. As explained in [Knuth], exponentiation by squaring
// is the most efficient algorithm when the Hamming weight is 2 or less. It
// isn't the most efficient for all other, uncommon, exponent values but any
// suboptimality is bounded by `PUBLIC_EXPONENT_MAX_VALUE`.
//
// This implementation is slightly simplified by taking advantage of the
// fact that we require the exponent to be a positive integer.
//
// [Knuth]: The Art of Computer Programming, Volume 2: Seminumerical
// Algorithms (3rd Edition), Section 4.6.3.
assert!(exponent >= 1);
assert!(exponent <= PUBLIC_EXPONENT_MAX_VALUE);
let mut acc = base.clone();
let mut bit = 1 << (64 - 1 - exponent.leading_zeros());
debug_assert!((exponent & bit) != 0);
while bit > 1 {
bit >>= 1;
acc = elem_squared(acc, m);
if (exponent & bit) != 0 {
acc = elem_mul_(&base, acc, m);
}
}
acc
}
// `M` represents the prime modulus for which the exponent is in the interval
// [1, `m` - 1).
pub struct PrivateExponent<M> {
limbs: BoxedLimbs<M>,
}
impl<M> PrivateExponent<M> {
pub fn from_be_bytes_padded(
input: untrusted::Input, p: &Modulus<M>,
) -> Result<Self, error::Unspecified> {
let dP = BoxedLimbs::from_be_bytes_padded_less_than(input, p)?;
// Proof that `dP < p - 1`:
//
// If `dP < p` then either `dP == p - 1` or `dP < p - 1`. Since `p` is
// odd, `p - 1` is even. `d` is odd, and an odd number modulo an even
// number is odd. Therefore `dP` must be odd. But then it cannot be
// `p - 1` and so we know `dP < p - 1`.
//
// Further we know `dP != 0` because `dP` is not even.
if limb::limbs_are_even_constant_time(&dP) != LimbMask::False {
return Err(error::Unspecified);
}
Ok(PrivateExponent { limbs: dP })
}
}
impl<M: Prime> PrivateExponent<M> {
// Returns `p - 2`.
fn for_flt(p: &Modulus<M>) -> Self {
let two = elem_add(p.one(), p.one(), p);
let p_minus_2 = elem_sub(p.zero(), &two, p);
PrivateExponent {
limbs: p_minus_2.limbs,
}
}
}
#[cfg(not(target_arch = "x86_64"))]
pub fn elem_exp_consttime<M>(
base: Elem<M, R>, exponent: &PrivateExponent<M>, m: &Modulus<M>,
) -> Result<Elem<M, Unencoded>, error::Unspecified> {
use crate::limb::Window;
const WINDOW_BITS: usize = 5;
const TABLE_ENTRIES: usize = 1 << WINDOW_BITS;
let num_limbs = m.limbs.len();
let mut table = vec![0; TABLE_ENTRIES * num_limbs];
fn gather<M>(table: &[Limb], i: Window, r: &mut Elem<M, R>) {
extern "C" {
fn LIMBS_select_512_32(
r: *mut Limb, table: *const Limb, num_limbs: c::size_t, i: Window,
) -> bssl::Result;
}
Result::from(unsafe {
LIMBS_select_512_32(r.limbs.as_mut_ptr(), table.as_ptr(), r.limbs.len(), i)
})
.unwrap();
}
fn power<M>(
table: &[Limb], i: Window, mut acc: Elem<M, R>, mut tmp: Elem<M, R>, m: &Modulus<M>,
) -> (Elem<M, R>, Elem<M, R>) {
for _ in 0..WINDOW_BITS {
acc = elem_squared(acc, &m.as_partial());
}
gather(table, i, &mut tmp);
let acc = elem_mul(&tmp, acc, m);
(acc, tmp)
}
let tmp = m.one();
let tmp = elem_mul(m.oneRR().as_ref(), tmp, m);
fn entry(table: &[Limb], i: usize, num_limbs: usize) -> &[Limb] {
&table[(i * num_limbs)..][..num_limbs]
}
fn entry_mut(table: &mut [Limb], i: usize, num_limbs: usize) -> &mut [Limb] {
&mut table[(i * num_limbs)..][..num_limbs]
}
let num_limbs = m.limbs.len();
entry_mut(&mut table, 0, num_limbs).copy_from_slice(&tmp.limbs);
entry_mut(&mut table, 1, num_limbs).copy_from_slice(&base.limbs);
for i in 2..TABLE_ENTRIES {
let (src1, src2) = if i % 2 == 0 {
(i / 2, i / 2)
} else {
(i - 1, 1)
};
let (previous, rest) = table.split_at_mut(num_limbs * i);
let src1 = entry(previous, src1, num_limbs);
let src2 = entry(previous, src2, num_limbs);
let dst = entry_mut(rest, 0, num_limbs);
limbs_mont_product(dst, src1, src2, &m.limbs, &m.n0);
}
let (r, _) = limb::fold_5_bit_windows(
&exponent.limbs,
|initial_window| {
let mut r = Elem {
limbs: base.limbs,
encoding: PhantomData,
};
gather(&table, initial_window, &mut r);
(r, tmp)
},
|(acc, tmp), window| power(&table, window, acc, tmp, m),
);
let r = r.into_unencoded(m);
Ok(r)
}
/// Uses Fermat's Little Theorem to calculate modular inverse in constant time.
pub fn elem_inverse_consttime<M: Prime>(
a: Elem<M, R>, m: &Modulus<M>,
) -> Result<Elem<M, Unencoded>, error::Unspecified> {
elem_exp_consttime(a, &PrivateExponent::for_flt(&m), m)
}
#[cfg(target_arch = "x86_64")]
pub fn elem_exp_consttime<M>(
base: Elem<M, R>, exponent: &PrivateExponent<M>, m: &Modulus<M>,
) -> Result<Elem<M, Unencoded>, error::Unspecified> {
// The x86_64 assembly was written under the assumption that the input data
// is aligned to `MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH` bytes, which was/is
// 64 in OpenSSL. Similarly, OpenSSL uses the x86_64 assembly functions by
// giving it only inputs `tmp`, `am`, and `np` that immediately follow the
// table. The code seems to "work" even when the inputs aren't exactly
// like that but the side channel defenses might not be as effective. All
// the awkwardness here stems from trying to use the assembly code like
// OpenSSL does.
use crate::limb::Window;
const WINDOW_BITS: usize = 5;
const TABLE_ENTRIES: usize = 1 << WINDOW_BITS;
let num_limbs = m.limbs.len();
const ALIGNMENT: usize = 64;
assert_eq!(ALIGNMENT % LIMB_BYTES, 0);
let mut table = vec![0; ((TABLE_ENTRIES + 3) * num_limbs) + ALIGNMENT];
let (table, state) = {
let misalignment = (table.as_ptr() as usize) % ALIGNMENT;
let table = &mut table[((ALIGNMENT - misalignment) / LIMB_BYTES)..];
assert_eq!((table.as_ptr() as usize) % ALIGNMENT, 0);
table.split_at_mut(TABLE_ENTRIES * num_limbs)
};
fn entry(table: &[Limb], i: usize, num_limbs: usize) -> &[Limb] {
&table[(i * num_limbs)..][..num_limbs]
}
fn entry_mut(table: &mut [Limb], i: usize, num_limbs: usize) -> &mut [Limb] {
&mut table[(i * num_limbs)..][..num_limbs]
}
const ACC: usize = 0; // `tmp` in OpenSSL
const BASE: usize = ACC + 1; // `am` in OpenSSL
const M: usize = BASE + 1; // `np` in OpenSSL
entry_mut(state, BASE, num_limbs).copy_from_slice(&base.limbs);
entry_mut(state, M, num_limbs).copy_from_slice(&m.limbs);
fn scatter(table: &mut [Limb], state: &[Limb], i: Window, num_limbs: usize) {
extern "C" {
fn GFp_bn_scatter5(a: *const Limb, a_len: c::size_t, table: *mut Limb, i: Window);
}
unsafe {
GFp_bn_scatter5(
entry(state, ACC, num_limbs).as_ptr(),
num_limbs,
table.as_mut_ptr(),
i,
)
}
}
fn gather(table: &[Limb], state: &mut [Limb], i: Window, num_limbs: usize) {
extern "C" {
fn GFp_bn_gather5(r: *mut Limb, a_len: c::size_t, table: *const Limb, i: Window);
}
unsafe {
GFp_bn_gather5(
entry_mut(state, ACC, num_limbs).as_mut_ptr(),
num_limbs,
table.as_ptr(),
i,
)
}
}
fn gather_square(table: &[Limb], state: &mut [Limb], n0: &N0, i: Window, num_limbs: usize) {
gather(table, state, i, num_limbs);
assert_eq!(ACC, 0);
let (acc, rest) = state.split_at_mut(num_limbs);
let m = entry(rest, M - 1, num_limbs);
limbs_mont_square(acc, m, n0);
}
fn gather_mul_base(table: &[Limb], state: &mut [Limb], n0: &N0, i: Window, num_limbs: usize) {
extern "C" {
fn GFp_bn_mul_mont_gather5(
rp: *mut Limb, ap: *const Limb, table: *const Limb, np: *const Limb, n0: &N0,
num: c::size_t, power: Window,
);
}
unsafe {
GFp_bn_mul_mont_gather5(
entry_mut(state, ACC, num_limbs).as_mut_ptr(),
entry(state, BASE, num_limbs).as_ptr(),
table.as_ptr(),
entry(state, M, num_limbs).as_ptr(),
n0,
num_limbs,
i,
);
}
}
fn power(table: &[Limb], state: &mut [Limb], n0: &N0, i: Window, num_limbs: usize) {
extern "C" {
fn GFp_bn_power5(
r: *mut Limb, a: *const Limb, table: *const Limb, n: *const Limb, n0: &N0,
num: c::size_t, i: Window,
);
}
unsafe {
GFp_bn_power5(
entry_mut(state, ACC, num_limbs).as_mut_ptr(),
entry_mut(state, ACC, num_limbs).as_mut_ptr(),
table.as_ptr(),
entry(state, M, num_limbs).as_ptr(),
n0,
num_limbs,
i,
);
}
}
// table[0] = base**0.
{
let acc = entry_mut(state, ACC, num_limbs);
acc[0] = 1;
limbs_mont_mul(acc, &m.oneRR.0.limbs, &m.limbs, &m.n0);
}
scatter(table, state, 0, num_limbs);
// table[1] = base**1.
entry_mut(state, ACC, num_limbs).copy_from_slice(&base.limbs);
scatter(table, state, 1, num_limbs);
for i in 2..(TABLE_ENTRIES as Window) {
if i % 2 == 0 {
// TODO: Optimize this to avoid gathering
gather_square(table, state, &m.n0, i / 2, num_limbs);
} else {
gather_mul_base(table, state, &m.n0, i - 1, num_limbs)
};
scatter(table, state, i, num_limbs);
}
let state = limb::fold_5_bit_windows(
&exponent.limbs,
|initial_window| {
gather(table, state, initial_window, num_limbs);
state
},
|state, window| {
power(table, state, &m.n0, window, num_limbs);
state
},
);
extern "C" {
fn GFp_bn_from_montgomery(
r: *mut Limb, a: *const Limb, not_used: *const Limb, n: *const Limb, n0: &N0,
num: c::size_t,
) -> bssl::Result;
}
Result::from(unsafe {
GFp_bn_from_montgomery(
entry_mut(state, ACC, num_limbs).as_mut_ptr(),
entry(state, ACC, num_limbs).as_ptr(),
core::ptr::null(),
entry(state, M, num_limbs).as_ptr(),
&m.n0,
num_limbs,
)
})?;
let mut r = Elem {
limbs: base.limbs,
encoding: PhantomData,
};
r.limbs.copy_from_slice(entry(state, ACC, num_limbs));
Ok(r)
}
/// Verified a == b**-1 (mod m), i.e. a**-1 == b (mod m).
pub fn verify_inverses_consttime<M>(
a: &Elem<M, R>, b: Elem<M, Unencoded>, m: &Modulus<M>,
) -> Result<(), error::Unspecified> {
if elem_mul(a, b, m).is_one() {
Ok(())
} else {
Err(error::Unspecified)
}
}
#[inline]
pub fn elem_verify_equal_consttime<M, E>(
a: &Elem<M, E>, b: &Elem<M, E>,
) -> Result<(), error::Unspecified> {
if limb::limbs_equal_limbs_consttime(&a.limbs, &b.limbs) == LimbMask::True {
Ok(())
} else {
Err(error::Unspecified)
}
}
/// Nonnegative integers.
pub struct Nonnegative {
limbs: Vec<Limb>,
}
impl Nonnegative {
pub fn from_be_bytes_with_bit_length(
input: untrusted::Input,
) -> Result<(Self, bits::BitLength), error::Unspecified> {
let mut limbs = vec![0; (input.len() + LIMB_BYTES - 1) / LIMB_BYTES];
// Rejects empty inputs.
limb::parse_big_endian_and_pad_consttime(input, &mut limbs)?;
while limbs.last() == Some(&0) {
let _ = limbs.pop();
}
let r_bits = limb::limbs_minimal_bits(&limbs);
Ok((Self { limbs }, r_bits))
}
#[inline]
pub fn is_odd(&self) -> bool {
limb::limbs_are_even_constant_time(&self.limbs) != LimbMask::True
}
pub fn verify_less_than(&self, other: &Self) -> Result<(), error::Unspecified> {
if !greater_than(other, self) {
return Err(error::Unspecified);
}
Ok(())
}
pub fn to_elem<M>(&self, m: &Modulus<M>) -> Result<Elem<M, Unencoded>, error::Unspecified> {
self.verify_less_than_modulus(&m)?;
let mut r = m.zero();
r.limbs[0..self.limbs.len()].copy_from_slice(&self.limbs);
Ok(r)
}
pub fn verify_less_than_modulus<M>(&self, m: &Modulus<M>) -> Result<(), error::Unspecified> {
if self.limbs.len() > m.limbs.len() {
return Err(error::Unspecified);
}
if self.limbs.len() == m.limbs.len() {
if limb::limbs_less_than_limbs_consttime(&self.limbs, &m.limbs) != LimbMask::True {
return Err(error::Unspecified);
}
}
return Ok(());
}
}
// Returns a > b.
fn greater_than(a: &Nonnegative, b: &Nonnegative) -> bool {
if a.limbs.len() == b.limbs.len() {
limb::limbs_less_than_limbs_vartime(&b.limbs, &a.limbs)
} else {
a.limbs.len() > b.limbs.len()
}
}
#[derive(Clone)]
#[repr(transparent)]
struct N0([Limb; 2]);
const N0_LIMBS_USED: usize = 64 / LIMB_BITS;
impl From<u64> for N0 {
#[inline]
fn from(n0: u64) -> Self {
#[cfg(target_pointer_width = "64")]
{
N0([n0, 0])
}
#[cfg(target_pointer_width = "32")]
{
N0([n0 as Limb, (n0 >> LIMB_BITS) as Limb])
}
}
}
/// r *= a
fn limbs_mont_mul(r: &mut [Limb], a: &[Limb], m: &[Limb], n0: &N0) {
debug_assert_eq!(r.len(), m.len());
debug_assert_eq!(a.len(), m.len());
unsafe {
GFp_bn_mul_mont(
r.as_mut_ptr(),
r.as_ptr(),
a.as_ptr(),
m.as_ptr(),
n0,
r.len(),
)
}
}
/// r = a * b
#[cfg(not(target_arch = "x86_64"))]
fn limbs_mont_product(r: &mut [Limb], a: &[Limb], b: &[Limb], m: &[Limb], n0: &N0) {
debug_assert_eq!(r.len(), m.len());
debug_assert_eq!(a.len(), m.len());
debug_assert_eq!(b.len(), m.len());
unsafe {
GFp_bn_mul_mont(
r.as_mut_ptr(),
a.as_ptr(),
b.as_ptr(),
m.as_ptr(),
n0,
r.len(),
)
}
}
/// r = r**2
fn limbs_mont_square(r: &mut [Limb], m: &[Limb], n0: &N0) {
debug_assert_eq!(r.len(), m.len());
unsafe {
GFp_bn_mul_mont(
r.as_mut_ptr(),
r.as_ptr(),
r.as_ptr(),
m.as_ptr(),
n0,
r.len(),
)
}
}
extern "C" {
// `r` and/or 'a' and/or 'b' may alias.
fn GFp_bn_mul_mont(
r: *mut Limb, a: *const Limb, b: *const Limb, n: *const Limb, n0: &N0, num_limbs: c::size_t,
);
}
#[cfg(test)]
mod tests {
use super::*;
use crate::test;
use untrusted;
// Type-level representation of an arbitrary modulus.
struct M {}
#[test]
fn test_elem_exp_consttime() {
test::from_file(
"src/rsa/bigint_elem_exp_consttime_tests.txt",
|section, test_case| {
assert_eq!(section, "");
let m = consume_modulus::<M>(test_case, "M");
let expected_result = consume_elem(test_case, "ModExp", &m);
let base = consume_elem(test_case, "A", &m);
let e = {
let bytes = test_case.consume_bytes("E");
PrivateExponent::from_be_bytes_padded(untrusted::Input::from(&bytes), &m)
.expect("valid exponent")
};
let base = into_encoded(base, &m);
let actual_result = elem_exp_consttime(base, &e, &m).unwrap();
assert_elem_eq(&actual_result, &expected_result);
Ok(())
},
)
}
#[test]
// TODO: fn test_elem_exp_vartime() using
// "src/rsa/bigint_elem_exp_vartime_tests.txt". See that file for details.
// In the meantime, the function is tested indirectly via the RSA
// verification and signing tests.
#[test]
fn test_elem_mul() {
test::from_file("src/rsa/bigint_elem_mul_tests.txt", |section, test_case| {
assert_eq!(section, "");
let m = consume_modulus::<M>(test_case, "M");
let expected_result = consume_elem(test_case, "ModMul", &m);
let a = consume_elem(test_case, "A", &m);
let b = consume_elem(test_case, "B", &m);
let b = into_encoded(b, &m);
let a = into_encoded(a, &m);
let actual_result = elem_mul(&a, b, &m);
let actual_result = actual_result.into_unencoded(&m);
assert_elem_eq(&actual_result, &expected_result);
Ok(())
})
}
#[test]
fn test_elem_squared() {
test::from_file(
"src/rsa/bigint_elem_squared_tests.txt",
|section, test_case| {
assert_eq!(section, "");
let m = consume_modulus::<M>(test_case, "M");
let expected_result = consume_elem(test_case, "ModSquare", &m);
let a = consume_elem(test_case, "A", &m);
let a = into_encoded(a, &m);
let actual_result = elem_squared(a, &m.as_partial());
let actual_result = actual_result.into_unencoded(&m);
assert_elem_eq(&actual_result, &expected_result);
Ok(())
},
)
}
#[test]
fn test_elem_reduced() {
test::from_file(
"src/rsa/bigint_elem_reduced_tests.txt",
|section, test_case| {
assert_eq!(section, "");
struct MM {}
unsafe impl SmallerModulus<MM> for M {}
unsafe impl NotMuchSmallerModulus<MM> for M {}
let m = consume_modulus::<M>(test_case, "M");
let expected_result = consume_elem(test_case, "R", &m);
let a =
consume_elem_unchecked::<MM>(test_case, "A", expected_result.limbs.len() * 2);
let actual_result = elem_reduced(&a, &m).unwrap();
let oneRR = m.oneRR();
let actual_result = elem_mul(oneRR.as_ref(), actual_result, &m);
assert_elem_eq(&actual_result, &expected_result);
Ok(())
},
)
}
#[test]
fn test_elem_reduced_once() {
test::from_file(
"src/rsa/bigint_elem_reduced_once_tests.txt",
|section, test_case| {
assert_eq!(section, "");
struct N {}
struct QQ {}
unsafe impl SmallerModulus<N> for QQ {}
unsafe impl SlightlySmallerModulus<N> for QQ {}
let qq = consume_modulus::<QQ>(test_case, "QQ");
let expected_result = consume_elem::<QQ>(test_case, "R", &qq);
let n = consume_modulus::<N>(test_case, "N");
let a = consume_elem::<N>(test_case, "A", &n);
let actual_result = elem_reduced_once(&a, &qq);
assert_elem_eq(&actual_result, &expected_result);
Ok(())
},
)
}
#[test]
fn test_modulus_debug() {
let (modulus, _) = Modulus::from_be_bytes_with_bit_length(untrusted::Input::from(
&vec![0xff; LIMB_BYTES * MODULUS_MIN_LIMBS],
))
.unwrap();
assert_eq!("Modulus", format!("{:?}", modulus));
}
#[test]
fn test_public_exponent_debug() {
let exponent =
PublicExponent::from_be_bytes(untrusted::Input::from(&[0x1, 0x00, 0x01]), 65537)
.unwrap();
assert_eq!("PublicExponent(65537)", format!("{:?}", exponent));
}
fn consume_elem<M>(
test_case: &mut test::TestCase, name: &str, m: &Modulus<M>,
) -> Elem<M, Unencoded> {
let value = test_case.consume_bytes(name);
Elem::from_be_bytes_padded(untrusted::Input::from(&value), m).unwrap()
}
fn consume_elem_unchecked<M>(
test_case: &mut test::TestCase, name: &str, num_limbs: usize,
) -> Elem<M, Unencoded> {
let value = consume_nonnegative(test_case, name);
let mut limbs = BoxedLimbs::zero(Width {
num_limbs,
m: PhantomData,
});
limbs[0..value.limbs.len()].copy_from_slice(&value.limbs);
Elem {
limbs,
encoding: PhantomData,
}
}
fn consume_modulus<M>(test_case: &mut test::TestCase, name: &str) -> Modulus<M> {
let value = test_case.consume_bytes(name);
let (value, _) =
Modulus::from_be_bytes_with_bit_length(untrusted::Input::from(&value)).unwrap();
value
}
fn consume_nonnegative(test_case: &mut test::TestCase, name: &str) -> Nonnegative {
let bytes = test_case.consume_bytes(name);
let (r, _r_bits) =
Nonnegative::from_be_bytes_with_bit_length(untrusted::Input::from(&bytes)).unwrap();
r
}
fn assert_elem_eq<M, E>(a: &Elem<M, E>, b: &Elem<M, E>) {
elem_verify_equal_consttime(&a, b).unwrap()
}
fn into_encoded<M>(a: Elem<M, Unencoded>, m: &Modulus<M>) -> Elem<M, R> {
elem_mul(m.oneRR().as_ref(), a, m)
}
}