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Deferred Montgomery reductions #20

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Deferred Montgomery reductions #20

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fea0013
Introduce FpWide type to handle unreduced products.
ebfull Nov 22, 2019
cff96bf
Defer Montgomery reductions during Fp2 multiplication.
ebfull Nov 22, 2019
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110 changes: 77 additions & 33 deletions src/fp.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -9,12 +9,58 @@ use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};

use crate::util::{adc, mac, sbb};

mod magnitudes;

pub use magnitudes::*;

// The internal representation of this type is six 64-bit unsigned
// integers in little-endian order. `Fp` values are always in
// Montgomery form; i.e., Scalar(a) = aR mod p, with R = 2^384.
#[derive(Copy, Clone)]
pub struct Fp([u64; 6]);

#[derive(Debug)]
pub struct FpWide([u64; 12]);

impl FpWide {
pub const fn add(&self, rhs: &FpWide) -> FpWide {
let (d0, carry) = adc(self.0[0], rhs.0[0], 0);
let (d1, carry) = adc(self.0[1], rhs.0[1], carry);
let (d2, carry) = adc(self.0[2], rhs.0[2], carry);
let (d3, carry) = adc(self.0[3], rhs.0[3], carry);
let (d4, carry) = adc(self.0[4], rhs.0[4], carry);
let (d5, carry) = adc(self.0[5], rhs.0[5], carry);
let (d6, carry) = adc(self.0[6], rhs.0[6], carry);
let (d7, carry) = adc(self.0[7], rhs.0[7], carry);
let (d8, carry) = adc(self.0[8], rhs.0[8], carry);
let (d9, carry) = adc(self.0[9], rhs.0[9], carry);
let (d10, carry) = adc(self.0[10], rhs.0[10], carry);
let (d11, _) = adc(self.0[11], rhs.0[11], carry);

FpWide([d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11])
}

/// Negates an element of magnitude 1 by subtracting it from a multiple of
/// the prime that is larger than this element could be (p^2), increasing
/// the magnitude by 1.
pub const fn negate(&self) -> FpWide {
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This should be called negate_u1. (I would use a generic, but it's not stable in Rust yet for const functions...)

let (d0, borrow) = sbb(<U1 as Magnitude>::P2[0], self.0[0], 0);
let (d1, borrow) = sbb(<U1 as Magnitude>::P2[1], self.0[1], borrow);
let (d2, borrow) = sbb(<U1 as Magnitude>::P2[2], self.0[2], borrow);
let (d3, borrow) = sbb(<U1 as Magnitude>::P2[3], self.0[3], borrow);
let (d4, borrow) = sbb(<U1 as Magnitude>::P2[4], self.0[4], borrow);
let (d5, borrow) = sbb(<U1 as Magnitude>::P2[5], self.0[5], borrow);
let (d6, borrow) = sbb(<U1 as Magnitude>::P2[6], self.0[6], borrow);
let (d7, borrow) = sbb(<U1 as Magnitude>::P2[7], self.0[7], borrow);
let (d8, borrow) = sbb(<U1 as Magnitude>::P2[8], self.0[8], borrow);
let (d9, borrow) = sbb(<U1 as Magnitude>::P2[9], self.0[9], borrow);
let (d10, borrow) = sbb(<U1 as Magnitude>::P2[10], self.0[10], borrow);
let (d11, _) = sbb(<U1 as Magnitude>::P2[11], self.0[11], borrow);

FpWide([d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11])
}
}

impl fmt::Debug for Fp {
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
let tmp = self.to_bytes();
Expand Down Expand Up @@ -199,9 +245,9 @@ impl Fp {
pub fn to_bytes(&self) -> [u8; 48] {
// Turn into canonical form by computing
// (a.R) / R = a
let tmp = Fp::montgomery_reduce(
let tmp = Fp::montgomery_reduce(FpWide([
self.0[0], self.0[1], self.0[2], self.0[3], self.0[4], self.0[5], 0, 0, 0, 0, 0, 0,
);
]));

let mut res = [0; 48];
res[0..8].copy_from_slice(&tmp.0[5].to_be_bytes());
Expand All @@ -223,9 +269,9 @@ impl Fp {
// (p - 1) // 2.

// First, because self is in Montgomery form we need to reduce it
let tmp = Fp::montgomery_reduce(
let tmp = Fp::montgomery_reduce(FpWide([
self.0[0], self.0[1], self.0[2], self.0[3], self.0[4], self.0[5], 0, 0, 0, 0, 0, 0,
);
]));

let (_, borrow) = sbb(tmp.0[0], 0xdcff7fffffffd556, 0);
let (_, borrow) = sbb(tmp.0[1], 0x0f55ffff58a9ffff, borrow);
Expand Down Expand Up @@ -369,32 +415,19 @@ impl Fp {
}

#[inline(always)]
const fn montgomery_reduce(
t0: u64,
t1: u64,
t2: u64,
t3: u64,
t4: u64,
t5: u64,
t6: u64,
t7: u64,
t8: u64,
t9: u64,
t10: u64,
t11: u64,
) -> Self {
pub const fn montgomery_reduce(t: FpWide) -> Self {
// The Montgomery reduction here is based on Algorithm 14.32 in
// Handbook of Applied Cryptography
// <http://cacr.uwaterloo.ca/hac/about/chap14.pdf>.

let k = t0.wrapping_mul(INV);
let (_, carry) = mac(t0, k, MODULUS[0], 0);
let (r1, carry) = mac(t1, k, MODULUS[1], carry);
let (r2, carry) = mac(t2, k, MODULUS[2], carry);
let (r3, carry) = mac(t3, k, MODULUS[3], carry);
let (r4, carry) = mac(t4, k, MODULUS[4], carry);
let (r5, carry) = mac(t5, k, MODULUS[5], carry);
let (r6, r7) = adc(t6, 0, carry);
let k = t.0[0].wrapping_mul(INV);
let (_, carry) = mac(t.0[0], k, MODULUS[0], 0);
let (r1, carry) = mac(t.0[1], k, MODULUS[1], carry);
let (r2, carry) = mac(t.0[2], k, MODULUS[2], carry);
let (r3, carry) = mac(t.0[3], k, MODULUS[3], carry);
let (r4, carry) = mac(t.0[4], k, MODULUS[4], carry);
let (r5, carry) = mac(t.0[5], k, MODULUS[5], carry);
let (r6, r7) = adc(t.0[6], 0, carry);

let k = r1.wrapping_mul(INV);
let (_, carry) = mac(r1, k, MODULUS[0], 0);
Expand All @@ -403,7 +436,7 @@ impl Fp {
let (r4, carry) = mac(r4, k, MODULUS[3], carry);
let (r5, carry) = mac(r5, k, MODULUS[4], carry);
let (r6, carry) = mac(r6, k, MODULUS[5], carry);
let (r7, r8) = adc(t7, r7, carry);
let (r7, r8) = adc(t.0[7], r7, carry);

let k = r2.wrapping_mul(INV);
let (_, carry) = mac(r2, k, MODULUS[0], 0);
Expand All @@ -412,7 +445,7 @@ impl Fp {
let (r5, carry) = mac(r5, k, MODULUS[3], carry);
let (r6, carry) = mac(r6, k, MODULUS[4], carry);
let (r7, carry) = mac(r7, k, MODULUS[5], carry);
let (r8, r9) = adc(t8, r8, carry);
let (r8, r9) = adc(t.0[8], r8, carry);

let k = r3.wrapping_mul(INV);
let (_, carry) = mac(r3, k, MODULUS[0], 0);
Expand All @@ -421,7 +454,7 @@ impl Fp {
let (r6, carry) = mac(r6, k, MODULUS[3], carry);
let (r7, carry) = mac(r7, k, MODULUS[4], carry);
let (r8, carry) = mac(r8, k, MODULUS[5], carry);
let (r9, r10) = adc(t9, r9, carry);
let (r9, r10) = adc(t.0[9], r9, carry);

let k = r4.wrapping_mul(INV);
let (_, carry) = mac(r4, k, MODULUS[0], 0);
Expand All @@ -430,7 +463,7 @@ impl Fp {
let (r7, carry) = mac(r7, k, MODULUS[3], carry);
let (r8, carry) = mac(r8, k, MODULUS[4], carry);
let (r9, carry) = mac(r9, k, MODULUS[5], carry);
let (r10, r11) = adc(t10, r10, carry);
let (r10, r11) = adc(t.0[10], r10, carry);

let k = r5.wrapping_mul(INV);
let (_, carry) = mac(r5, k, MODULUS[0], 0);
Expand All @@ -439,7 +472,7 @@ impl Fp {
let (r8, carry) = mac(r8, k, MODULUS[3], carry);
let (r9, carry) = mac(r9, k, MODULUS[4], carry);
let (r10, carry) = mac(r10, k, MODULUS[5], carry);
let (r11, _) = adc(t11, r11, carry);
let (r11, _) = adc(t.0[11], r11, carry);

// Attempt to subtract the modulus, to ensure the value
// is smaller than the modulus.
Expand All @@ -448,6 +481,11 @@ impl Fp {

#[inline]
pub const fn mul(&self, rhs: &Fp) -> Fp {
Self::montgomery_reduce(self.mul_unreduced(rhs))
}

#[inline]
pub const fn mul_unreduced(&self, rhs: &Fp) -> FpWide {
let (t0, carry) = mac(0, self.0[0], rhs.0[0], 0);
let (t1, carry) = mac(0, self.0[0], rhs.0[1], carry);
let (t2, carry) = mac(0, self.0[0], rhs.0[2], carry);
Expand Down Expand Up @@ -490,12 +528,18 @@ impl Fp {
let (t9, carry) = mac(t9, self.0[5], rhs.0[4], carry);
let (t10, t11) = mac(t10, self.0[5], rhs.0[5], carry);

Self::montgomery_reduce(t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11)
FpWide([t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11])
}

/// Squares this element.
#[inline]
pub const fn square(&self) -> Self {
Self::montgomery_reduce(self.square_unreduced())
}

/// Squares this element, returning the wide value.
#[inline]
pub const fn square_unreduced(&self) -> FpWide {
let (t1, carry) = mac(0, self.0[0], self.0[1], 0);
let (t2, carry) = mac(0, self.0[0], self.0[2], carry);
let (t3, carry) = mac(0, self.0[0], self.0[3], carry);
Expand Down Expand Up @@ -541,7 +585,7 @@ impl Fp {
let (t10, carry) = mac(t10, self.0[5], self.0[5], carry);
let (t11, _) = adc(t11, 0, carry);

Self::montgomery_reduce(t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11)
FpWide([t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11])
}
}

Expand Down
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