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Refactor multiply implementation to eliminate reduce() call

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Henry de Valence 2018-01-19 18:23:41 -08:00
parent ec50ec96f7
commit 0e9b8e0a6e
1 changed files with 69 additions and 22 deletions
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91
src/backend/u64/field.rs
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@ -108,7 +108,19 @@ impl<'a, 'b> Mul<&'b FieldElement64> for &'a FieldElement64 {
let a: &[u64; 5] = &self.0;
let b: &[u64; 5] = &_rhs.0;
// 64-bit precomputations to avoid 128-bit multiplications
// Precondition: assume input limbs a[i], b[i] are bounded as
//
// a[i], b[i] < 2^(51 + b)
//
// where b is a real parameter measuring the "bit excess" of the limbs.
// 64-bit precomputations to avoid 128-bit multiplications.
//
// This fits into a u64 whenever 51 + b + lg(19) < 64.
//
// Since 51 + b + lg(19) < 51 + 4.25 + b
// = 55.25 + b,
// this fits if b < 8.75.
let b1_19 = b[1] * 19;
let b2_19 = b[2] * 19;
let b3_19 = b[3] * 19;
@ -121,34 +133,69 @@ impl<'a, 'b> Mul<&'b FieldElement64> for &'a FieldElement64 {
let mut c3: u128 = m(a[3],b[0]) + m(a[2],b[1]) + m(a[1],b[2]) + m(a[0],b[3]) + m(a[4],b4_19);
let mut c4: u128 = m(a[4],b[0]) + m(a[3],b[1]) + m(a[2],b[2]) + m(a[1],b[3]) + m(a[0],b[4]);
// Now c[i] < 2^2b * (1+i + (4-i)*19) < 2^(2b + lg(1+4*19)) < 2^(2b + 6.27)
// where b is the bitlength of the input limbs.
// The carry (c[i] >> 51) fits into a u64 iff 2b+6.27 < 64+51 iff b <= 54.
// After the first carry pass, all c[i] fit into u64.
// How big are the c[i]? We have
//
// c[i] < 2^(102 + 2*b) * (1+i + (4-i)*19)
// < 2^(102 + lg(1 + 4*19) + 2*b)
// < 2^(108.27 + 2*b)
//
// The carry (c[i] >> 51) fits into a u64 when
// 108.27 + 2*b - 51 < 64
// 2*b < 6.73
// b < 3.365.
//
// So we require b < 3 to ensure this fits.
debug_assert!(a[0] < (1 << 54)); debug_assert!(b[0] < (1 << 54));
debug_assert!(a[1] < (1 << 54)); debug_assert!(b[1] < (1 << 54));
debug_assert!(a[2] < (1 << 54)); debug_assert!(b[2] < (1 << 54));
debug_assert!(a[3] < (1 << 54)); debug_assert!(b[3] < (1 << 54));
debug_assert!(a[4] < (1 << 54)); debug_assert!(b[4] < (1 << 54));
// The 128-bit output limbs are stored in two 64-bit registers
// (low/high part). By rebinding the names after carrying, we
// inform LLVM that the values have shrunk, so it can
// efficiently allocate registers.
let low_51_bit_mask = (1u64 << 51) - 1;
c1 += (c0 >> 51) as u128;
let mut c0: u64 = (c0 as u64) & low_51_bit_mask;
c2 += (c1 >> 51) as u128;
let c1: u64 = (c1 as u64) & low_51_bit_mask;
c3 += (c2 >> 51) as u128;
let c2: u64 = (c2 as u64) & low_51_bit_mask;
c4 += (c3 >> 51) as u128;
let c3: u64 = (c3 as u64) & low_51_bit_mask;
c0 += ((c4 >> 51) as u64) * 19;
let c4: u64 = (c4 as u64) & low_51_bit_mask;
// Casting to u64 and back tells the compiler that the carry is
// bounded by 2^64, so that the addition is a u128 + u64 rather
// than u128 + u128.
FieldElement64::reduce([c0,c1,c2,c3,c4])
const LOW_51_BIT_MASK: u64 = (1u64 << 51) - 1;
let mut out = [0u64; 5];
c1 += ((c0 >> 51) as u64) as u128;
out[0] = (c0 as u64) & LOW_51_BIT_MASK;
c2 += ((c1 >> 51) as u64) as u128;
out[1] = (c1 as u64) & LOW_51_BIT_MASK;
c3 += ((c2 >> 51) as u64) as u128;
out[2] = (c2 as u64) & LOW_51_BIT_MASK;
c4 += ((c3 >> 51) as u64) as u128;
out[3] = (c3 as u64) & LOW_51_BIT_MASK;
let carry: u64 = (c4 >> 51) as u64;
out[4] = (c4 as u64) & LOW_51_BIT_MASK;
// To see that this does not overflow, we need out[0] + carry * 19 < 2^64.
//
// c4 < a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + (carry from c3)
// < 5*(2^(51 + b) * 2^(51 + b)) + (carry from c3)
// < 2^(102 + 2*b + lg(5)) + 2^64.
//
// When b < 3 we get
//
// c4 < 2^110.33 so that carry < 2^59.33
//
// so that
//
// out[0] + carry * 19 < 2^51 + 19 * 2^59.33 < 2^63.58
//
// and there is no overflow.
out[0] = out[0] + carry * 19;
// Now out[1] < 2^51 + 2^(64 -51) = 2^51 + 2^13 < 2^(51 + eps).
out[1] += out[0] >> 51;
out[0] &= LOW_51_BIT_MASK;
// Now out[i] < 2^(51 + eps) for all i.
FieldElement64(out)
}
}