265 lines
8.4 KiB
Go
265 lines
8.4 KiB
Go
// Copyright (c) 2017 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package field
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import "math/bits"
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// uint128 holds a 128-bit number as two 64-bit limbs, for use with the
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// bits.Mul64 and bits.Add64 intrinsics.
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type uint128 struct {
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lo, hi uint64
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}
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// mul64 returns a * b.
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func mul64(a, b uint64) uint128 {
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hi, lo := bits.Mul64(a, b)
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return uint128{lo, hi}
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}
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// addMul64 returns v + a * b.
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func addMul64(v uint128, a, b uint64) uint128 {
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hi, lo := bits.Mul64(a, b)
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lo, c := bits.Add64(lo, v.lo, 0)
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hi, _ = bits.Add64(hi, v.hi, c)
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return uint128{lo, hi}
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}
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// shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits.
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func shiftRightBy51(a uint128) uint64 {
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return (a.hi << (64 - 51)) | (a.lo >> 51)
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}
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func feMulGeneric(v, a, b *Element) {
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a0 := a.l0
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a1 := a.l1
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a2 := a.l2
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a3 := a.l3
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a4 := a.l4
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b0 := b.l0
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b1 := b.l1
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b2 := b.l2
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b3 := b.l3
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b4 := b.l4
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// Limb multiplication works like pen-and-paper columnar multiplication, but
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// with 51-bit limbs instead of digits.
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//
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// a4 a3 a2 a1 a0 x
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// b4 b3 b2 b1 b0 =
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// ------------------------
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// a4b0 a3b0 a2b0 a1b0 a0b0 +
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// a4b1 a3b1 a2b1 a1b1 a0b1 +
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// a4b2 a3b2 a2b2 a1b2 a0b2 +
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// a4b3 a3b3 a2b3 a1b3 a0b3 +
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// a4b4 a3b4 a2b4 a1b4 a0b4 =
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// ----------------------------------------------
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// r8 r7 r6 r5 r4 r3 r2 r1 r0
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//
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// We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to
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// reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5,
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// r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc.
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//
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// Reduction can be carried out simultaneously to multiplication. For
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// example, we do not compute r5: whenever the result of a multiplication
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// belongs to r5, like a1b4, we multiply it by 19 and add the result to r0.
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//
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// a4b0 a3b0 a2b0 a1b0 a0b0 +
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// a3b1 a2b1 a1b1 a0b1 19×a4b1 +
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// a2b2 a1b2 a0b2 19×a4b2 19×a3b2 +
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// a1b3 a0b3 19×a4b3 19×a3b3 19×a2b3 +
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// a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4 =
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// --------------------------------------
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// r4 r3 r2 r1 r0
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//
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// Finally we add up the columns into wide, overlapping limbs.
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a1_19 := a1 * 19
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a2_19 := a2 * 19
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a3_19 := a3 * 19
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a4_19 := a4 * 19
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// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
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r0 := mul64(a0, b0)
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r0 = addMul64(r0, a1_19, b4)
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r0 = addMul64(r0, a2_19, b3)
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r0 = addMul64(r0, a3_19, b2)
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r0 = addMul64(r0, a4_19, b1)
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// r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
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r1 := mul64(a0, b1)
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r1 = addMul64(r1, a1, b0)
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r1 = addMul64(r1, a2_19, b4)
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r1 = addMul64(r1, a3_19, b3)
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r1 = addMul64(r1, a4_19, b2)
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// r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
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r2 := mul64(a0, b2)
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r2 = addMul64(r2, a1, b1)
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r2 = addMul64(r2, a2, b0)
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r2 = addMul64(r2, a3_19, b4)
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r2 = addMul64(r2, a4_19, b3)
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// r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
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r3 := mul64(a0, b3)
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r3 = addMul64(r3, a1, b2)
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r3 = addMul64(r3, a2, b1)
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r3 = addMul64(r3, a3, b0)
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r3 = addMul64(r3, a4_19, b4)
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// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
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r4 := mul64(a0, b4)
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r4 = addMul64(r4, a1, b3)
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r4 = addMul64(r4, a2, b2)
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r4 = addMul64(r4, a3, b1)
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r4 = addMul64(r4, a4, b0)
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// After the multiplication, we need to reduce (carry) the five coefficients
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// to obtain a result with limbs that are at most slightly larger than 2⁵¹,
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// to respect the Element invariant.
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//
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// Overall, the reduction works the same as carryPropagate, except with
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// wider inputs: we take the carry for each coefficient by shifting it right
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// by 51, and add it to the limb above it. The top carry is multiplied by 19
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// according to the reduction identity and added to the lowest limb.
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//
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// The largest coefficient (r0) will be at most 111 bits, which guarantees
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// that all carries are at most 111 - 51 = 60 bits, which fits in a uint64.
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//
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// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
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// r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²)
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// r0 < (1 + 19 × 4) × 2⁵² × 2⁵²
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// r0 < 2⁷ × 2⁵² × 2⁵²
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// r0 < 2¹¹¹
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//
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// Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most
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// 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and
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// allows us to easily apply the reduction identity.
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//
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// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
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// r4 < 5 × 2⁵² × 2⁵²
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// r4 < 2¹⁰⁷
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//
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c0 := shiftRightBy51(r0)
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c1 := shiftRightBy51(r1)
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c2 := shiftRightBy51(r2)
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c3 := shiftRightBy51(r3)
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c4 := shiftRightBy51(r4)
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rr0 := r0.lo&maskLow51Bits + c4*19
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rr1 := r1.lo&maskLow51Bits + c0
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rr2 := r2.lo&maskLow51Bits + c1
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rr3 := r3.lo&maskLow51Bits + c2
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rr4 := r4.lo&maskLow51Bits + c3
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// Now all coefficients fit into 64-bit registers but are still too large to
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// be passed around as a Element. We therefore do one last carry chain,
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// where the carries will be small enough to fit in the wiggle room above 2⁵¹.
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*v = Element{rr0, rr1, rr2, rr3, rr4}
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v.carryPropagate()
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}
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func feSquareGeneric(v, a *Element) {
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l0 := a.l0
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l1 := a.l1
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l2 := a.l2
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l3 := a.l3
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l4 := a.l4
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// Squaring works precisely like multiplication above, but thanks to its
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// symmetry we get to group a few terms together.
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//
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// l4 l3 l2 l1 l0 x
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// l4 l3 l2 l1 l0 =
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// ------------------------
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// l4l0 l3l0 l2l0 l1l0 l0l0 +
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// l4l1 l3l1 l2l1 l1l1 l0l1 +
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// l4l2 l3l2 l2l2 l1l2 l0l2 +
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// l4l3 l3l3 l2l3 l1l3 l0l3 +
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// l4l4 l3l4 l2l4 l1l4 l0l4 =
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// ----------------------------------------------
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// r8 r7 r6 r5 r4 r3 r2 r1 r0
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//
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// l4l0 l3l0 l2l0 l1l0 l0l0 +
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// l3l1 l2l1 l1l1 l0l1 19×l4l1 +
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// l2l2 l1l2 l0l2 19×l4l2 19×l3l2 +
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// l1l3 l0l3 19×l4l3 19×l3l3 19×l2l3 +
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// l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4 =
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// --------------------------------------
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// r4 r3 r2 r1 r0
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//
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// With precomputed 2×, 19×, and 2×19× terms, we can compute each limb with
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// only three Mul64 and four Add64, instead of five and eight.
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l0_2 := l0 * 2
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l1_2 := l1 * 2
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l1_38 := l1 * 38
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l2_38 := l2 * 38
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l3_38 := l3 * 38
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l3_19 := l3 * 19
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l4_19 := l4 * 19
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// r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3)
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r0 := mul64(l0, l0)
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r0 = addMul64(r0, l1_38, l4)
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r0 = addMul64(r0, l2_38, l3)
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// r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
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r1 := mul64(l0_2, l1)
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r1 = addMul64(r1, l2_38, l4)
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r1 = addMul64(r1, l3_19, l3)
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// r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4
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r2 := mul64(l0_2, l2)
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r2 = addMul64(r2, l1, l1)
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r2 = addMul64(r2, l3_38, l4)
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// r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
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r3 := mul64(l0_2, l3)
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r3 = addMul64(r3, l1_2, l2)
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r3 = addMul64(r3, l4_19, l4)
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// r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2
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r4 := mul64(l0_2, l4)
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r4 = addMul64(r4, l1_2, l3)
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r4 = addMul64(r4, l2, l2)
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c0 := shiftRightBy51(r0)
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c1 := shiftRightBy51(r1)
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c2 := shiftRightBy51(r2)
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c3 := shiftRightBy51(r3)
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c4 := shiftRightBy51(r4)
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rr0 := r0.lo&maskLow51Bits + c4*19
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rr1 := r1.lo&maskLow51Bits + c0
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rr2 := r2.lo&maskLow51Bits + c1
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rr3 := r3.lo&maskLow51Bits + c2
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rr4 := r4.lo&maskLow51Bits + c3
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*v = Element{rr0, rr1, rr2, rr3, rr4}
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v.carryPropagate()
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}
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// carryPropagate brings the limbs below 52 bits by applying the reduction
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// identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry. TODO inline
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func (v *Element) carryPropagateGeneric() *Element {
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c0 := v.l0 >> 51
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c1 := v.l1 >> 51
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c2 := v.l2 >> 51
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c3 := v.l3 >> 51
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c4 := v.l4 >> 51
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v.l0 = v.l0&maskLow51Bits + c4*19
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v.l1 = v.l1&maskLow51Bits + c0
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v.l2 = v.l2&maskLow51Bits + c1
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v.l3 = v.l3&maskLow51Bits + c2
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v.l4 = v.l4&maskLow51Bits + c3
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return v
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}
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