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Diffstat (limited to 'vendor/filippo.io/edwards25519/field/fe_generic.go')
| -rw-r--r-- | vendor/filippo.io/edwards25519/field/fe_generic.go | 266 |
1 files changed, 266 insertions, 0 deletions
diff --git a/vendor/filippo.io/edwards25519/field/fe_generic.go b/vendor/filippo.io/edwards25519/field/fe_generic.go new file mode 100644 index 0000000..86f5fd9 --- /dev/null +++ b/vendor/filippo.io/edwards25519/field/fe_generic.go @@ -0,0 +1,266 @@ +// Copyright (c) 2017 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package field + +import "math/bits" + +// uint128 holds a 128-bit number as two 64-bit limbs, for use with the +// bits.Mul64 and bits.Add64 intrinsics. +type uint128 struct { + lo, hi uint64 +} + +// mul64 returns a * b. +func mul64(a, b uint64) uint128 { + hi, lo := bits.Mul64(a, b) + return uint128{lo, hi} +} + +// addMul64 returns v + a * b. +func addMul64(v uint128, a, b uint64) uint128 { + hi, lo := bits.Mul64(a, b) + lo, c := bits.Add64(lo, v.lo, 0) + hi, _ = bits.Add64(hi, v.hi, c) + return uint128{lo, hi} +} + +// shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits. +func shiftRightBy51(a uint128) uint64 { + return (a.hi << (64 - 51)) | (a.lo >> 51) +} + +func feMulGeneric(v, a, b *Element) { + a0 := a.l0 + a1 := a.l1 + a2 := a.l2 + a3 := a.l3 + a4 := a.l4 + + b0 := b.l0 + b1 := b.l1 + b2 := b.l2 + b3 := b.l3 + b4 := b.l4 + + // Limb multiplication works like pen-and-paper columnar multiplication, but + // with 51-bit limbs instead of digits. + // + // a4 a3 a2 a1 a0 x + // b4 b3 b2 b1 b0 = + // ------------------------ + // a4b0 a3b0 a2b0 a1b0 a0b0 + + // a4b1 a3b1 a2b1 a1b1 a0b1 + + // a4b2 a3b2 a2b2 a1b2 a0b2 + + // a4b3 a3b3 a2b3 a1b3 a0b3 + + // a4b4 a3b4 a2b4 a1b4 a0b4 = + // ---------------------------------------------- + // r8 r7 r6 r5 r4 r3 r2 r1 r0 + // + // We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to + // reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5, + // r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc. + // + // Reduction can be carried out simultaneously to multiplication. For + // example, we do not compute r5: whenever the result of a multiplication + // belongs to r5, like a1b4, we multiply it by 19 and add the result to r0. + // + // a4b0 a3b0 a2b0 a1b0 a0b0 + + // a3b1 a2b1 a1b1 a0b1 19×a4b1 + + // a2b2 a1b2 a0b2 19×a4b2 19×a3b2 + + // a1b3 a0b3 19×a4b3 19×a3b3 19×a2b3 + + // a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4 = + // -------------------------------------- + // r4 r3 r2 r1 r0 + // + // Finally we add up the columns into wide, overlapping limbs. + + a1_19 := a1 * 19 + a2_19 := a2 * 19 + a3_19 := a3 * 19 + a4_19 := a4 * 19 + + // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1) + r0 := mul64(a0, b0) + r0 = addMul64(r0, a1_19, b4) + r0 = addMul64(r0, a2_19, b3) + r0 = addMul64(r0, a3_19, b2) + r0 = addMul64(r0, a4_19, b1) + + // r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2) + r1 := mul64(a0, b1) + r1 = addMul64(r1, a1, b0) + r1 = addMul64(r1, a2_19, b4) + r1 = addMul64(r1, a3_19, b3) + r1 = addMul64(r1, a4_19, b2) + + // r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3) + r2 := mul64(a0, b2) + r2 = addMul64(r2, a1, b1) + r2 = addMul64(r2, a2, b0) + r2 = addMul64(r2, a3_19, b4) + r2 = addMul64(r2, a4_19, b3) + + // r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4 + r3 := mul64(a0, b3) + r3 = addMul64(r3, a1, b2) + r3 = addMul64(r3, a2, b1) + r3 = addMul64(r3, a3, b0) + r3 = addMul64(r3, a4_19, b4) + + // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0 + r4 := mul64(a0, b4) + r4 = addMul64(r4, a1, b3) + r4 = addMul64(r4, a2, b2) + r4 = addMul64(r4, a3, b1) + r4 = addMul64(r4, a4, b0) + + // After the multiplication, we need to reduce (carry) the five coefficients + // to obtain a result with limbs that are at most slightly larger than 2⁵¹, + // to respect the Element invariant. + // + // Overall, the reduction works the same as carryPropagate, except with + // wider inputs: we take the carry for each coefficient by shifting it right + // by 51, and add it to the limb above it. The top carry is multiplied by 19 + // according to the reduction identity and added to the lowest limb. + // + // The largest coefficient (r0) will be at most 111 bits, which guarantees + // that all carries are at most 111 - 51 = 60 bits, which fits in a uint64. + // + // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1) + // r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²) + // r0 < (1 + 19 × 4) × 2⁵² × 2⁵² + // r0 < 2⁷ × 2⁵² × 2⁵² + // r0 < 2¹¹¹ + // + // Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most + // 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and + // allows us to easily apply the reduction identity. + // + // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0 + // r4 < 5 × 2⁵² × 2⁵² + // r4 < 2¹⁰⁷ + // + + c0 := shiftRightBy51(r0) + c1 := shiftRightBy51(r1) + c2 := shiftRightBy51(r2) + c3 := shiftRightBy51(r3) + c4 := shiftRightBy51(r4) + + rr0 := r0.lo&maskLow51Bits + c4*19 + rr1 := r1.lo&maskLow51Bits + c0 + rr2 := r2.lo&maskLow51Bits + c1 + rr3 := r3.lo&maskLow51Bits + c2 + rr4 := r4.lo&maskLow51Bits + c3 + + // Now all coefficients fit into 64-bit registers but are still too large to + // be passed around as an Element. We therefore do one last carry chain, + // where the carries will be small enough to fit in the wiggle room above 2⁵¹. + *v = Element{rr0, rr1, rr2, rr3, rr4} + v.carryPropagate() +} + +func feSquareGeneric(v, a *Element) { + l0 := a.l0 + l1 := a.l1 + l2 := a.l2 + l3 := a.l3 + l4 := a.l4 + + // Squaring works precisely like multiplication above, but thanks to its + // symmetry we get to group a few terms together. + // + // l4 l3 l2 l1 l0 x + // l4 l3 l2 l1 l0 = + // ------------------------ + // l4l0 l3l0 l2l0 l1l0 l0l0 + + // l4l1 l3l1 l2l1 l1l1 l0l1 + + // l4l2 l3l2 l2l2 l1l2 l0l2 + + // l4l3 l3l3 l2l3 l1l3 l0l3 + + // l4l4 l3l4 l2l4 l1l4 l0l4 = + // ---------------------------------------------- + // r8 r7 r6 r5 r4 r3 r2 r1 r0 + // + // l4l0 l3l0 l2l0 l1l0 l0l0 + + // l3l1 l2l1 l1l1 l0l1 19×l4l1 + + // l2l2 l1l2 l0l2 19×l4l2 19×l3l2 + + // l1l3 l0l3 19×l4l3 19×l3l3 19×l2l3 + + // l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4 = + // -------------------------------------- + // r4 r3 r2 r1 r0 + // + // With precomputed 2×, 19×, and 2×19× terms, we can compute each limb with + // only three Mul64 and four Add64, instead of five and eight. + + l0_2 := l0 * 2 + l1_2 := l1 * 2 + + l1_38 := l1 * 38 + l2_38 := l2 * 38 + l3_38 := l3 * 38 + + l3_19 := l3 * 19 + l4_19 := l4 * 19 + + // r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3) + r0 := mul64(l0, l0) + r0 = addMul64(r0, l1_38, l4) + r0 = addMul64(r0, l2_38, l3) + + // r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3 + r1 := mul64(l0_2, l1) + r1 = addMul64(r1, l2_38, l4) + r1 = addMul64(r1, l3_19, l3) + + // r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4 + r2 := mul64(l0_2, l2) + r2 = addMul64(r2, l1, l1) + r2 = addMul64(r2, l3_38, l4) + + // r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4 + r3 := mul64(l0_2, l3) + r3 = addMul64(r3, l1_2, l2) + r3 = addMul64(r3, l4_19, l4) + + // r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2 + r4 := mul64(l0_2, l4) + r4 = addMul64(r4, l1_2, l3) + r4 = addMul64(r4, l2, l2) + + c0 := shiftRightBy51(r0) + c1 := shiftRightBy51(r1) + c2 := shiftRightBy51(r2) + c3 := shiftRightBy51(r3) + c4 := shiftRightBy51(r4) + + rr0 := r0.lo&maskLow51Bits + c4*19 + rr1 := r1.lo&maskLow51Bits + c0 + rr2 := r2.lo&maskLow51Bits + c1 + rr3 := r3.lo&maskLow51Bits + c2 + rr4 := r4.lo&maskLow51Bits + c3 + + *v = Element{rr0, rr1, rr2, rr3, rr4} + v.carryPropagate() +} + +// carryPropagateGeneric brings the limbs below 52 bits by applying the reduction +// identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry. +func (v *Element) carryPropagateGeneric() *Element { + c0 := v.l0 >> 51 + c1 := v.l1 >> 51 + c2 := v.l2 >> 51 + c3 := v.l3 >> 51 + c4 := v.l4 >> 51 + + // c4 is at most 64 - 51 = 13 bits, so c4*19 is at most 18 bits, and + // the final l0 will be at most 52 bits. Similarly for the rest. + v.l0 = v.l0&maskLow51Bits + c4*19 + v.l1 = v.l1&maskLow51Bits + c0 + v.l2 = v.l2&maskLow51Bits + c1 + v.l3 = v.l3&maskLow51Bits + c2 + v.l4 = v.l4&maskLow51Bits + c3 + + return v +} |
