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 // 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 a 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() } // carryPropagate brings the limbs below 52 bits by applying the reduction // identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry. TODO inline 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 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 }