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@@ -12,20 +12,42 @@ 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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+// mul returns a * b.
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+func mul(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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+// addMul returns v + a * b.
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+func addMul(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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+// mul19 returns v * 19.
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+func mul19(v uint64) uint64 {
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+ // Using this approach seems to yield better optimizations than *19.
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+ return v + (v+v<<3)<<1
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+}
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+
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+// addMul19 returns v + 19 * a * b, where a and b are at most 52 bits.
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+func addMul19(v uint128, a, b uint64) uint128 {
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+ hi, lo := bits.Mul64(mul19(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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+
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+// addMul38 returns v + 38 * a * b, where a and b are at most 52 bits.
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+func addMul38(v uint128, a, b uint64) uint128 {
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+ hi, lo := bits.Mul64(mul19(a), b*2)
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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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+
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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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@@ -76,45 +98,40 @@ func feMulGeneric(v, a, b *Element) {
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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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-
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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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+ r0 := mul(a0, b0)
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+ r0 = addMul19(r0, a1, b4)
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+ r0 = addMul19(r0, a2, b3)
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+ r0 = addMul19(r0, a3, b2)
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+ r0 = addMul19(r0, a4, 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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+ r1 := mul(a0, b1)
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+ r1 = addMul(r1, a1, b0)
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+ r1 = addMul19(r1, a2, b4)
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+ r1 = addMul19(r1, a3, b3)
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+ r1 = addMul19(r1, a4, 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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+ r2 := mul(a0, b2)
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+ r2 = addMul(r2, a1, b1)
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+ r2 = addMul(r2, a2, b0)
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+ r2 = addMul19(r2, a3, b4)
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+ r2 = addMul19(r2, a4, 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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+ r3 := mul(a0, b3)
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+ r3 = addMul(r3, a1, b2)
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+ r3 = addMul(r3, a2, b1)
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+ r3 = addMul(r3, a3, b0)
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+ r3 = addMul19(r3, a4, 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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+ r4 := mul(a0, b4)
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+ r4 = addMul(r4, a1, b3)
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+ r4 = addMul(r4, a2, b2)
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+ r4 = addMul(r4, a3, b1)
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+ r4 = addMul(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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@@ -149,7 +166,7 @@ func feMulGeneric(v, a, b *Element) {
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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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+ rr0 := r0.lo&maskLow51Bits + mul19(c4)
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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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@@ -158,8 +175,12 @@ func feMulGeneric(v, a, b *Element) {
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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 an 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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+ v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
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+ v.l1 = rr1&maskLow51Bits + rr0>>51
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+ v.l2 = rr2&maskLow51Bits + rr1>>51
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+ v.l3 = rr3&maskLow51Bits + rr2>>51
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+ v.l4 = rr4&maskLow51Bits + rr3>>51
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}
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func feSquareGeneric(v, a *Element) {
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@@ -190,44 +211,31 @@ func feSquareGeneric(v, a *Element) {
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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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-
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- l0_2 := l0 * 2
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- l1_2 := l1 * 2
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-
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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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-
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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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+ r0 := mul(l0, l0)
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+ r0 = addMul38(r0, l1, l4)
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+ r0 = addMul38(r0, l2, 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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+ r1 := mul(l0*2, l1)
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+ r1 = addMul38(r1, l2, l4)
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+ r1 = addMul19(r1, l3, 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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+ r2 := mul(l0*2, l2)
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+ r2 = addMul(r2, l1, l1)
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+ r2 = addMul38(r2, l3, 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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+ r3 := mul(l0*2, l3)
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+ r3 = addMul(r3, l1*2, l2)
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+ r3 = addMul19(r3, l4, 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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+ r4 := mul(l0*2, l4)
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+ r4 = addMul(r4, l1*2, l3)
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+ r4 = addMul(r4, l2, l2)
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c0 := shiftRightBy51(r0)
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c1 := shiftRightBy51(r1)
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@@ -235,32 +243,30 @@ func feSquareGeneric(v, a *Element) {
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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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+ rr0 := r0.lo&maskLow51Bits + mul19(c4)
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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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+ v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
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+ v.l1 = rr1&maskLow51Bits + rr0>>51
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+ v.l2 = rr2&maskLow51Bits + rr1>>51
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+ v.l3 = rr3&maskLow51Bits + rr2>>51
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+ v.l4 = rr4&maskLow51Bits + rr3>>51
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}
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-// carryPropagateGeneric brings the limbs below 52 bits by applying the reduction
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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.
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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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-
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- // c4 is at most 64 - 51 = 13 bits, so c4*19 is at most 18 bits, and
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+func (v *Element) carryPropagate() *Element {
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+ // (l4>>51) is at most 64 - 51 = 13 bits, so (l4>>51)*19 is at most 18 bits, and
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// the final l0 will be at most 52 bits. Similarly for the rest.
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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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+ l0 := v.l0
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+ v.l0 = v.l0&maskLow51Bits + mul19(v.l4>>51)
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+ v.l4 = v.l4&maskLow51Bits + v.l3>>51
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+ v.l3 = v.l3&maskLow51Bits + v.l2>>51
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+ v.l2 = v.l2&maskLow51Bits + v.l1>>51
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+ v.l1 = v.l1&maskLow51Bits + l0>>51
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return v
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}
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