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bls12: addition chain for inversion in Fp #296

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Oct 8, 2021
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bls12: addition chain for inversion in Fp #296

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73 changes: 63 additions & 10 deletions ecc/bls12381/ff/fp.go
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Original file line number Diff line number Diff line change
Expand Up @@ -41,7 +41,6 @@ func (z *Fp) Add(x, y *Fp) { fiatFpMontAdd(&z.i, &x.i, &y.i) }
func (z *Fp) Sub(x, y *Fp) { fiatFpMontSub(&z.i, &x.i, &y.i) }
func (z *Fp) Mul(x, y *Fp) { fiatFpMontMul(&z.i, &x.i, &y.i) }
func (z *Fp) Sqr(x *Fp) { fiatFpMontSquare(&z.i, &x.i) }
func (z *Fp) Inv(x *Fp) { z.ExpVarTime(x, fpOrderMinus2[:]) }
func (z *Fp) toMont(in *fpRaw) { fiatFpMontMul(&z.i, in, &fpRSquare) }
func (z Fp) fromMont() (out fpRaw) { fiatFpMontMul(&out, &z.i, &fpMont{1}); return }
func (z Fp) Sgn0() int { return int(z.fromMont()[0]) & 1 }
Expand Down Expand Up @@ -119,6 +118,69 @@ func (z *Fp) SetString(s string) error {

func fiatFpMontCmovznzU64(z *uint64, b, x, y uint64) { cselectU64(z, b, x, y) }

func (z *Fp) Inv(x *Fp) {
// Addition chain found using mmcloughlin/addchain: v0.3.0
// McLoughlin, Michael Ben. (2021). https://doi.org/10.5281/zenodo.4758226
var i2, i4, i8, i9, i11, i13, i17, i20, i25, i26, i52, i54, i55, i77, i79,
i86, i93, i103, i105, i119, i123, i137, i149, i151, i169, i177, i191,
i195, i208, i215, i225, i229, i235, i245, i255 Fp

i2.Sqr(x)
i4.Sqr(&i2)
i8.Sqr(&i4)
i9.Mul(&i8, x)
i11.Mul(&i9, &i2)
i13.Mul(&i11, &i2)
i17.Mul(&i13, &i4)
i20.Mul(&i11, &i9)
i25.Mul(&i17, &i8)
i26.Mul(&i25, x)
i52.Sqr(&i26)
i54.Mul(&i52, &i2)
i55.Mul(&i54, x)
i77.Mul(&i52, &i25)
i79.Mul(&i77, &i2)
i86.Mul(&i77, &i8)
i93.Mul(&i86, &i8)
i103.Mul(&i77, &i26)
i105.Mul(&i103, &i2)
i119.Mul(&i93, &i26)
i123.Mul(&i119, &i4)
i137.Mul(&i86, &i52)
i149.Mul(&i123, &i26)
i151.Mul(&i149, &i2)
i169.Mul(&i149, &i20)
i177.Mul(&i169, &i8)
i191.Mul(&i137, &i54)
i195.Mul(&i191, &i4)
i208.Mul(&i195, &i13)
i215.Mul(&i195, &i20)
i225.Mul(&i208, &i17)
i229.Mul(&i225, &i4)
i235.Mul(&i215, &i20)
i245.Mul(&i225, &i20)
i255.Mul(&i235, &i20)
z.Mul(&i225, &i191)

for _, s := range []struct {
l int
x *Fp
}{
{8, &i17}, {11, &i245}, {11, &i229}, {8, &i255}, {7, &i77}, {9, &i105},
{10, &i177}, {7, &i93}, {9, &i123}, {6, &i25}, {11, &i105}, {9, &i235},
{10, &i215}, {6, &i25}, {10, &i119}, {9, &i151}, {11, &i79}, {10, &i225},
{9, &i137}, {9, &i191}, {8, &i103}, {10, &i195}, {9, &i149}, {12, &i123},
{5, &i11}, {11, &i123}, {7, &i9}, {13, &i245}, {9, &i191}, {8, &i255},
{8, &i235}, {11, &i169}, {8, &i255}, {8, &i255}, {6, &i55}, {10, &i255},
{9, &i255}, {8, &i255}, {8, &i255}, {8, &i255}, {7, &i86}, {9, &i169},
} {
for i := 0; i < s.l; i++ {
z.Sqr(z)
}
z.Mul(z, s.x)
}
}

var (
// fpOrder is the order of the Fp field (big-endian).
fpOrder = [FpSize]byte{
Expand All @@ -129,15 +191,6 @@ var (
0x1e, 0xab, 0xff, 0xfe, 0xb1, 0x53, 0xff, 0xff,
0xb9, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xaa, 0xab,
}
// fpOrderMinus2 is the fpOrder minus two used for inversion (big-endian).
fpOrderMinus2 = [FpSize]byte{
0x1a, 0x01, 0x11, 0xea, 0x39, 0x7f, 0xe6, 0x9a,
0x4b, 0x1b, 0xa7, 0xb6, 0x43, 0x4b, 0xac, 0xd7,
0x64, 0x77, 0x4b, 0x84, 0xf3, 0x85, 0x12, 0xbf,
0x67, 0x30, 0xd2, 0xa0, 0xf6, 0xb0, 0xf6, 0x24,
0x1e, 0xab, 0xff, 0xfe, 0xb1, 0x53, 0xff, 0xff,
0xb9, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xaa, 0xa9,
}
// fpOrderPlus1Div2 is the half of (fpOrder plus one) used for lexicographically order (big-endian).
fpOrderPlus1Div2 = [FpSize]byte{
0x0d, 0x00, 0x88, 0xf5, 0x1c, 0xbf, 0xf3, 0x4d,
Expand Down