P2Pool/edwards25519
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edwards25519/scalar.go
Filippo Valsorda d3569cbbb3 all: flatten the package and make FieldElement opaque 
For the license changes, see gtank/ristretto255-private#28 and
gtank/ristretto255#32, that contribute all code in those repositories to
the Go project under the Google CLA.
2020-09-28 14:18:44 +02:00

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// Copyright (c) 2016 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 edwards25519
import (
"crypto/subtle"
"encoding/binary"
"errors"
)
// A Scalar is an integer modulo
//
// l = 2^252 + 27742317777372353535851937790883648493
//
// represented as a little-endian byte string.
type Scalar [32]byte
var (
scZero = Scalar([32]byte{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0})
scOne = Scalar([32]byte{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0})
// sage: l = GF(2**252 + 27742317777372353535851937790883648493)
// sage: l(-1).lift().digits(256)
scMinusOne = Scalar([32]byte{236, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16})
)
// Add sets s = x + y mod l and returns s.
func (s *Scalar) Add(x, y *Scalar) *Scalar {
// s = 1 * x + y mod l
scMulAdd(s, &scOne, x, y)
return s
}
// Sub sets s = x - y mod l and returns s.
func (s *Scalar) Sub(x, y *Scalar) *Scalar {
// s = -1 * y + x
scMulAdd(s, &scMinusOne, y, x)
return s
}
// Neg sets s = -x mod l and returns s.
func (s *Scalar) Neg(x *Scalar) *Scalar {
scMulAdd(s, &scMinusOne, x, &scZero)
return s
}
// Mul sets s = x * y mod l and returns s.
func (s *Scalar) Mul(x, y *Scalar) *Scalar {
scMulAdd(s, x, y, &scZero)
return s
}
// FromUniformBytes sets s to an uniformly distributed value given 64 uniformly
// distributed random bytes.
func (s *Scalar) FromUniformBytes(x []byte) *Scalar {
if len(x) != 64 {
panic("scalar: invalid uniform input length")
}
var wideBytes [64]byte
copy(wideBytes[:], x[:])
scReduce(s, &wideBytes)
return s
}
// FromCanonicalBytes sets s = x, where x is a 32 bytes little-endian encoding
// of s. If x is not a canonical encoding of s, FromCanonicalBytes returns an
// error and the receiver is unchanged.
func (s *Scalar) FromCanonicalBytes(x []byte) error {
if len(x) != 32 {
panic("scalar: invalid scalar length")
}
if !scMinimal(x) {
return errors.New("invalid scalar encoding")
}
copy(s[:], x)
return nil
}
// reduce reduces s mod l returns it.
func (s *Scalar) reduce() *Scalar {
var wideBytes [64]byte
copy(wideBytes[:], s[:])
scReduce(s, &wideBytes)
return s
}
// Bytes appends a 32 bytes little-endian encoding of s to b.
func (s *Scalar) Bytes(b []byte) []byte {
t := *s
t.reduce()
res, out := sliceForAppend(b, 32)
copy(out, t[:])
return res
}
// Equal returns 1 if s and t are equal, and 0 otherwise.
func (s *Scalar) Equal(t *Scalar) int {
var ss, st [32]byte
t.Bytes(st[:0])
s.Bytes(ss[:0])
return subtle.ConstantTimeCompare(ss[:], st[:])
}
func load3(in []byte) int64 {
r := int64(in[0])
r |= int64(in[1]) << 8
r |= int64(in[2]) << 16
return r
}
func load4(in []byte) int64 {
r := int64(in[0])
r |= int64(in[1]) << 8
r |= int64(in[2]) << 16
r |= int64(in[3]) << 24
return r
}
// Input:
// a[0]+256*a[1]+...+256^31*a[31] = a
// b[0]+256*b[1]+...+256^31*b[31] = b
// c[0]+256*c[1]+...+256^31*c[31] = c
//
// Output:
// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
// where l = 2^252 + 27742317777372353535851937790883648493.
func scMulAdd(s, a, b, c *Scalar) {
a0 := 2097151 & load3(a[:])
a1 := 2097151 & (load4(a[2:]) >> 5)
a2 := 2097151 & (load3(a[5:]) >> 2)
a3 := 2097151 & (load4(a[7:]) >> 7)
a4 := 2097151 & (load4(a[10:]) >> 4)
a5 := 2097151 & (load3(a[13:]) >> 1)
a6 := 2097151 & (load4(a[15:]) >> 6)
a7 := 2097151 & (load3(a[18:]) >> 3)
a8 := 2097151 & load3(a[21:])
a9 := 2097151 & (load4(a[23:]) >> 5)
a10 := 2097151 & (load3(a[26:]) >> 2)
a11 := (load4(a[28:]) >> 7)
b0 := 2097151 & load3(b[:])
b1 := 2097151 & (load4(b[2:]) >> 5)
b2 := 2097151 & (load3(b[5:]) >> 2)
b3 := 2097151 & (load4(b[7:]) >> 7)
b4 := 2097151 & (load4(b[10:]) >> 4)
b5 := 2097151 & (load3(b[13:]) >> 1)
b6 := 2097151 & (load4(b[15:]) >> 6)
b7 := 2097151 & (load3(b[18:]) >> 3)
b8 := 2097151 & load3(b[21:])
b9 := 2097151 & (load4(b[23:]) >> 5)
b10 := 2097151 & (load3(b[26:]) >> 2)
b11 := (load4(b[28:]) >> 7)
c0 := 2097151 & load3(c[:])
c1 := 2097151 & (load4(c[2:]) >> 5)
c2 := 2097151 & (load3(c[5:]) >> 2)
c3 := 2097151 & (load4(c[7:]) >> 7)
c4 := 2097151 & (load4(c[10:]) >> 4)
c5 := 2097151 & (load3(c[13:]) >> 1)
c6 := 2097151 & (load4(c[15:]) >> 6)
c7 := 2097151 & (load3(c[18:]) >> 3)
c8 := 2097151 & load3(c[21:])
c9 := 2097151 & (load4(c[23:]) >> 5)
c10 := 2097151 & (load3(c[26:]) >> 2)
c11 := (load4(c[28:]) >> 7)
var carry [23]int64
s0 := c0 + a0*b0
s1 := c1 + a0*b1 + a1*b0
s2 := c2 + a0*b2 + a1*b1 + a2*b0
s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0
s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0
s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0
s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0
s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0
s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0
s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0
s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0
s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0
s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1
s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2
s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3
s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4
s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5
s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6
s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7
s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8
s20 := a9*b11 + a10*b10 + a11*b9
s21 := a10*b11 + a11*b10
s22 := a11 * b11
s23 := int64(0)
carry[0] = (s0 + (1 << 20)) >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[2] = (s2 + (1 << 20)) >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[4] = (s4 + (1 << 20)) >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[6] = (s6 + (1 << 20)) >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[8] = (s8 + (1 << 20)) >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[10] = (s10 + (1 << 20)) >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[12] = (s12 + (1 << 20)) >> 21
s13 += carry[12]
s12 -= carry[12] << 21
carry[14] = (s14 + (1 << 20)) >> 21
s15 += carry[14]
s14 -= carry[14] << 21
carry[16] = (s16 + (1 << 20)) >> 21
s17 += carry[16]
s16 -= carry[16] << 21
carry[18] = (s18 + (1 << 20)) >> 21
s19 += carry[18]
s18 -= carry[18] << 21
carry[20] = (s20 + (1 << 20)) >> 21
s21 += carry[20]
s20 -= carry[20] << 21
carry[22] = (s22 + (1 << 20)) >> 21
s23 += carry[22]
s22 -= carry[22] << 21
carry[1] = (s1 + (1 << 20)) >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[3] = (s3 + (1 << 20)) >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[5] = (s5 + (1 << 20)) >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[7] = (s7 + (1 << 20)) >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[9] = (s9 + (1 << 20)) >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[11] = (s11 + (1 << 20)) >> 21
s12 += carry[11]
s11 -= carry[11] << 21
carry[13] = (s13 + (1 << 20)) >> 21
s14 += carry[13]
s13 -= carry[13] << 21
carry[15] = (s15 + (1 << 20)) >> 21
s16 += carry[15]
s15 -= carry[15] << 21
carry[17] = (s17 + (1 << 20)) >> 21
s18 += carry[17]
s17 -= carry[17] << 21
carry[19] = (s19 + (1 << 20)) >> 21
s20 += carry[19]
s19 -= carry[19] << 21
carry[21] = (s21 + (1 << 20)) >> 21
s22 += carry[21]
s21 -= carry[21] << 21
s11 += s23 * 666643
s12 += s23 * 470296
s13 += s23 * 654183
s14 -= s23 * 997805
s15 += s23 * 136657
s16 -= s23 * 683901
s23 = 0
s10 += s22 * 666643
s11 += s22 * 470296
s12 += s22 * 654183
s13 -= s22 * 997805
s14 += s22 * 136657
s15 -= s22 * 683901
s22 = 0
s9 += s21 * 666643
s10 += s21 * 470296
s11 += s21 * 654183
s12 -= s21 * 997805
s13 += s21 * 136657
s14 -= s21 * 683901
s21 = 0
s8 += s20 * 666643
s9 += s20 * 470296
s10 += s20 * 654183
s11 -= s20 * 997805
s12 += s20 * 136657
s13 -= s20 * 683901
s20 = 0
s7 += s19 * 666643
s8 += s19 * 470296
s9 += s19 * 654183
s10 -= s19 * 997805
s11 += s19 * 136657
s12 -= s19 * 683901
s19 = 0
s6 += s18 * 666643
s7 += s18 * 470296
s8 += s18 * 654183
s9 -= s18 * 997805
s10 += s18 * 136657
s11 -= s18 * 683901
s18 = 0
carry[6] = (s6 + (1 << 20)) >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[8] = (s8 + (1 << 20)) >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[10] = (s10 + (1 << 20)) >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[12] = (s12 + (1 << 20)) >> 21
s13 += carry[12]
s12 -= carry[12] << 21
carry[14] = (s14 + (1 << 20)) >> 21
s15 += carry[14]
s14 -= carry[14] << 21
carry[16] = (s16 + (1 << 20)) >> 21
s17 += carry[16]
s16 -= carry[16] << 21
carry[7] = (s7 + (1 << 20)) >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[9] = (s9 + (1 << 20)) >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[11] = (s11 + (1 << 20)) >> 21
s12 += carry[11]
s11 -= carry[11] << 21
carry[13] = (s13 + (1 << 20)) >> 21
s14 += carry[13]
s13 -= carry[13] << 21
carry[15] = (s15 + (1 << 20)) >> 21
s16 += carry[15]
s15 -= carry[15] << 21
s5 += s17 * 666643
s6 += s17 * 470296
s7 += s17 * 654183
s8 -= s17 * 997805
s9 += s17 * 136657
s10 -= s17 * 683901
s17 = 0
s4 += s16 * 666643
s5 += s16 * 470296
s6 += s16 * 654183
s7 -= s16 * 997805
s8 += s16 * 136657
s9 -= s16 * 683901
s16 = 0
s3 += s15 * 666643
s4 += s15 * 470296
s5 += s15 * 654183
s6 -= s15 * 997805
s7 += s15 * 136657
s8 -= s15 * 683901
s15 = 0
s2 += s14 * 666643
s3 += s14 * 470296
s4 += s14 * 654183
s5 -= s14 * 997805
s6 += s14 * 136657
s7 -= s14 * 683901
s14 = 0
s1 += s13 * 666643
s2 += s13 * 470296
s3 += s13 * 654183
s4 -= s13 * 997805
s5 += s13 * 136657
s6 -= s13 * 683901
s13 = 0
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = (s0 + (1 << 20)) >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[2] = (s2 + (1 << 20)) >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[4] = (s4 + (1 << 20)) >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[6] = (s6 + (1 << 20)) >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[8] = (s8 + (1 << 20)) >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[10] = (s10 + (1 << 20)) >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[1] = (s1 + (1 << 20)) >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[3] = (s3 + (1 << 20)) >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[5] = (s5 + (1 << 20)) >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[7] = (s7 + (1 << 20)) >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[9] = (s9 + (1 << 20)) >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[11] = (s11 + (1 << 20)) >> 21
s12 += carry[11]
s11 -= carry[11] << 21
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = s0 >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[1] = s1 >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[2] = s2 >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[3] = s3 >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[4] = s4 >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[5] = s5 >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[6] = s6 >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[7] = s7 >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[8] = s8 >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[9] = s9 >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[10] = s10 >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[11] = s11 >> 21
s12 += carry[11]
s11 -= carry[11] << 21
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = s0 >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[1] = s1 >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[2] = s2 >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[3] = s3 >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[4] = s4 >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[5] = s5 >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[6] = s6 >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[7] = s7 >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[8] = s8 >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[9] = s9 >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[10] = s10 >> 21
s11 += carry[10]
s10 -= carry[10] << 21
s[0] = byte(s0 >> 0)
s[1] = byte(s0 >> 8)
s[2] = byte((s0 >> 16) | (s1 << 5))
s[3] = byte(s1 >> 3)
s[4] = byte(s1 >> 11)
s[5] = byte((s1 >> 19) | (s2 << 2))
s[6] = byte(s2 >> 6)
s[7] = byte((s2 >> 14) | (s3 << 7))
s[8] = byte(s3 >> 1)
s[9] = byte(s3 >> 9)
s[10] = byte((s3 >> 17) | (s4 << 4))
s[11] = byte(s4 >> 4)
s[12] = byte(s4 >> 12)
s[13] = byte((s4 >> 20) | (s5 << 1))
s[14] = byte(s5 >> 7)
s[15] = byte((s5 >> 15) | (s6 << 6))
s[16] = byte(s6 >> 2)
s[17] = byte(s6 >> 10)
s[18] = byte((s6 >> 18) | (s7 << 3))
s[19] = byte(s7 >> 5)
s[20] = byte(s7 >> 13)
s[21] = byte(s8 >> 0)
s[22] = byte(s8 >> 8)
s[23] = byte((s8 >> 16) | (s9 << 5))
s[24] = byte(s9 >> 3)
s[25] = byte(s9 >> 11)
s[26] = byte((s9 >> 19) | (s10 << 2))
s[27] = byte(s10 >> 6)
s[28] = byte((s10 >> 14) | (s11 << 7))
s[29] = byte(s11 >> 1)
s[30] = byte(s11 >> 9)
s[31] = byte(s11 >> 17)
}
// Input:
// s[0]+256*s[1]+...+256^63*s[63] = s
//
// Output:
// s[0]+256*s[1]+...+256^31*s[31] = s mod l
// where l = 2^252 + 27742317777372353535851937790883648493.
func scReduce(out *Scalar, s *[64]byte) {
s0 := 2097151 & load3(s[:])
s1 := 2097151 & (load4(s[2:]) >> 5)
s2 := 2097151 & (load3(s[5:]) >> 2)
s3 := 2097151 & (load4(s[7:]) >> 7)
s4 := 2097151 & (load4(s[10:]) >> 4)
s5 := 2097151 & (load3(s[13:]) >> 1)
s6 := 2097151 & (load4(s[15:]) >> 6)
s7 := 2097151 & (load3(s[18:]) >> 3)
s8 := 2097151 & load3(s[21:])
s9 := 2097151 & (load4(s[23:]) >> 5)
s10 := 2097151 & (load3(s[26:]) >> 2)
s11 := 2097151 & (load4(s[28:]) >> 7)
s12 := 2097151 & (load4(s[31:]) >> 4)
s13 := 2097151 & (load3(s[34:]) >> 1)
s14 := 2097151 & (load4(s[36:]) >> 6)
s15 := 2097151 & (load3(s[39:]) >> 3)
s16 := 2097151 & load3(s[42:])
s17 := 2097151 & (load4(s[44:]) >> 5)
s18 := 2097151 & (load3(s[47:]) >> 2)
s19 := 2097151 & (load4(s[49:]) >> 7)
s20 := 2097151 & (load4(s[52:]) >> 4)
s21 := 2097151 & (load3(s[55:]) >> 1)
s22 := 2097151 & (load4(s[57:]) >> 6)
s23 := (load4(s[60:]) >> 3)
s11 += s23 * 666643
s12 += s23 * 470296
s13 += s23 * 654183
s14 -= s23 * 997805
s15 += s23 * 136657
s16 -= s23 * 683901
s23 = 0
s10 += s22 * 666643
s11 += s22 * 470296
s12 += s22 * 654183
s13 -= s22 * 997805
s14 += s22 * 136657
s15 -= s22 * 683901
s22 = 0
s9 += s21 * 666643
s10 += s21 * 470296
s11 += s21 * 654183
s12 -= s21 * 997805
s13 += s21 * 136657
s14 -= s21 * 683901
s21 = 0
s8 += s20 * 666643
s9 += s20 * 470296
s10 += s20 * 654183
s11 -= s20 * 997805
s12 += s20 * 136657
s13 -= s20 * 683901
s20 = 0
s7 += s19 * 666643
s8 += s19 * 470296
s9 += s19 * 654183
s10 -= s19 * 997805
s11 += s19 * 136657
s12 -= s19 * 683901
s19 = 0
s6 += s18 * 666643
s7 += s18 * 470296
s8 += s18 * 654183
s9 -= s18 * 997805
s10 += s18 * 136657
s11 -= s18 * 683901
s18 = 0
var carry [17]int64
carry[6] = (s6 + (1 << 20)) >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[8] = (s8 + (1 << 20)) >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[10] = (s10 + (1 << 20)) >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[12] = (s12 + (1 << 20)) >> 21
s13 += carry[12]
s12 -= carry[12] << 21
carry[14] = (s14 + (1 << 20)) >> 21
s15 += carry[14]
s14 -= carry[14] << 21
carry[16] = (s16 + (1 << 20)) >> 21
s17 += carry[16]
s16 -= carry[16] << 21
carry[7] = (s7 + (1 << 20)) >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[9] = (s9 + (1 << 20)) >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[11] = (s11 + (1 << 20)) >> 21
s12 += carry[11]
s11 -= carry[11] << 21
carry[13] = (s13 + (1 << 20)) >> 21
s14 += carry[13]
s13 -= carry[13] << 21
carry[15] = (s15 + (1 << 20)) >> 21
s16 += carry[15]
s15 -= carry[15] << 21
s5 += s17 * 666643
s6 += s17 * 470296
s7 += s17 * 654183
s8 -= s17 * 997805
s9 += s17 * 136657
s10 -= s17 * 683901
s17 = 0
s4 += s16 * 666643
s5 += s16 * 470296
s6 += s16 * 654183
s7 -= s16 * 997805
s8 += s16 * 136657
s9 -= s16 * 683901
s16 = 0
s3 += s15 * 666643
s4 += s15 * 470296
s5 += s15 * 654183
s6 -= s15 * 997805
s7 += s15 * 136657
s8 -= s15 * 683901
s15 = 0
s2 += s14 * 666643
s3 += s14 * 470296
s4 += s14 * 654183
s5 -= s14 * 997805
s6 += s14 * 136657
s7 -= s14 * 683901
s14 = 0
s1 += s13 * 666643
s2 += s13 * 470296
s3 += s13 * 654183
s4 -= s13 * 997805
s5 += s13 * 136657
s6 -= s13 * 683901
s13 = 0
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = (s0 + (1 << 20)) >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[2] = (s2 + (1 << 20)) >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[4] = (s4 + (1 << 20)) >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[6] = (s6 + (1 << 20)) >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[8] = (s8 + (1 << 20)) >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[10] = (s10 + (1 << 20)) >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[1] = (s1 + (1 << 20)) >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[3] = (s3 + (1 << 20)) >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[5] = (s5 + (1 << 20)) >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[7] = (s7 + (1 << 20)) >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[9] = (s9 + (1 << 20)) >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[11] = (s11 + (1 << 20)) >> 21
s12 += carry[11]
s11 -= carry[11] << 21
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = s0 >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[1] = s1 >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[2] = s2 >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[3] = s3 >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[4] = s4 >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[5] = s5 >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[6] = s6 >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[7] = s7 >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[8] = s8 >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[9] = s9 >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[10] = s10 >> 21
s11 += carry[10]
s10 -= carry[10] << 21
carry[11] = s11 >> 21
s12 += carry[11]
s11 -= carry[11] << 21
s0 += s12 * 666643
s1 += s12 * 470296
s2 += s12 * 654183
s3 -= s12 * 997805
s4 += s12 * 136657
s5 -= s12 * 683901
s12 = 0
carry[0] = s0 >> 21
s1 += carry[0]
s0 -= carry[0] << 21
carry[1] = s1 >> 21
s2 += carry[1]
s1 -= carry[1] << 21
carry[2] = s2 >> 21
s3 += carry[2]
s2 -= carry[2] << 21
carry[3] = s3 >> 21
s4 += carry[3]
s3 -= carry[3] << 21
carry[4] = s4 >> 21
s5 += carry[4]
s4 -= carry[4] << 21
carry[5] = s5 >> 21
s6 += carry[5]
s5 -= carry[5] << 21
carry[6] = s6 >> 21
s7 += carry[6]
s6 -= carry[6] << 21
carry[7] = s7 >> 21
s8 += carry[7]
s7 -= carry[7] << 21
carry[8] = s8 >> 21
s9 += carry[8]
s8 -= carry[8] << 21
carry[9] = s9 >> 21
s10 += carry[9]
s9 -= carry[9] << 21
carry[10] = s10 >> 21
s11 += carry[10]
s10 -= carry[10] << 21
out[0] = byte(s0 >> 0)
out[1] = byte(s0 >> 8)
out[2] = byte((s0 >> 16) | (s1 << 5))
out[3] = byte(s1 >> 3)
out[4] = byte(s1 >> 11)
out[5] = byte((s1 >> 19) | (s2 << 2))
out[6] = byte(s2 >> 6)
out[7] = byte((s2 >> 14) | (s3 << 7))
out[8] = byte(s3 >> 1)
out[9] = byte(s3 >> 9)
out[10] = byte((s3 >> 17) | (s4 << 4))
out[11] = byte(s4 >> 4)
out[12] = byte(s4 >> 12)
out[13] = byte((s4 >> 20) | (s5 << 1))
out[14] = byte(s5 >> 7)
out[15] = byte((s5 >> 15) | (s6 << 6))
out[16] = byte(s6 >> 2)
out[17] = byte(s6 >> 10)
out[18] = byte((s6 >> 18) | (s7 << 3))
out[19] = byte(s7 >> 5)
out[20] = byte(s7 >> 13)
out[21] = byte(s8 >> 0)
out[22] = byte(s8 >> 8)
out[23] = byte((s8 >> 16) | (s9 << 5))
out[24] = byte(s9 >> 3)
out[25] = byte(s9 >> 11)
out[26] = byte((s9 >> 19) | (s10 << 2))
out[27] = byte(s10 >> 6)
out[28] = byte((s10 >> 14) | (s11 << 7))
out[29] = byte(s11 >> 1)
out[30] = byte(s11 >> 9)
out[31] = byte(s11 >> 17)
}
// order is the order of Curve25519 in little-endian form.
var order = [4]uint64{0x5812631a5cf5d3ed, 0x14def9dea2f79cd6, 0, 0x1000000000000000}
// scMinimal returns true if the given scalar is less than the order of the
// curve.
func scMinimal(sc []byte) bool {
for i := 3; ; i-- {
v := binary.LittleEndian.Uint64(sc[i*8:])
if v > order[i] {
return false
} else if v < order[i] {
break
} else if i == 0 {
return false
}
}
return true
}
// NonAdjacentForm computes a width-w non-adjacent form for this scalar.
func (s *Scalar) NonAdjacentForm(w uint) [256]int8 {
// This implementation is adapted from the one
// in curve25519-dalek and is documented there:
// https://github.com/dalek-cryptography/curve25519-dalek/blob/f630041af28e9a405255f98a8a93adca18e4315b/src/scalar.rs#L800-L871
if s[31] > 127 {
panic("scalar has high bit set illegally")
}
if w < 2 {
panic("w must be at least 2 by the definition of NAF")
} else if w > 8 {
panic("NAF digits must fit in int8")
}
var naf [256]int8
var digits [5]uint64
for i := 0; i < 4; i++ {
digits[i] = binary.LittleEndian.Uint64(s[i*8:])
}
width := uint64(1 << w)
windowMask := uint64(width - 1)
pos := uint(0)
carry := uint64(0)
for pos < 256 {
indexU64 := pos / 64
indexBit := pos % 64
var bitBuf uint64
if indexBit < 64-w {
// This window's bits are contained in a single u64
bitBuf = digits[indexU64] >> indexBit
} else {
// Combine the current 64 bits with bits from the next 64
bitBuf = (digits[indexU64] >> indexBit) | (digits[1+indexU64] << (64 - indexBit))
}
// Add carry into the current window
window := carry + (bitBuf & windowMask)
if window&1 == 0 {
// If the window value is even, preserve the carry and continue.
// Why is the carry preserved?
// If carry == 0 and window & 1 == 0,
// then the next carry should be 0
// If carry == 1 and window & 1 == 0,
// then bit_buf & 1 == 1 so the next carry should be 1
pos += 1
continue
}
if window < width/2 {
carry = 0
naf[pos] = int8(window)
} else {
carry = 1
naf[pos] = int8(window) - int8(width)
}
pos += w
}
return naf
}
func (s *Scalar) SignedRadix16() [64]int8 {
if s[31] > 127 {
panic("scalar has high bit set illegally")
}
var digits [64]int8
// Compute unsigned radix-16 digits:
for i := 0; i < 32; i++ {
digits[2*i] = int8(s[i] & 15)
digits[2*i+1] = int8((s[i] >> 4) & 15)
}
// Recenter coefficients:
for i := 0; i < 63; i++ {
carry := (digits[i] + 8) >> 4
digits[i] -= carry << 4
digits[i+1] += carry
}
return digits
}
// Given k > 0, set s = s**(2*i).
func (s *Scalar) pow2k(k int) {
for i := 0; i < k; i++ {
s.Mul(s, s)
}
}
// Inv sets s to the inverse of a nonzero scalar v and returns s.
func (s *Scalar) Inv(t *Scalar) *Scalar {
// Uses a hardcoded sliding window of width 4.
var table [8]Scalar
var tt Scalar
tt.Mul(t, t)
table[0] = *t
for i := 0; i < 7; i++ {
table[i+1].Mul(&table[i], &tt)
}
// Now table = [t**1, t**3, t**7, t**11, t**13, t**15]
// so t**k = t[k/2] for odd k
// To compute the sliding window digits, use the following Sage script:
// sage: import itertools
// sage: def sliding_window(w,k):
// ....: digits = []
// ....: while k > 0:
// ....: if k % 2 == 1:
// ....: kmod = k % (2**w)
// ....: digits.append(kmod)
// ....: k = k - kmod
// ....: else:
// ....: digits.append(0)
// ....: k = k // 2
// ....: return digits
// Now we can compute s roughly as follows:
// sage: s = 1
// sage: for coeff in reversed(sliding_window(4,l-2)):
// ....: s = s*s
// ....: if coeff > 0 :
// ....: s = s*t**coeff
// This works on one bit at a time, with many runs of zeros.
// The digits can be collapsed into [(count, coeff)] as follows:
// sage: [(len(list(group)),d) for d,group in itertools.groupby(sliding_window(4,l-2))]
// Entries of the form (k, 0) turn into pow2k(k)
// Entries of the form (1, coeff) turn into a squaring and then a table lookup.
// We can fold the squaring into the previous pow2k(k) as pow2k(k+1).
*s = table[1/2]
s.pow2k(127 + 1)
s.Mul(s, &table[1/2])
s.pow2k(4 + 1)
s.Mul(s, &table[9/2])
s.pow2k(3 + 1)
s.Mul(s, &table[11/2])
s.pow2k(3 + 1)
s.Mul(s, &table[13/2])
s.pow2k(3 + 1)
s.Mul(s, &table[15/2])
s.pow2k(4 + 1)
s.Mul(s, &table[7/2])
s.pow2k(4 + 1)
s.Mul(s, &table[15/2])
s.pow2k(3 + 1)
s.Mul(s, &table[5/2])
s.pow2k(3 + 1)
s.Mul(s, &table[1/2])
s.pow2k(4 + 1)
s.Mul(s, &table[15/2])
s.pow2k(4 + 1)
s.Mul(s, &table[15/2])
s.pow2k(4 + 1)
s.Mul(s, &table[7/2])
s.pow2k(3 + 1)
s.Mul(s, &table[3/2])
s.pow2k(4 + 1)
s.Mul(s, &table[11/2])
s.pow2k(5 + 1)
s.Mul(s, &table[11/2])
s.pow2k(9 + 1)
s.Mul(s, &table[9/2])
s.pow2k(3 + 1)
s.Mul(s, &table[3/2])
s.pow2k(4 + 1)
s.Mul(s, &table[3/2])
s.pow2k(4 + 1)
s.Mul(s, &table[3/2])
s.pow2k(4 + 1)
s.Mul(s, &table[9/2])
s.pow2k(3 + 1)
s.Mul(s, &table[7/2])
s.pow2k(3 + 1)
s.Mul(s, &table[3/2])
s.pow2k(3 + 1)
s.Mul(s, &table[13/2])
s.pow2k(3 + 1)
s.Mul(s, &table[7/2])
s.pow2k(4 + 1)
s.Mul(s, &table[9/2])
s.pow2k(3 + 1)
s.Mul(s, &table[15/2])
s.pow2k(4 + 1)
s.Mul(s, &table[11/2])
return s
}
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