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edwards25519.go
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edwards25519.go
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// Copyright 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 "encoding/binary"
// This code is a port of the public domain, “ref10” implementation of ed25519
// from SUPERCOP.
// FieldElement represents an element of the field GF(2^255 - 19). An element
// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
// t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
// context.
type FieldElement [10]int32
var zero FieldElement
func FeZero(fe *FieldElement) {
copy(fe[:], zero[:])
}
func FeOne(fe *FieldElement) {
FeZero(fe)
fe[0] = 1
}
func FeAdd(dst, a, b *FieldElement) {
dst[0] = a[0] + b[0]
dst[1] = a[1] + b[1]
dst[2] = a[2] + b[2]
dst[3] = a[3] + b[3]
dst[4] = a[4] + b[4]
dst[5] = a[5] + b[5]
dst[6] = a[6] + b[6]
dst[7] = a[7] + b[7]
dst[8] = a[8] + b[8]
dst[9] = a[9] + b[9]
}
func FeSub(dst, a, b *FieldElement) {
dst[0] = a[0] - b[0]
dst[1] = a[1] - b[1]
dst[2] = a[2] - b[2]
dst[3] = a[3] - b[3]
dst[4] = a[4] - b[4]
dst[5] = a[5] - b[5]
dst[6] = a[6] - b[6]
dst[7] = a[7] - b[7]
dst[8] = a[8] - b[8]
dst[9] = a[9] - b[9]
}
func FeCopy(dst, src *FieldElement) {
copy(dst[:], src[:])
}
// Replace (f,g) with (g,g) if b == 1;
// replace (f,g) with (f,g) if b == 0.
//
// Preconditions: b in {0,1}.
func FeCMove(f, g *FieldElement, b int32) {
b = -b
f[0] ^= b & (f[0] ^ g[0])
f[1] ^= b & (f[1] ^ g[1])
f[2] ^= b & (f[2] ^ g[2])
f[3] ^= b & (f[3] ^ g[3])
f[4] ^= b & (f[4] ^ g[4])
f[5] ^= b & (f[5] ^ g[5])
f[6] ^= b & (f[6] ^ g[6])
f[7] ^= b & (f[7] ^ g[7])
f[8] ^= b & (f[8] ^ g[8])
f[9] ^= b & (f[9] ^ g[9])
}
func load3(in []byte) int64 {
var r int64 = int64(in[0])
r |= int64(in[1]) << 8
r |= int64(in[2]) << 16
return r
}
func load4(in []byte) int64 {
var r int64 = int64(in[0])
r |= int64(in[1]) << 8
r |= int64(in[2]) << 16
r |= int64(in[3]) << 24
return r
}
func FeFromBytes(dst *FieldElement, src *[32]byte) {
h0 := load4(src[:])
h1 := load3(src[4:]) << 6
h2 := load3(src[7:]) << 5
h3 := load3(src[10:]) << 3
h4 := load3(src[13:]) << 2
h5 := load4(src[16:])
h6 := load3(src[20:]) << 7
h7 := load3(src[23:]) << 5
h8 := load3(src[26:]) << 4
h9 := (load3(src[29:]) & 8388607) << 2
FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
}
// FeToBytes marshals h to s.
// Preconditions:
// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
//
// Write p=2^255-19; q=floor(h/p).
// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
//
// Proof:
// Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
// Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4.
//
// Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
// Then 0<y<1.
//
// Write r=h-pq.
// Have 0<=r<=p-1=2^255-20.
// Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
//
// Write x=r+19(2^-255)r+y.
// Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
//
// Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
// so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
func FeToBytes(s *[32]byte, h *FieldElement) {
var carry [10]int32
q := (19*h[9] + (1 << 24)) >> 25
q = (h[0] + q) >> 26
q = (h[1] + q) >> 25
q = (h[2] + q) >> 26
q = (h[3] + q) >> 25
q = (h[4] + q) >> 26
q = (h[5] + q) >> 25
q = (h[6] + q) >> 26
q = (h[7] + q) >> 25
q = (h[8] + q) >> 26
q = (h[9] + q) >> 25
// Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20.
h[0] += 19 * q
// Goal: Output h-2^255 q, which is between 0 and 2^255-20.
carry[0] = h[0] >> 26
h[1] += carry[0]
h[0] -= carry[0] << 26
carry[1] = h[1] >> 25
h[2] += carry[1]
h[1] -= carry[1] << 25
carry[2] = h[2] >> 26
h[3] += carry[2]
h[2] -= carry[2] << 26
carry[3] = h[3] >> 25
h[4] += carry[3]
h[3] -= carry[3] << 25
carry[4] = h[4] >> 26
h[5] += carry[4]
h[4] -= carry[4] << 26
carry[5] = h[5] >> 25
h[6] += carry[5]
h[5] -= carry[5] << 25
carry[6] = h[6] >> 26
h[7] += carry[6]
h[6] -= carry[6] << 26
carry[7] = h[7] >> 25
h[8] += carry[7]
h[7] -= carry[7] << 25
carry[8] = h[8] >> 26
h[9] += carry[8]
h[8] -= carry[8] << 26
carry[9] = h[9] >> 25
h[9] -= carry[9] << 25
// h10 = carry9
// Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
// Have h[0]+...+2^230 h[9] between 0 and 2^255-1;
// evidently 2^255 h10-2^255 q = 0.
// Goal: Output h[0]+...+2^230 h[9].
s[0] = byte(h[0] >> 0)
s[1] = byte(h[0] >> 8)
s[2] = byte(h[0] >> 16)
s[3] = byte((h[0] >> 24) | (h[1] << 2))
s[4] = byte(h[1] >> 6)
s[5] = byte(h[1] >> 14)
s[6] = byte((h[1] >> 22) | (h[2] << 3))
s[7] = byte(h[2] >> 5)
s[8] = byte(h[2] >> 13)
s[9] = byte((h[2] >> 21) | (h[3] << 5))
s[10] = byte(h[3] >> 3)
s[11] = byte(h[3] >> 11)
s[12] = byte((h[3] >> 19) | (h[4] << 6))
s[13] = byte(h[4] >> 2)
s[14] = byte(h[4] >> 10)
s[15] = byte(h[4] >> 18)
s[16] = byte(h[5] >> 0)
s[17] = byte(h[5] >> 8)
s[18] = byte(h[5] >> 16)
s[19] = byte((h[5] >> 24) | (h[6] << 1))
s[20] = byte(h[6] >> 7)
s[21] = byte(h[6] >> 15)
s[22] = byte((h[6] >> 23) | (h[7] << 3))
s[23] = byte(h[7] >> 5)
s[24] = byte(h[7] >> 13)
s[25] = byte((h[7] >> 21) | (h[8] << 4))
s[26] = byte(h[8] >> 4)
s[27] = byte(h[8] >> 12)
s[28] = byte((h[8] >> 20) | (h[9] << 6))
s[29] = byte(h[9] >> 2)
s[30] = byte(h[9] >> 10)
s[31] = byte(h[9] >> 18)
}
func FeIsNegative(f *FieldElement) byte {
var s [32]byte
FeToBytes(&s, f)
return s[0] & 1
}
func FeIsNonZero(f *FieldElement) int32 {
var s [32]byte
FeToBytes(&s, f)
var x uint8
for _, b := range s {
x |= b
}
x |= x >> 4
x |= x >> 2
x |= x >> 1
return int32(x & 1)
}
// FeNeg sets h = -f
//
// Preconditions:
// |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
//
// Postconditions:
// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
func FeNeg(h, f *FieldElement) {
h[0] = -f[0]
h[1] = -f[1]
h[2] = -f[2]
h[3] = -f[3]
h[4] = -f[4]
h[5] = -f[5]
h[6] = -f[6]
h[7] = -f[7]
h[8] = -f[8]
h[9] = -f[9]
}
func FeCombine(h *FieldElement, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) {
var c0, c1, c2, c3, c4, c5, c6, c7, c8, c9 int64
/*
|h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
|h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9
*/
c0 = (h0 + (1 << 25)) >> 26
h1 += c0
h0 -= c0 << 26
c4 = (h4 + (1 << 25)) >> 26
h5 += c4
h4 -= c4 << 26
/* |h0| <= 2^25 */
/* |h4| <= 2^25 */
/* |h1| <= 1.51*2^58 */
/* |h5| <= 1.51*2^58 */
c1 = (h1 + (1 << 24)) >> 25
h2 += c1
h1 -= c1 << 25
c5 = (h5 + (1 << 24)) >> 25
h6 += c5
h5 -= c5 << 25
/* |h1| <= 2^24; from now on fits into int32 */
/* |h5| <= 2^24; from now on fits into int32 */
/* |h2| <= 1.21*2^59 */
/* |h6| <= 1.21*2^59 */
c2 = (h2 + (1 << 25)) >> 26
h3 += c2
h2 -= c2 << 26
c6 = (h6 + (1 << 25)) >> 26
h7 += c6
h6 -= c6 << 26
/* |h2| <= 2^25; from now on fits into int32 unchanged */
/* |h6| <= 2^25; from now on fits into int32 unchanged */
/* |h3| <= 1.51*2^58 */
/* |h7| <= 1.51*2^58 */
c3 = (h3 + (1 << 24)) >> 25
h4 += c3
h3 -= c3 << 25
c7 = (h7 + (1 << 24)) >> 25
h8 += c7
h7 -= c7 << 25
/* |h3| <= 2^24; from now on fits into int32 unchanged */
/* |h7| <= 2^24; from now on fits into int32 unchanged */
/* |h4| <= 1.52*2^33 */
/* |h8| <= 1.52*2^33 */
c4 = (h4 + (1 << 25)) >> 26
h5 += c4
h4 -= c4 << 26
c8 = (h8 + (1 << 25)) >> 26
h9 += c8
h8 -= c8 << 26
/* |h4| <= 2^25; from now on fits into int32 unchanged */
/* |h8| <= 2^25; from now on fits into int32 unchanged */
/* |h5| <= 1.01*2^24 */
/* |h9| <= 1.51*2^58 */
c9 = (h9 + (1 << 24)) >> 25
h0 += c9 * 19
h9 -= c9 << 25
/* |h9| <= 2^24; from now on fits into int32 unchanged */
/* |h0| <= 1.8*2^37 */
c0 = (h0 + (1 << 25)) >> 26
h1 += c0
h0 -= c0 << 26
/* |h0| <= 2^25; from now on fits into int32 unchanged */
/* |h1| <= 1.01*2^24 */
h[0] = int32(h0)
h[1] = int32(h1)
h[2] = int32(h2)
h[3] = int32(h3)
h[4] = int32(h4)
h[5] = int32(h5)
h[6] = int32(h6)
h[7] = int32(h7)
h[8] = int32(h8)
h[9] = int32(h9)
}
// FeMul calculates h = f * g
// Can overlap h with f or g.
//
// Preconditions:
// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
//
// Postconditions:
// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
//
// Notes on implementation strategy:
//
// Using schoolbook multiplication.
// Karatsuba would save a little in some cost models.
//
// Most multiplications by 2 and 19 are 32-bit precomputations;
// cheaper than 64-bit postcomputations.
//
// There is one remaining multiplication by 19 in the carry chain;
// one *19 precomputation can be merged into this,
// but the resulting data flow is considerably less clean.
//
// There are 12 carries below.
// 10 of them are 2-way parallelizable and vectorizable.
// Can get away with 11 carries, but then data flow is much deeper.
//
// With tighter constraints on inputs, can squeeze carries into int32.
func FeMul(h, f, g *FieldElement) {
f0 := int64(f[0])
f1 := int64(f[1])
f2 := int64(f[2])
f3 := int64(f[3])
f4 := int64(f[4])
f5 := int64(f[5])
f6 := int64(f[6])
f7 := int64(f[7])
f8 := int64(f[8])
f9 := int64(f[9])
f1_2 := int64(2 * f[1])
f3_2 := int64(2 * f[3])
f5_2 := int64(2 * f[5])
f7_2 := int64(2 * f[7])
f9_2 := int64(2 * f[9])
g0 := int64(g[0])
g1 := int64(g[1])
g2 := int64(g[2])
g3 := int64(g[3])
g4 := int64(g[4])
g5 := int64(g[5])
g6 := int64(g[6])
g7 := int64(g[7])
g8 := int64(g[8])
g9 := int64(g[9])
g1_19 := int64(19 * g[1]) /* 1.4*2^29 */
g2_19 := int64(19 * g[2]) /* 1.4*2^30; still ok */
g3_19 := int64(19 * g[3])
g4_19 := int64(19 * g[4])
g5_19 := int64(19 * g[5])
g6_19 := int64(19 * g[6])
g7_19 := int64(19 * g[7])
g8_19 := int64(19 * g[8])
g9_19 := int64(19 * g[9])
h0 := f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19
h1 := f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19
h2 := f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19
h3 := f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19
h4 := f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19
h5 := f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19
h6 := f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19
h7 := f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19
h8 := f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19
h9 := f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0
FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
}
func feSquare(f *FieldElement) (h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) {
f0 := int64(f[0])
f1 := int64(f[1])
f2 := int64(f[2])
f3 := int64(f[3])
f4 := int64(f[4])
f5 := int64(f[5])
f6 := int64(f[6])
f7 := int64(f[7])
f8 := int64(f[8])
f9 := int64(f[9])
f0_2 := int64(2 * f[0])
f1_2 := int64(2 * f[1])
f2_2 := int64(2 * f[2])
f3_2 := int64(2 * f[3])
f4_2 := int64(2 * f[4])
f5_2 := int64(2 * f[5])
f6_2 := int64(2 * f[6])
f7_2 := int64(2 * f[7])
f5_38 := 38 * f5 // 1.31*2^30
f6_19 := 19 * f6 // 1.31*2^30
f7_38 := 38 * f7 // 1.31*2^30
f8_19 := 19 * f8 // 1.31*2^30
f9_38 := 38 * f9 // 1.31*2^30
h0 = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38
h1 = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19
h2 = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19
h3 = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38
h4 = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38
h5 = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19
h6 = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19
h7 = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38
h8 = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38
h9 = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5
return
}
// FeSquare calculates h = f*f. Can overlap h with f.
//
// Preconditions:
// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
//
// Postconditions:
// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
func FeSquare(h, f *FieldElement) {
h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f)
FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
}
// FeSquare2 sets h = 2 * f * f
//
// Can overlap h with f.
//
// Preconditions:
// |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
//
// Postconditions:
// |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
// See fe_mul.c for discussion of implementation strategy.
func FeSquare2(h, f *FieldElement) {
h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f)
h0 += h0
h1 += h1
h2 += h2
h3 += h3
h4 += h4
h5 += h5
h6 += h6
h7 += h7
h8 += h8
h9 += h9
FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
}
func FeInvert(out, z *FieldElement) {
var t0, t1, t2, t3 FieldElement
var i int
FeSquare(&t0, z) // 2^1
FeSquare(&t1, &t0) // 2^2
for i = 1; i < 2; i++ { // 2^3
FeSquare(&t1, &t1)
}
FeMul(&t1, z, &t1) // 2^3 + 2^0
FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0
FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1
FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0
FeSquare(&t2, &t1) // 5,4,3,2,1
for i = 1; i < 5; i++ { // 9,8,7,6,5
FeSquare(&t2, &t2)
}
FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0
FeSquare(&t2, &t1) // 10..1
for i = 1; i < 10; i++ { // 19..10
FeSquare(&t2, &t2)
}
FeMul(&t2, &t2, &t1) // 19..0
FeSquare(&t3, &t2) // 20..1
for i = 1; i < 20; i++ { // 39..20
FeSquare(&t3, &t3)
}
FeMul(&t2, &t3, &t2) // 39..0
FeSquare(&t2, &t2) // 40..1
for i = 1; i < 10; i++ { // 49..10
FeSquare(&t2, &t2)
}
FeMul(&t1, &t2, &t1) // 49..0
FeSquare(&t2, &t1) // 50..1
for i = 1; i < 50; i++ { // 99..50
FeSquare(&t2, &t2)
}
FeMul(&t2, &t2, &t1) // 99..0
FeSquare(&t3, &t2) // 100..1
for i = 1; i < 100; i++ { // 199..100
FeSquare(&t3, &t3)
}
FeMul(&t2, &t3, &t2) // 199..0
FeSquare(&t2, &t2) // 200..1
for i = 1; i < 50; i++ { // 249..50
FeSquare(&t2, &t2)
}
FeMul(&t1, &t2, &t1) // 249..0
FeSquare(&t1, &t1) // 250..1
for i = 1; i < 5; i++ { // 254..5
FeSquare(&t1, &t1)
}
FeMul(out, &t1, &t0) // 254..5,3,1,0
}
func fePow22523(out, z *FieldElement) {
var t0, t1, t2 FieldElement
var i int
FeSquare(&t0, z)
for i = 1; i < 1; i++ {
FeSquare(&t0, &t0)
}
FeSquare(&t1, &t0)
for i = 1; i < 2; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t1, z, &t1)
FeMul(&t0, &t0, &t1)
FeSquare(&t0, &t0)
for i = 1; i < 1; i++ {
FeSquare(&t0, &t0)
}
FeMul(&t0, &t1, &t0)
FeSquare(&t1, &t0)
for i = 1; i < 5; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t0, &t1, &t0)
FeSquare(&t1, &t0)
for i = 1; i < 10; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t1, &t1, &t0)
FeSquare(&t2, &t1)
for i = 1; i < 20; i++ {
FeSquare(&t2, &t2)
}
FeMul(&t1, &t2, &t1)
FeSquare(&t1, &t1)
for i = 1; i < 10; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t0, &t1, &t0)
FeSquare(&t1, &t0)
for i = 1; i < 50; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t1, &t1, &t0)
FeSquare(&t2, &t1)
for i = 1; i < 100; i++ {
FeSquare(&t2, &t2)
}
FeMul(&t1, &t2, &t1)
FeSquare(&t1, &t1)
for i = 1; i < 50; i++ {
FeSquare(&t1, &t1)
}
FeMul(&t0, &t1, &t0)
FeSquare(&t0, &t0)
for i = 1; i < 2; i++ {
FeSquare(&t0, &t0)
}
FeMul(out, &t0, z)
}
// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 *
// y^2 where d = -121665/121666.
//
// Several representations are used:
// ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z
// ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
// CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
// PreComputedGroupElement: (y+x,y-x,2dxy)
type ProjectiveGroupElement struct {
X, Y, Z FieldElement
}
type ExtendedGroupElement struct {
X, Y, Z, T FieldElement
}
type CompletedGroupElement struct {
X, Y, Z, T FieldElement
}
type PreComputedGroupElement struct {
yPlusX, yMinusX, xy2d FieldElement
}
type CachedGroupElement struct {
yPlusX, yMinusX, Z, T2d FieldElement
}
func (p *ProjectiveGroupElement) Zero() {
FeZero(&p.X)
FeOne(&p.Y)
FeOne(&p.Z)
}
func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) {
var t0 FieldElement
FeSquare(&r.X, &p.X)
FeSquare(&r.Z, &p.Y)
FeSquare2(&r.T, &p.Z)
FeAdd(&r.Y, &p.X, &p.Y)
FeSquare(&t0, &r.Y)
FeAdd(&r.Y, &r.Z, &r.X)
FeSub(&r.Z, &r.Z, &r.X)
FeSub(&r.X, &t0, &r.Y)
FeSub(&r.T, &r.T, &r.Z)
}
func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) {
var recip, x, y FieldElement
FeInvert(&recip, &p.Z)
FeMul(&x, &p.X, &recip)
FeMul(&y, &p.Y, &recip)
FeToBytes(s, &y)
s[31] ^= FeIsNegative(&x) << 7
}
func (p *ExtendedGroupElement) Zero() {
FeZero(&p.X)
FeOne(&p.Y)
FeOne(&p.Z)
FeZero(&p.T)
}
func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) {
var q ProjectiveGroupElement
p.ToProjective(&q)
q.Double(r)
}
func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) {
FeAdd(&r.yPlusX, &p.Y, &p.X)
FeSub(&r.yMinusX, &p.Y, &p.X)
FeCopy(&r.Z, &p.Z)
FeMul(&r.T2d, &p.T, &d2)
}
func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) {
FeCopy(&r.X, &p.X)
FeCopy(&r.Y, &p.Y)
FeCopy(&r.Z, &p.Z)
}
func (p *ExtendedGroupElement) ToBytes(s *[32]byte) {
var recip, x, y FieldElement
FeInvert(&recip, &p.Z)
FeMul(&x, &p.X, &recip)
FeMul(&y, &p.Y, &recip)
FeToBytes(s, &y)
s[31] ^= FeIsNegative(&x) << 7
}
func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool {
var u, v, v3, vxx, check FieldElement
FeFromBytes(&p.Y, s)
FeOne(&p.Z)
FeSquare(&u, &p.Y)
FeMul(&v, &u, &d)
FeSub(&u, &u, &p.Z) // y = y^2-1
FeAdd(&v, &v, &p.Z) // v = dy^2+1
FeSquare(&v3, &v)
FeMul(&v3, &v3, &v) // v3 = v^3
FeSquare(&p.X, &v3)
FeMul(&p.X, &p.X, &v)
FeMul(&p.X, &p.X, &u) // x = uv^7
fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8)
FeMul(&p.X, &p.X, &v3)
FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8)
var tmpX, tmp2 [32]byte
FeSquare(&vxx, &p.X)
FeMul(&vxx, &vxx, &v)
FeSub(&check, &vxx, &u) // vx^2-u
if FeIsNonZero(&check) == 1 {
FeAdd(&check, &vxx, &u) // vx^2+u
if FeIsNonZero(&check) == 1 {
return false
}
FeMul(&p.X, &p.X, &SqrtM1)
FeToBytes(&tmpX, &p.X)
for i, v := range tmpX {
tmp2[31-i] = v
}
}
if FeIsNegative(&p.X) != (s[31] >> 7) {
FeNeg(&p.X, &p.X)
}
FeMul(&p.T, &p.X, &p.Y)
return true
}
func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) {
FeMul(&r.X, &p.X, &p.T)
FeMul(&r.Y, &p.Y, &p.Z)
FeMul(&r.Z, &p.Z, &p.T)
}
func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) {
FeMul(&r.X, &p.X, &p.T)
FeMul(&r.Y, &p.Y, &p.Z)
FeMul(&r.Z, &p.Z, &p.T)
FeMul(&r.T, &p.X, &p.Y)
}
func (p *PreComputedGroupElement) Zero() {
FeOne(&p.yPlusX)
FeOne(&p.yMinusX)
FeZero(&p.xy2d)
}
func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) {
var t0 FieldElement
FeAdd(&r.X, &p.Y, &p.X)
FeSub(&r.Y, &p.Y, &p.X)
FeMul(&r.Z, &r.X, &q.yPlusX)
FeMul(&r.Y, &r.Y, &q.yMinusX)
FeMul(&r.T, &q.T2d, &p.T)
FeMul(&r.X, &p.Z, &q.Z)
FeAdd(&t0, &r.X, &r.X)
FeSub(&r.X, &r.Z, &r.Y)
FeAdd(&r.Y, &r.Z, &r.Y)
FeAdd(&r.Z, &t0, &r.T)
FeSub(&r.T, &t0, &r.T)
}
func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) {
var t0 FieldElement
FeAdd(&r.X, &p.Y, &p.X)
FeSub(&r.Y, &p.Y, &p.X)
FeMul(&r.Z, &r.X, &q.yMinusX)
FeMul(&r.Y, &r.Y, &q.yPlusX)
FeMul(&r.T, &q.T2d, &p.T)
FeMul(&r.X, &p.Z, &q.Z)
FeAdd(&t0, &r.X, &r.X)
FeSub(&r.X, &r.Z, &r.Y)
FeAdd(&r.Y, &r.Z, &r.Y)
FeSub(&r.Z, &t0, &r.T)
FeAdd(&r.T, &t0, &r.T)
}
func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) {
var t0 FieldElement
FeAdd(&r.X, &p.Y, &p.X)
FeSub(&r.Y, &p.Y, &p.X)
FeMul(&r.Z, &r.X, &q.yPlusX)
FeMul(&r.Y, &r.Y, &q.yMinusX)
FeMul(&r.T, &q.xy2d, &p.T)
FeAdd(&t0, &p.Z, &p.Z)
FeSub(&r.X, &r.Z, &r.Y)
FeAdd(&r.Y, &r.Z, &r.Y)
FeAdd(&r.Z, &t0, &r.T)
FeSub(&r.T, &t0, &r.T)
}
func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) {
var t0 FieldElement
FeAdd(&r.X, &p.Y, &p.X)
FeSub(&r.Y, &p.Y, &p.X)
FeMul(&r.Z, &r.X, &q.yMinusX)
FeMul(&r.Y, &r.Y, &q.yPlusX)
FeMul(&r.T, &q.xy2d, &p.T)
FeAdd(&t0, &p.Z, &p.Z)
FeSub(&r.X, &r.Z, &r.Y)
FeAdd(&r.Y, &r.Z, &r.Y)
FeSub(&r.Z, &t0, &r.T)
FeAdd(&r.T, &t0, &r.T)
}
func slide(r *[256]int8, a *[32]byte) {
for i := range r {
r[i] = int8(1 & (a[i>>3] >> uint(i&7)))
}
for i := range r {
if r[i] != 0 {
for b := 1; b <= 6 && i+b < 256; b++ {
if r[i+b] != 0 {
if r[i]+(r[i+b]<<uint(b)) <= 15 {
r[i] += r[i+b] << uint(b)
r[i+b] = 0
} else if r[i]-(r[i+b]<<uint(b)) >= -15 {
r[i] -= r[i+b] << uint(b)
for k := i + b; k < 256; k++ {
if r[k] == 0 {
r[k] = 1
break
}
r[k] = 0
}
} else {
break
}
}
}
}
}
}
// GeDoubleScalarMultVartime sets r = a*A + b*B
// where a = a[0]+256*a[1]+...+256^31 a[31].
// and b = b[0]+256*b[1]+...+256^31 b[31].
// B is the Ed25519 base point (x,4/5) with x positive.
func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) {
var aSlide, bSlide [256]int8
var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A
var t CompletedGroupElement
var u, A2 ExtendedGroupElement
var i int
slide(&aSlide, a)
slide(&bSlide, b)
A.ToCached(&Ai[0])
A.Double(&t)
t.ToExtended(&A2)
for i := 0; i < 7; i++ {
geAdd(&t, &A2, &Ai[i])
t.ToExtended(&u)
u.ToCached(&Ai[i+1])
}
r.Zero()
for i = 255; i >= 0; i-- {
if aSlide[i] != 0 || bSlide[i] != 0 {
break
}
}
for ; i >= 0; i-- {
r.Double(&t)
if aSlide[i] > 0 {
t.ToExtended(&u)
geAdd(&t, &u, &Ai[aSlide[i]/2])
} else if aSlide[i] < 0 {
t.ToExtended(&u)
geSub(&t, &u, &Ai[(-aSlide[i])/2])
}
if bSlide[i] > 0 {
t.ToExtended(&u)
geMixedAdd(&t, &u, &bi[bSlide[i]/2])
} else if bSlide[i] < 0 {
t.ToExtended(&u)
geMixedSub(&t, &u, &bi[(-bSlide[i])/2])
}
t.ToProjective(r)
}
}
// equal returns 1 if b == c and 0 otherwise, assuming that b and c are
// non-negative.
func equal(b, c int32) int32 {
x := uint32(b ^ c)
x--
return int32(x >> 31)
}
// negative returns 1 if b < 0 and 0 otherwise.
func negative(b int32) int32 {
return (b >> 31) & 1
}
func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) {
FeCMove(&t.yPlusX, &u.yPlusX, b)
FeCMove(&t.yMinusX, &u.yMinusX, b)
FeCMove(&t.xy2d, &u.xy2d, b)
}
func selectPoint(t *PreComputedGroupElement, pos int32, b int32) {
var minusT PreComputedGroupElement
bNegative := negative(b)
bAbs := b - (((-bNegative) & b) << 1)
t.Zero()
for i := int32(0); i < 8; i++ {
PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1))
}
FeCopy(&minusT.yPlusX, &t.yMinusX)
FeCopy(&minusT.yMinusX, &t.yPlusX)
FeNeg(&minusT.xy2d, &t.xy2d)
PreComputedGroupElementCMove(t, &minusT, bNegative)
}
// GeScalarMultBase computes h = a*B, where
// a = a[0]+256*a[1]+...+256^31 a[31]
// B is the Ed25519 base point (x,4/5) with x positive.
//
// Preconditions:
// a[31] <= 127
func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) {
var e [64]int8
for i, v := range a {
e[2*i] = int8(v & 15)
e[2*i+1] = int8((v >> 4) & 15)
}
// each e[i] is between 0 and 15 and e[63] is between 0 and 7.
carry := int8(0)
for i := 0; i < 63; i++ {
e[i] += carry
carry = (e[i] + 8) >> 4
e[i] -= carry << 4
}
e[63] += carry
// each e[i] is between -8 and 8.
h.Zero()
var t PreComputedGroupElement
var r CompletedGroupElement
for i := int32(1); i < 64; i += 2 {
selectPoint(&t, i/2, int32(e[i]))
geMixedAdd(&r, h, &t)
r.ToExtended(h)
}
var s ProjectiveGroupElement
h.Double(&r)
r.ToProjective(&s)
s.Double(&r)