edwards25519.go

// 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
	r = int64(in[0])
	r |= int64(in[1]) << 8
	r |= int64(in[2]) << 16
	return r
}

func load4(in []byte) int64 {
	var r int64
	r = 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)
	r.ToProjective(&s)
	s.Double(&r)
	r.ToProjective(&s)
	s.Double(&r)
	r.ToExtended(h)

	for i := int32(0); i < 64; i += 2 {
		selectPoint(&t, i/2, int32(e[i]))
		geMixedAdd(&r, h, &t)
		r.ToExtended(h)
	}
}

// The scalars are GF(2^252 + 27742317777372353535851937790883648493).

// 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 *[32]byte) {
	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 *[32]byte, 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(scalar *[32]byte) bool {
	for i := 3; ; i-- {
		v := binary.LittleEndian.Uint64(scalar[i*8:])
		if v > order[i] {
			return false
		} else if v < order[i] {
			break
		} else if i == 0 {
			return false
		}
	}

	return true
}