extra25519.go

// Copyright 2013 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 extra25519

import (
	"crypto/sha512"

	"github.com/ooni/psiphon/oopsi/github.com/agl/ed25519/edwards25519"
)

// PrivateKeyToCurve25519 converts an ed25519 private key into a corresponding
// curve25519 private key such that the resulting curve25519 public key will
// equal the result from PublicKeyToCurve25519.
func PrivateKeyToCurve25519(curve25519Private *[32]byte, privateKey *[64]byte) {
	h := sha512.New()
	h.Write(privateKey[:32])
	digest := h.Sum(nil)

	digest[0] &= 248
	digest[31] &= 127
	digest[31] |= 64

	copy(curve25519Private[:], digest)
}

func edwardsToMontgomeryX(outX, y *edwards25519.FieldElement) {
	// We only need the x-coordinate of the curve25519 point, which I'll
	// call u. The isomorphism is u=(y+1)/(1-y), since y=Y/Z, this gives
	// u=(Y+Z)/(Z-Y). We know that Z=1, thus u=(Y+1)/(1-Y).
	var oneMinusY edwards25519.FieldElement
	edwards25519.FeOne(&oneMinusY)
	edwards25519.FeSub(&oneMinusY, &oneMinusY, y)
	edwards25519.FeInvert(&oneMinusY, &oneMinusY)

	edwards25519.FeOne(outX)
	edwards25519.FeAdd(outX, outX, y)

	edwards25519.FeMul(outX, outX, &oneMinusY)
}

// PublicKeyToCurve25519 converts an Ed25519 public key into the curve25519
// public key that would be generated from the same private key.
func PublicKeyToCurve25519(curve25519Public *[32]byte, publicKey *[32]byte) bool {
	var A edwards25519.ExtendedGroupElement
	if !A.FromBytes(publicKey) {
		return false
	}

	// A.Z = 1 as a postcondition of FromBytes.
	var x edwards25519.FieldElement
	edwardsToMontgomeryX(&x, &A.Y)
	edwards25519.FeToBytes(curve25519Public, &x)
	return true
}

// sqrtMinusAPlus2 is sqrt(-(486662+2))
var sqrtMinusAPlus2 = edwards25519.FieldElement{
	-12222970, -8312128, -11511410, 9067497, -15300785, -241793, 25456130, 14121551, -12187136, 3972024,
}

// sqrtMinusHalf is sqrt(-1/2)
var sqrtMinusHalf = edwards25519.FieldElement{
	-17256545, 3971863, 28865457, -1750208, 27359696, -16640980, 12573105, 1002827, -163343, 11073975,
}

// halfQMinus1Bytes is (2^255-20)/2 expressed in little endian form.
var halfQMinus1Bytes = [32]byte{
	0xf6, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x3f,
}

// feBytesLess returns one if a <= b and zero otherwise.
func feBytesLE(a, b *[32]byte) int32 {
	equalSoFar := int32(-1)
	greater := int32(0)

	for i := uint(31); i < 32; i-- {
		x := int32(a[i])
		y := int32(b[i])

		greater = (^equalSoFar & greater) | (equalSoFar & ((x - y) >> 31))
		equalSoFar = equalSoFar & (((x ^ y) - 1) >> 31)
	}

	return int32(^equalSoFar & 1 & greater)
}

// ScalarBaseMult computes a curve25519 public key from a private key and also
// a uniform representative for that public key. Note that this function will
// fail and return false for about half of private keys.
// See http://elligator.cr.yp.to/elligator-20130828.pdf.
func ScalarBaseMult(publicKey, representative, privateKey *[32]byte) bool {
	var maskedPrivateKey [32]byte
	copy(maskedPrivateKey[:], privateKey[:])

	maskedPrivateKey[0] &= 248
	maskedPrivateKey[31] &= 127
	maskedPrivateKey[31] |= 64

	var A edwards25519.ExtendedGroupElement
	edwards25519.GeScalarMultBase(&A, &maskedPrivateKey)

	var inv1 edwards25519.FieldElement
	edwards25519.FeSub(&inv1, &A.Z, &A.Y)
	edwards25519.FeMul(&inv1, &inv1, &A.X)
	edwards25519.FeInvert(&inv1, &inv1)

	var t0, u edwards25519.FieldElement
	edwards25519.FeMul(&u, &inv1, &A.X)
	edwards25519.FeAdd(&t0, &A.Y, &A.Z)
	edwards25519.FeMul(&u, &u, &t0)

	var v edwards25519.FieldElement
	edwards25519.FeMul(&v, &t0, &inv1)
	edwards25519.FeMul(&v, &v, &A.Z)
	edwards25519.FeMul(&v, &v, &sqrtMinusAPlus2)

	var b edwards25519.FieldElement
	edwards25519.FeAdd(&b, &u, &edwards25519.A)

	var c, b3, b7, b8 edwards25519.FieldElement
	edwards25519.FeSquare(&b3, &b)   // 2
	edwards25519.FeMul(&b3, &b3, &b) // 3
	edwards25519.FeSquare(&c, &b3)   // 6
	edwards25519.FeMul(&b7, &c, &b)  // 7
	edwards25519.FeMul(&b8, &b7, &b) // 8
	edwards25519.FeMul(&c, &b7, &u)
	q58(&c, &c)

	var chi edwards25519.FieldElement
	edwards25519.FeSquare(&chi, &c)
	edwards25519.FeSquare(&chi, &chi)

	edwards25519.FeSquare(&t0, &u)
	edwards25519.FeMul(&chi, &chi, &t0)

	edwards25519.FeSquare(&t0, &b7) // 14
	edwards25519.FeMul(&chi, &chi, &t0)
	edwards25519.FeNeg(&chi, &chi)

	var chiBytes [32]byte
	edwards25519.FeToBytes(&chiBytes, &chi)
	// chi[1] is either 0 or 0xff
	if chiBytes[1] == 0xff {
		return false
	}

	// Calculate r1 = sqrt(-u/(2*(u+A)))
	var r1 edwards25519.FieldElement
	edwards25519.FeMul(&r1, &c, &u)
	edwards25519.FeMul(&r1, &r1, &b3)
	edwards25519.FeMul(&r1, &r1, &sqrtMinusHalf)

	var maybeSqrtM1 edwards25519.FieldElement
	edwards25519.FeSquare(&t0, &r1)
	edwards25519.FeMul(&t0, &t0, &b)
	edwards25519.FeAdd(&t0, &t0, &t0)
	edwards25519.FeAdd(&t0, &t0, &u)

	edwards25519.FeOne(&maybeSqrtM1)
	edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0))
	edwards25519.FeMul(&r1, &r1, &maybeSqrtM1)

	// Calculate r = sqrt(-(u+A)/(2u))
	var r edwards25519.FieldElement
	edwards25519.FeSquare(&t0, &c)   // 2
	edwards25519.FeMul(&t0, &t0, &c) // 3
	edwards25519.FeSquare(&t0, &t0)  // 6
	edwards25519.FeMul(&r, &t0, &c)  // 7

	edwards25519.FeSquare(&t0, &u)   // 2
	edwards25519.FeMul(&t0, &t0, &u) // 3
	edwards25519.FeMul(&r, &r, &t0)

	edwards25519.FeSquare(&t0, &b8)   // 16
	edwards25519.FeMul(&t0, &t0, &b8) // 24
	edwards25519.FeMul(&t0, &t0, &b)  // 25
	edwards25519.FeMul(&r, &r, &t0)
	edwards25519.FeMul(&r, &r, &sqrtMinusHalf)

	edwards25519.FeSquare(&t0, &r)
	edwards25519.FeMul(&t0, &t0, &u)
	edwards25519.FeAdd(&t0, &t0, &t0)
	edwards25519.FeAdd(&t0, &t0, &b)
	edwards25519.FeOne(&maybeSqrtM1)
	edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0))
	edwards25519.FeMul(&r, &r, &maybeSqrtM1)

	var vBytes [32]byte
	edwards25519.FeToBytes(&vBytes, &v)
	vInSquareRootImage := feBytesLE(&vBytes, &halfQMinus1Bytes)
	edwards25519.FeCMove(&r, &r1, vInSquareRootImage)

	edwards25519.FeToBytes(publicKey, &u)
	edwards25519.FeToBytes(representative, &r)
	return true
}

// q58 calculates out = z^((p-5)/8).
func q58(out, z *edwards25519.FieldElement) {
	var t1, t2, t3 edwards25519.FieldElement
	var i int

	edwards25519.FeSquare(&t1, z)     // 2^1
	edwards25519.FeMul(&t1, &t1, z)   // 2^1 + 2^0
	edwards25519.FeSquare(&t1, &t1)   // 2^2 + 2^1
	edwards25519.FeSquare(&t2, &t1)   // 2^3 + 2^2
	edwards25519.FeSquare(&t2, &t2)   // 2^4 + 2^3
	edwards25519.FeMul(&t2, &t2, &t1) // 4,3,2,1
	edwards25519.FeMul(&t1, &t2, z)   // 4..0
	edwards25519.FeSquare(&t2, &t1)   // 5..1
	for i = 1; i < 5; i++ {           // 9,8,7,6,5
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0
	edwards25519.FeSquare(&t2, &t1)   // 10..1
	for i = 1; i < 10; i++ {          // 19..10
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t2, &t2, &t1) // 19..0
	edwards25519.FeSquare(&t3, &t2)   // 20..1
	for i = 1; i < 20; i++ {          // 39..20
		edwards25519.FeSquare(&t3, &t3)
	}
	edwards25519.FeMul(&t2, &t3, &t2) // 39..0
	edwards25519.FeSquare(&t2, &t2)   // 40..1
	for i = 1; i < 10; i++ {          // 49..10
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 49..0
	edwards25519.FeSquare(&t2, &t1)   // 50..1
	for i = 1; i < 50; i++ {          // 99..50
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t2, &t2, &t1) // 99..0
	edwards25519.FeSquare(&t3, &t2)   // 100..1
	for i = 1; i < 100; i++ {         // 199..100
		edwards25519.FeSquare(&t3, &t3)
	}
	edwards25519.FeMul(&t2, &t3, &t2) // 199..0
	edwards25519.FeSquare(&t2, &t2)   // 200..1
	for i = 1; i < 50; i++ {          // 249..50
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 249..0
	edwards25519.FeSquare(&t1, &t1)   // 250..1
	edwards25519.FeSquare(&t1, &t1)   // 251..2
	edwards25519.FeMul(out, &t1, z)   // 251..2,0
}

// chi calculates out = z^((p-1)/2). The result is either 1, 0, or -1 depending
// on whether z is a non-zero square, zero, or a non-square.
func chi(out, z *edwards25519.FieldElement) {
	var t0, t1, t2, t3 edwards25519.FieldElement
	var i int

	edwards25519.FeSquare(&t0, z)     // 2^1
	edwards25519.FeMul(&t1, &t0, z)   // 2^1 + 2^0
	edwards25519.FeSquare(&t0, &t1)   // 2^2 + 2^1
	edwards25519.FeSquare(&t2, &t0)   // 2^3 + 2^2
	edwards25519.FeSquare(&t2, &t2)   // 4,3
	edwards25519.FeMul(&t2, &t2, &t0) // 4,3,2,1
	edwards25519.FeMul(&t1, &t2, z)   // 4..0
	edwards25519.FeSquare(&t2, &t1)   // 5..1
	for i = 1; i < 5; i++ {           // 9,8,7,6,5
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0
	edwards25519.FeSquare(&t2, &t1)   // 10..1
	for i = 1; i < 10; i++ {          // 19..10
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t2, &t2, &t1) // 19..0
	edwards25519.FeSquare(&t3, &t2)   // 20..1
	for i = 1; i < 20; i++ {          // 39..20
		edwards25519.FeSquare(&t3, &t3)
	}
	edwards25519.FeMul(&t2, &t3, &t2) // 39..0
	edwards25519.FeSquare(&t2, &t2)   // 40..1
	for i = 1; i < 10; i++ {          // 49..10
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 49..0
	edwards25519.FeSquare(&t2, &t1)   // 50..1
	for i = 1; i < 50; i++ {          // 99..50
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t2, &t2, &t1) // 99..0
	edwards25519.FeSquare(&t3, &t2)   // 100..1
	for i = 1; i < 100; i++ {         // 199..100
		edwards25519.FeSquare(&t3, &t3)
	}
	edwards25519.FeMul(&t2, &t3, &t2) // 199..0
	edwards25519.FeSquare(&t2, &t2)   // 200..1
	for i = 1; i < 50; i++ {          // 249..50
		edwards25519.FeSquare(&t2, &t2)
	}
	edwards25519.FeMul(&t1, &t2, &t1) // 249..0
	edwards25519.FeSquare(&t1, &t1)   // 250..1
	for i = 1; i < 4; i++ {           // 253..4
		edwards25519.FeSquare(&t1, &t1)
	}
	edwards25519.FeMul(out, &t1, &t0) // 253..4,2,1
}

// RepresentativeToPublicKey converts a uniform representative value for a
// curve25519 public key, as produced by ScalarBaseMult, to a curve25519 public
// key.
func RepresentativeToPublicKey(publicKey, representative *[32]byte) {
	var rr2, v, e edwards25519.FieldElement
	edwards25519.FeFromBytes(&rr2, representative)

	edwards25519.FeSquare2(&rr2, &rr2)
	rr2[0]++
	edwards25519.FeInvert(&rr2, &rr2)
	edwards25519.FeMul(&v, &edwards25519.A, &rr2)
	edwards25519.FeNeg(&v, &v)

	var v2, v3 edwards25519.FieldElement
	edwards25519.FeSquare(&v2, &v)
	edwards25519.FeMul(&v3, &v, &v2)
	edwards25519.FeAdd(&e, &v3, &v)
	edwards25519.FeMul(&v2, &v2, &edwards25519.A)
	edwards25519.FeAdd(&e, &v2, &e)
	chi(&e, &e)
	var eBytes [32]byte
	edwards25519.FeToBytes(&eBytes, &e)
	// eBytes[1] is either 0 (for e = 1) or 0xff (for e = -1)
	eIsMinus1 := int32(eBytes[1]) & 1
	var negV edwards25519.FieldElement
	edwards25519.FeNeg(&negV, &v)
	edwards25519.FeCMove(&v, &negV, eIsMinus1)

	edwards25519.FeZero(&v2)
	edwards25519.FeCMove(&v2, &edwards25519.A, eIsMinus1)
	edwards25519.FeSub(&v, &v, &v2)

	edwards25519.FeToBytes(publicKey, &v)
}