go.crypto/curve25519: rewrite generic code to be constant time.
Previously curve25519 contained a constant-time, optimised amd64 implementation and
a generic implemenation that used math/big and that was not constant-time.
This change contains a Go port of the public domain, "ref10" implementation from
SUPERCOP. This has the advantage of being faster and constant-time.
R=golang-dev, bradfitz
CC=golang-dev
https://golang.org/cl/13343045
diff --git a/curve25519/curve25519.go b/curve25519/curve25519.go
index f1c6729..5b7b9bc 100644
--- a/curve25519/curve25519.go
+++ b/curve25519/curve25519.go
@@ -1,136 +1,841 @@
-// Copyright 2012 The Go Authors. All rights reserved.
+// 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.
// We have a implementation in amd64 assembly so this code is only run on
-// non-amd64 platforms. The amd64 assembly does not support gccgo.
+// non-amd64 platforms. The amd64 assembly does not support gccgo.
// +build !amd64 gccgo
package curve25519
-import (
- "math/big"
-)
+// This code is a port of the public domain, "ref10" implementation of
+// curve25519 from SUPERCOP 20130419 by D. J. Bernstein.
-// p is the prime order of the underlying field: 2^255-19
-var p *big.Int
+// 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
-// pMinus2 is p-2
-var pMinus2 *big.Int
-
-// a is a parameter of the elliptic curve: 486662
-var a *big.Int
-
-func init() {
- p, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed", 16)
- pMinus2, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeb", 16)
- a = new(big.Int).SetInt64(486662)
+func feZero(fe *fieldElement) {
+ for i := range fe {
+ fe[i] = 0
+ }
}
-// context contains state shared throughout the computation, including scratch
-// variables to save on allocation.
-type context struct {
- tmp1, tmp2, tmp3, tmp4 *big.Int
- x1 *big.Int
+func feOne(fe *fieldElement) {
+ feZero(fe)
+ fe[0] = 1
}
-// add sets (outx, outz) to the sum of two points in the elliptic curve group.
-// See http://hyperelliptic.org/EFD/g1p/auto-montgom-xz.html#diffadd-dadd-1987-m
-// outx and outz should not alias any of the other inputs.
-func (c *context) add(outx, outz, xn, zn, xm, zm *big.Int) {
- // x₃ = 4(x·x′ - z·z′)² · z1
- // (z1 == 1 here)
- c.tmp1.Mul(xn, xm)
- c.tmp2.Mul(zn, zm)
- c.tmp3.Sub(c.tmp1, c.tmp2)
- outx.Mul(c.tmp3, c.tmp3)
- outx.Lsh(outx, 2)
- outx.Mod(outx, p)
-
- // z₃ = 4(x·z′ - z·x′)² · x1
- // (x1 == 1 here)
- c.tmp1.Mul(xm, zn)
- c.tmp2.Mul(zm, xn)
- c.tmp3.Sub(c.tmp1, c.tmp2)
- outz.Mul(c.tmp3, c.tmp3)
- outz.Mul(outz, c.x1)
- outz.Lsh(outz, 2)
- outz.Mod(outz, p)
-
- return
+func feAdd(dst, a, b *fieldElement) {
+ for i := range dst {
+ dst[i] = a[i] + b[i]
+ }
}
-// double sets (outx, outz) to 2*(x,z) in the elliptic curve group. See
-// http://hyperelliptic.org/EFD/g1p/auto-montgom-xz.html#doubling-dbl-1987-m
-// outx and outz should not alias any of the other inputs.
-func (c *context) double(outx, outz, x, z *big.Int) {
- // x₂ = (x² - z²)²
- c.tmp1.Mul(x, x)
- c.tmp2.Mul(z, z)
- c.tmp3.Sub(c.tmp1, c.tmp2)
- outx.Mul(c.tmp3, c.tmp3)
- outx.Mod(outx, p)
+func feSub(dst, a, b *fieldElement) {
+ for i := range dst {
+ dst[i] = a[i] - b[i]
+ }
+}
- // z₂ = 4xz·(x² + Axz + z²)
- c.tmp3.Add(c.tmp1, c.tmp2)
- c.tmp1.Mul(x, z)
- c.tmp2.Mul(c.tmp1, a)
- outz.Add(c.tmp3, c.tmp2)
- c.tmp2.Lsh(c.tmp1, 2)
- outz.Mul(outz, c.tmp2)
- outz.Mod(outz, p)
+func feCopy(dst, src *fieldElement) {
+ for i := range dst {
+ dst[i] = src[i]
+ }
+}
- return
+// feCSwap replaces (f,g) with (g,f) if b == 1; replaces (f,g) with (f,g) if b == 0.
+//
+// Preconditions: b in {0,1}.
+func feCSwap(f, g *fieldElement, b int32) {
+ var x fieldElement
+ b = -b
+ for i := range x {
+ x[i] = b & (f[i] ^ g[i])
+ }
+
+ for i := range f {
+ f[i] ^= x[i]
+ }
+ for i := range g {
+ g[i] ^= x[i]
+ }
+}
+
+// load3 reads a 24-bit, little-endian value from in.
+func load3(in []byte) int64 {
+ var r int64
+ r = int64(in[0])
+ r |= int64(in[1]) << 8
+ r |= int64(in[2]) << 16
+ return r
+}
+
+// load4 reads a 32-bit, little-endian value from in.
+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:]) << 2
+
+ var carry [10]int64
+ carry[9] = (h9 + 1<<24) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+ carry[1] = (h1 + 1<<24) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[3] = (h3 + 1<<24) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[5] = (h5 + 1<<24) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+ carry[7] = (h7 + 1<<24) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[0] = (h0 + 1<<25) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[2] = (h2 + 1<<25) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[4] = (h4 + 1<<25) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[6] = (h6 + 1<<25) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+ carry[8] = (h8 + 1<<25) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ dst[0] = int32(h0)
+ dst[1] = int32(h1)
+ dst[2] = int32(h2)
+ dst[3] = int32(h3)
+ dst[4] = int32(h4)
+ dst[5] = int32(h5)
+ dst[6] = int32(h6)
+ dst[7] = int32(h7)
+ dst[8] = int32(h8)
+ dst[9] = int32(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)
+}
+
+// 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 := f[0]
+ f1 := f[1]
+ f2 := f[2]
+ f3 := f[3]
+ f4 := f[4]
+ f5 := f[5]
+ f6 := f[6]
+ f7 := f[7]
+ f8 := f[8]
+ f9 := f[9]
+ g0 := g[0]
+ g1 := g[1]
+ g2 := g[2]
+ g3 := g[3]
+ g4 := g[4]
+ g5 := g[5]
+ g6 := g[6]
+ g7 := g[7]
+ g8 := g[8]
+ g9 := g[9]
+ g1_19 := 19 * g1 // 1.4*2^29
+ g2_19 := 19 * g2 // 1.4*2^30; still ok
+ g3_19 := 19 * g3
+ g4_19 := 19 * g4
+ g5_19 := 19 * g5
+ g6_19 := 19 * g6
+ g7_19 := 19 * g7
+ g8_19 := 19 * g8
+ g9_19 := 19 * g9
+ f1_2 := 2 * f1
+ f3_2 := 2 * f3
+ f5_2 := 2 * f5
+ f7_2 := 2 * f7
+ f9_2 := 2 * f9
+ f0g0 := int64(f0) * int64(g0)
+ f0g1 := int64(f0) * int64(g1)
+ f0g2 := int64(f0) * int64(g2)
+ f0g3 := int64(f0) * int64(g3)
+ f0g4 := int64(f0) * int64(g4)
+ f0g5 := int64(f0) * int64(g5)
+ f0g6 := int64(f0) * int64(g6)
+ f0g7 := int64(f0) * int64(g7)
+ f0g8 := int64(f0) * int64(g8)
+ f0g9 := int64(f0) * int64(g9)
+ f1g0 := int64(f1) * int64(g0)
+ f1g1_2 := int64(f1_2) * int64(g1)
+ f1g2 := int64(f1) * int64(g2)
+ f1g3_2 := int64(f1_2) * int64(g3)
+ f1g4 := int64(f1) * int64(g4)
+ f1g5_2 := int64(f1_2) * int64(g5)
+ f1g6 := int64(f1) * int64(g6)
+ f1g7_2 := int64(f1_2) * int64(g7)
+ f1g8 := int64(f1) * int64(g8)
+ f1g9_38 := int64(f1_2) * int64(g9_19)
+ f2g0 := int64(f2) * int64(g0)
+ f2g1 := int64(f2) * int64(g1)
+ f2g2 := int64(f2) * int64(g2)
+ f2g3 := int64(f2) * int64(g3)
+ f2g4 := int64(f2) * int64(g4)
+ f2g5 := int64(f2) * int64(g5)
+ f2g6 := int64(f2) * int64(g6)
+ f2g7 := int64(f2) * int64(g7)
+ f2g8_19 := int64(f2) * int64(g8_19)
+ f2g9_19 := int64(f2) * int64(g9_19)
+ f3g0 := int64(f3) * int64(g0)
+ f3g1_2 := int64(f3_2) * int64(g1)
+ f3g2 := int64(f3) * int64(g2)
+ f3g3_2 := int64(f3_2) * int64(g3)
+ f3g4 := int64(f3) * int64(g4)
+ f3g5_2 := int64(f3_2) * int64(g5)
+ f3g6 := int64(f3) * int64(g6)
+ f3g7_38 := int64(f3_2) * int64(g7_19)
+ f3g8_19 := int64(f3) * int64(g8_19)
+ f3g9_38 := int64(f3_2) * int64(g9_19)
+ f4g0 := int64(f4) * int64(g0)
+ f4g1 := int64(f4) * int64(g1)
+ f4g2 := int64(f4) * int64(g2)
+ f4g3 := int64(f4) * int64(g3)
+ f4g4 := int64(f4) * int64(g4)
+ f4g5 := int64(f4) * int64(g5)
+ f4g6_19 := int64(f4) * int64(g6_19)
+ f4g7_19 := int64(f4) * int64(g7_19)
+ f4g8_19 := int64(f4) * int64(g8_19)
+ f4g9_19 := int64(f4) * int64(g9_19)
+ f5g0 := int64(f5) * int64(g0)
+ f5g1_2 := int64(f5_2) * int64(g1)
+ f5g2 := int64(f5) * int64(g2)
+ f5g3_2 := int64(f5_2) * int64(g3)
+ f5g4 := int64(f5) * int64(g4)
+ f5g5_38 := int64(f5_2) * int64(g5_19)
+ f5g6_19 := int64(f5) * int64(g6_19)
+ f5g7_38 := int64(f5_2) * int64(g7_19)
+ f5g8_19 := int64(f5) * int64(g8_19)
+ f5g9_38 := int64(f5_2) * int64(g9_19)
+ f6g0 := int64(f6) * int64(g0)
+ f6g1 := int64(f6) * int64(g1)
+ f6g2 := int64(f6) * int64(g2)
+ f6g3 := int64(f6) * int64(g3)
+ f6g4_19 := int64(f6) * int64(g4_19)
+ f6g5_19 := int64(f6) * int64(g5_19)
+ f6g6_19 := int64(f6) * int64(g6_19)
+ f6g7_19 := int64(f6) * int64(g7_19)
+ f6g8_19 := int64(f6) * int64(g8_19)
+ f6g9_19 := int64(f6) * int64(g9_19)
+ f7g0 := int64(f7) * int64(g0)
+ f7g1_2 := int64(f7_2) * int64(g1)
+ f7g2 := int64(f7) * int64(g2)
+ f7g3_38 := int64(f7_2) * int64(g3_19)
+ f7g4_19 := int64(f7) * int64(g4_19)
+ f7g5_38 := int64(f7_2) * int64(g5_19)
+ f7g6_19 := int64(f7) * int64(g6_19)
+ f7g7_38 := int64(f7_2) * int64(g7_19)
+ f7g8_19 := int64(f7) * int64(g8_19)
+ f7g9_38 := int64(f7_2) * int64(g9_19)
+ f8g0 := int64(f8) * int64(g0)
+ f8g1 := int64(f8) * int64(g1)
+ f8g2_19 := int64(f8) * int64(g2_19)
+ f8g3_19 := int64(f8) * int64(g3_19)
+ f8g4_19 := int64(f8) * int64(g4_19)
+ f8g5_19 := int64(f8) * int64(g5_19)
+ f8g6_19 := int64(f8) * int64(g6_19)
+ f8g7_19 := int64(f8) * int64(g7_19)
+ f8g8_19 := int64(f8) * int64(g8_19)
+ f8g9_19 := int64(f8) * int64(g9_19)
+ f9g0 := int64(f9) * int64(g0)
+ f9g1_38 := int64(f9_2) * int64(g1_19)
+ f9g2_19 := int64(f9) * int64(g2_19)
+ f9g3_38 := int64(f9_2) * int64(g3_19)
+ f9g4_19 := int64(f9) * int64(g4_19)
+ f9g5_38 := int64(f9_2) * int64(g5_19)
+ f9g6_19 := int64(f9) * int64(g6_19)
+ f9g7_38 := int64(f9_2) * int64(g7_19)
+ f9g8_19 := int64(f9) * int64(g8_19)
+ f9g9_38 := int64(f9_2) * int64(g9_19)
+ h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38
+ h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19
+ h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38
+ h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19
+ h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38
+ h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19
+ h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38
+ h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19
+ h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38
+ h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0
+ var carry [10]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
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ // |h0| <= 2^25
+ // |h4| <= 2^25
+ // |h1| <= 1.51*2^58
+ // |h5| <= 1.51*2^58
+
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 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
+
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 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
+
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 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
+
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 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
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+ // |h9| <= 2^24; from now on fits into int32 unchanged
+ // |h0| <= 1.8*2^37
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 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)
+}
+
+// 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) {
+ f0 := f[0]
+ f1 := f[1]
+ f2 := f[2]
+ f3 := f[3]
+ f4 := f[4]
+ f5 := f[5]
+ f6 := f[6]
+ f7 := f[7]
+ f8 := f[8]
+ f9 := f[9]
+ f0_2 := 2 * f0
+ f1_2 := 2 * f1
+ f2_2 := 2 * f2
+ f3_2 := 2 * f3
+ f4_2 := 2 * f4
+ f5_2 := 2 * f5
+ f6_2 := 2 * f6
+ f7_2 := 2 * f7
+ 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
+ f0f0 := int64(f0) * int64(f0)
+ f0f1_2 := int64(f0_2) * int64(f1)
+ f0f2_2 := int64(f0_2) * int64(f2)
+ f0f3_2 := int64(f0_2) * int64(f3)
+ f0f4_2 := int64(f0_2) * int64(f4)
+ f0f5_2 := int64(f0_2) * int64(f5)
+ f0f6_2 := int64(f0_2) * int64(f6)
+ f0f7_2 := int64(f0_2) * int64(f7)
+ f0f8_2 := int64(f0_2) * int64(f8)
+ f0f9_2 := int64(f0_2) * int64(f9)
+ f1f1_2 := int64(f1_2) * int64(f1)
+ f1f2_2 := int64(f1_2) * int64(f2)
+ f1f3_4 := int64(f1_2) * int64(f3_2)
+ f1f4_2 := int64(f1_2) * int64(f4)
+ f1f5_4 := int64(f1_2) * int64(f5_2)
+ f1f6_2 := int64(f1_2) * int64(f6)
+ f1f7_4 := int64(f1_2) * int64(f7_2)
+ f1f8_2 := int64(f1_2) * int64(f8)
+ f1f9_76 := int64(f1_2) * int64(f9_38)
+ f2f2 := int64(f2) * int64(f2)
+ f2f3_2 := int64(f2_2) * int64(f3)
+ f2f4_2 := int64(f2_2) * int64(f4)
+ f2f5_2 := int64(f2_2) * int64(f5)
+ f2f6_2 := int64(f2_2) * int64(f6)
+ f2f7_2 := int64(f2_2) * int64(f7)
+ f2f8_38 := int64(f2_2) * int64(f8_19)
+ f2f9_38 := int64(f2) * int64(f9_38)
+ f3f3_2 := int64(f3_2) * int64(f3)
+ f3f4_2 := int64(f3_2) * int64(f4)
+ f3f5_4 := int64(f3_2) * int64(f5_2)
+ f3f6_2 := int64(f3_2) * int64(f6)
+ f3f7_76 := int64(f3_2) * int64(f7_38)
+ f3f8_38 := int64(f3_2) * int64(f8_19)
+ f3f9_76 := int64(f3_2) * int64(f9_38)
+ f4f4 := int64(f4) * int64(f4)
+ f4f5_2 := int64(f4_2) * int64(f5)
+ f4f6_38 := int64(f4_2) * int64(f6_19)
+ f4f7_38 := int64(f4) * int64(f7_38)
+ f4f8_38 := int64(f4_2) * int64(f8_19)
+ f4f9_38 := int64(f4) * int64(f9_38)
+ f5f5_38 := int64(f5) * int64(f5_38)
+ f5f6_38 := int64(f5_2) * int64(f6_19)
+ f5f7_76 := int64(f5_2) * int64(f7_38)
+ f5f8_38 := int64(f5_2) * int64(f8_19)
+ f5f9_76 := int64(f5_2) * int64(f9_38)
+ f6f6_19 := int64(f6) * int64(f6_19)
+ f6f7_38 := int64(f6) * int64(f7_38)
+ f6f8_38 := int64(f6_2) * int64(f8_19)
+ f6f9_38 := int64(f6) * int64(f9_38)
+ f7f7_38 := int64(f7) * int64(f7_38)
+ f7f8_38 := int64(f7_2) * int64(f8_19)
+ f7f9_76 := int64(f7_2) * int64(f9_38)
+ f8f8_19 := int64(f8) * int64(f8_19)
+ f8f9_38 := int64(f8) * int64(f9_38)
+ f9f9_38 := int64(f9) * int64(f9_38)
+ h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38
+ h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38
+ h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19
+ h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38
+ h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38
+ h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38
+ h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19
+ h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38
+ h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38
+ h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2
+ var carry [10]int64
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+
+ 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)
+}
+
+// feMul121666 calculates h = f * 121666. 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 feMul121666(h, f *fieldElement) {
+ h0 := int64(f[0]) * 121666
+ h1 := int64(f[1]) * 121666
+ h2 := int64(f[2]) * 121666
+ h3 := int64(f[3]) * 121666
+ h4 := int64(f[4]) * 121666
+ h5 := int64(f[5]) * 121666
+ h6 := int64(f[6]) * 121666
+ h7 := int64(f[7]) * 121666
+ h8 := int64(f[8]) * 121666
+ h9 := int64(f[9]) * 121666
+ var carry [10]int64
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ 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)
+}
+
+// feInvert sets out = z^-1.
+func feInvert(out, z *fieldElement) {
+ var t0, t1, t2, t3 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(&t2, &t0)
+ for i = 1; i < 1; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t1, &t2)
+ feSquare(&t2, &t1)
+ for i = 1; i < 5; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t2, &t1)
+ for i = 1; i < 10; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t2, &t2, &t1)
+ feSquare(&t3, &t2)
+ for i = 1; i < 20; i++ {
+ feSquare(&t3, &t3)
+ }
+ feMul(&t2, &t3, &t2)
+ feSquare(&t2, &t2)
+ for i = 1; i < 10; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t2, &t1)
+ for i = 1; i < 50; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t2, &t2, &t1)
+ feSquare(&t3, &t2)
+ for i = 1; i < 100; i++ {
+ feSquare(&t3, &t3)
+ }
+ feMul(&t2, &t3, &t2)
+ feSquare(&t2, &t2)
+ for i = 1; i < 50; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t1, &t1)
+ for i = 1; i < 5; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(out, &t1, &t0)
}
func scalarMult(out, in, base *[32]byte) {
- var baseReversed, inCopy [32]byte
- for i := 0; i < 32; i++ {
- baseReversed[31-i] = base[i]
- inCopy[i] = in[i]
+ var e [32]byte
+
+ copy(e[:], in[:])
+ e[0] &= 248
+ e[31] &= 127
+ e[31] |= 64
+
+ var x1, x2, z2, x3, z3, tmp0, tmp1 fieldElement
+ feFromBytes(&x1, base)
+ feOne(&x2)
+ feCopy(&x3, &x1)
+ feOne(&z3)
+
+ swap := int32(0)
+ for pos := 254; pos >= 0; pos-- {
+ b := e[pos/8] >> uint(pos&7)
+ b &= 1
+ swap ^= int32(b)
+ feCSwap(&x2, &x3, swap)
+ feCSwap(&z2, &z3, swap)
+ swap = int32(b)
+
+ feSub(&tmp0, &x3, &z3)
+ feSub(&tmp1, &x2, &z2)
+ feAdd(&x2, &x2, &z2)
+ feAdd(&z2, &x3, &z3)
+ feMul(&z3, &tmp0, &x2)
+ feMul(&z2, &z2, &tmp1)
+ feSquare(&tmp0, &tmp1)
+ feSquare(&tmp1, &x2)
+ feAdd(&x3, &z3, &z2)
+ feSub(&z2, &z3, &z2)
+ feMul(&x2, &tmp1, &tmp0)
+ feSub(&tmp1, &tmp1, &tmp0)
+ feSquare(&z2, &z2)
+ feMul121666(&z3, &tmp1)
+ feSquare(&x3, &x3)
+ feAdd(&tmp0, &tmp0, &z3)
+ feMul(&z3, &x1, &z2)
+ feMul(&z2, &tmp1, &tmp0)
}
- inCopy[31] &= 127
- inCopy[31] |= 64
- inCopy[0] &= 248
+ feCSwap(&x2, &x3, swap)
+ feCSwap(&z2, &z3, swap)
- c := &context{new(big.Int), new(big.Int), new(big.Int), new(big.Int), nil}
- c.x1 = new(big.Int).SetBytes(baseReversed[:])
-
- x1 := new(big.Int).SetInt64(1)
- z1 := new(big.Int)
- x2 := new(big.Int).Set(c.x1)
- z2 := new(big.Int).SetInt64(1)
- outx := new(big.Int)
- outz := new(big.Int)
-
- for i := 0; i < 32; i++ {
- b := inCopy[31-i]
- for j := 0; j < 8; j++ {
- if b&0x80 != 0 {
- c.add(outx, outz, x1, z1, x2, z2)
- x1, z1, outx, outz = outx, outz, x1, z1
- c.double(outx, outz, x2, z2)
- x2, z2, outx, outz = outx, outz, x2, z2
- } else {
- c.add(outx, outz, x1, z1, x2, z2)
- x2, z2, outx, outz = outx, outz, x2, z2
- c.double(outx, outz, x1, z1)
- x1, z1, outx, outz = outx, outz, x1, z1
- }
- b <<= 1
- }
- }
-
- c.tmp1.Exp(z1, pMinus2, p)
- c.tmp2.Mul(x1, c.tmp1)
- c.tmp3.Mod(c.tmp2, p)
-
- outReversed := c.tmp3.Bytes()
- for i := 0; i < len(outReversed); i++ {
- out[i] = outReversed[len(outReversed)-(1+i)]
- }
- for i := len(outReversed); i < 32; i++ {
- out[i] = 0
- }
+ feInvert(&z2, &z2)
+ feMul(&x2, &x2, &z2)
+ feToBytes(out, &x2)
}
