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[tor-commits] [obfs4/master] common: Replace the extra25519 import with an internal package
commit 2d8f3c8bbfd7a7ca931738a64c2f6e97b7332d9e
Author: Yawning Angel <yawning@xxxxxxxxxxxxxxx>
Date: Fri Apr 10 11:36:29 2020 +0000
common: Replace the extra25519 import with an internal package
I really didn't want to do this, but this should make `go get` work
again, and maybe people will leave me alone.
---
ChangeLog | 1 +
common/ntor/ntor.go | 9 +-
go.mod | 3 +-
go.sum | 2 -
internal/README-extra25519.md | 21 +
internal/edwards25519/const.go | 1411 +++++++++++++++++++++++++
internal/edwards25519/edwards25519.go | 1773 ++++++++++++++++++++++++++++++++
internal/extra25519/extra25519.go | 291 ++++++
internal/extra25519/extra25519_test.go | 61 ++
9 files changed, 3565 insertions(+), 7 deletions(-)
diff --git a/ChangeLog b/ChangeLog
index fb2372f..eb4efc7 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,4 +1,5 @@
Changes in version 0.0.12 - UNRELEASED:
+ - Replace the extra25519 import with an internal package.
Changes in version 0.0.11 - 2019-06-21:
- Update my e-mail address.
diff --git a/common/ntor/ntor.go b/common/ntor/ntor.go
index 15f7f38..851cc1e 100644
--- a/common/ntor/ntor.go
+++ b/common/ntor/ntor.go
@@ -43,10 +43,11 @@ import (
"fmt"
"io"
- "github.com/agl/ed25519/extra25519"
- "gitlab.com/yawning/obfs4.git/common/csrand"
"golang.org/x/crypto/curve25519"
"golang.org/x/crypto/hkdf"
+
+ "gitlab.com/yawning/obfs4.git/common/csrand"
+ "gitlab.com/yawning/obfs4.git/internal/extra25519"
)
const (
@@ -203,7 +204,7 @@ func (repr *Representative) Bytes() *[RepresentativeLength]byte {
func (repr *Representative) ToPublic() *PublicKey {
pub := new(PublicKey)
- extra25519.RepresentativeToPublicKey(pub.Bytes(), repr.Bytes())
+ extra25519.UnsafeBrokenRepresentativeToPublicKey(pub.Bytes(), repr.Bytes())
return pub
}
@@ -275,7 +276,7 @@ func NewKeypair(elligator bool) (*Keypair, error) {
if elligator {
// Apply the Elligator transform. This fails ~50% of the time.
- if !extra25519.ScalarBaseMult(keypair.public.Bytes(),
+ if !extra25519.UnsafeBrokenScalarBaseMult(keypair.public.Bytes(),
keypair.representative.Bytes(),
keypair.private.Bytes()) {
continue
diff --git a/go.mod b/go.mod
index 888ea8c..a90d67d 100644
--- a/go.mod
+++ b/go.mod
@@ -2,9 +2,10 @@ module gitlab.com/yawning/obfs4.git
require (
git.torproject.org/pluggable-transports/goptlib.git v1.0.0
- github.com/agl/ed25519 v0.0.0-20170116200512-5312a6153412
github.com/dchest/siphash v1.2.1
gitlab.com/yawning/utls.git v0.0.11-1
golang.org/x/crypto v0.0.0-20190325154230-a5d413f7728c
golang.org/x/net v0.0.0-20190328230028-74de082e2cca
)
+
+go 1.13
diff --git a/go.sum b/go.sum
index 68b4e92..bffa15e 100644
--- a/go.sum
+++ b/go.sum
@@ -2,8 +2,6 @@ git.schwanenlied.me/yawning/bsaes.git v0.0.0-20190320102049-26d1add596b6 h1:zOrl
git.schwanenlied.me/yawning/bsaes.git v0.0.0-20190320102049-26d1add596b6/go.mod h1:BWqTsj8PgcPriQJGl7el20J/7TuT1d/hSyFDXMEpoEo=
git.torproject.org/pluggable-transports/goptlib.git v1.0.0 h1:ElTwFFPKf/tA6x5nuIk9g49JZzS4T5WN+eTQTjqd00A=
git.torproject.org/pluggable-transports/goptlib.git v1.0.0/go.mod h1:YT4XMSkuEXbtqlydr9+OxqFAyspUv0Gr9qhM3B++o/Q=
-github.com/agl/ed25519 v0.0.0-20170116200512-5312a6153412 h1:w1UutsfOrms1J05zt7ISrnJIXKzwaspym5BTKGx93EI=
-github.com/agl/ed25519 v0.0.0-20170116200512-5312a6153412/go.mod h1:WPjqKcmVOxf0XSf3YxCJs6N6AOSrOx3obionmG7T0y0=
github.com/dchest/siphash v1.2.1 h1:4cLinnzVJDKxTCl9B01807Yiy+W7ZzVHj/KIroQRvT4=
github.com/dchest/siphash v1.2.1/go.mod h1:q+IRvb2gOSrUnYoPqHiyHXS0FOBBOdl6tONBlVnOnt4=
github.com/dsnet/compress v0.0.1 h1:PlZu0n3Tuv04TzpfPbrnI0HW/YwodEXDS+oPKahKF0Q=
diff --git a/internal/README-extra25519.md b/internal/README-extra25519.md
new file mode 100644
index 0000000..f02418c
--- /dev/null
+++ b/internal/README-extra25519.md
@@ -0,0 +1,21 @@
+This includes a copy of the edwards25519 and extra25519 packages authored
+by agl, that formerly lived at github.com/agl/ed25519 as of the commit
+5312a61534124124185d41f09206b9fef1d88403 with the following changes:
+
+ * Import paths fixed up.
+
+ * The unused Ed25519->X25519 key conversion routines were removed.
+
+ * `UnsafeBroken` was prefixed to the routines that are known to be
+ severely flawed.
+
+The only reason this is being done (despite agl's wishes that the code
+base dies, which I wanted to respect) is so people stop bothering me
+about it.
+
+Do not ask me questions about this.
+Do not use it in other projects.
+Do not use it in anything new.
+Do not expect me to maintain this beyond ensuring it continues to build.
+
+All I want is to be left alone.
diff --git a/internal/edwards25519/const.go b/internal/edwards25519/const.go
new file mode 100644
index 0000000..ea5b77a
--- /dev/null
+++ b/internal/edwards25519/const.go
@@ -0,0 +1,1411 @@
+// 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 edwards25519
+
+var d = FieldElement{
+ -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116,
+}
+
+var d2 = FieldElement{
+ -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199,
+}
+
+var SqrtM1 = FieldElement{
+ -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482,
+}
+
+var A = FieldElement{
+ 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+}
+
+var bi = [8]PreComputedGroupElement{
+ {
+ FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605},
+ FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378},
+ FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546},
+ },
+ {
+ FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024},
+ FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574},
+ FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357},
+ },
+ {
+ FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380},
+ FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306},
+ FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942},
+ },
+ {
+ FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766},
+ FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701},
+ FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300},
+ },
+ {
+ FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877},
+ FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951},
+ FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784},
+ },
+ {
+ FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436},
+ FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918},
+ FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877},
+ },
+ {
+ FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800},
+ FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305},
+ FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300},
+ },
+ {
+ FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876},
+ FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619},
+ FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683},
+ },
+}
+
+var base = [32][8]PreComputedGroupElement{
+ {
+ {
+ FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605},
+ FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378},
+ FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546},
+ },
+ {
+ FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303},
+ FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081},
+ FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697},
+ },
+ {
+ FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024},
+ FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574},
+ FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357},
+ },
+ {
+ FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540},
+ FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397},
+ FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325},
+ },
+ {
+ FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380},
+ FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306},
+ FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942},
+ },
+ {
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+ FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652},
+ },
+ {
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+ FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300},
+ },
+ {
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+ },
+ },
+ {
+ {
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+ FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660},
+ },
+ {
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+ },
+ {
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+ FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628},
+ },
+ {
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+ FieldElement{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653},
+ },
+ {
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+ },
+ {
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+ FieldElement{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, -1508144, -4795045, -17169265, 4904953},
+ },
+ {
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+ FieldElement{-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 285431, 2763829, 15736322, 4143876},
+ },
+ {
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+ },
+ },
+ {
+ {
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+ FieldElement{-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 9208236, 15886429, 16489664},
+ },
+ {
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+ FieldElement{-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, -30341101, -7336386, 13847711, 5387222},
+ },
+ {
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+ },
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+ },
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+ },
+ {
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+ },
+ {
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+ FieldElement{-11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 27491595, -4612359, 3179268, -9478891},
+ },
+ {
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+ },
+ },
+ {
+ {
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+ FieldElement{-11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2014099, -9050574, -2369172, -5877341},
+ },
+ {
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+ FieldElement{-28241164, -8072475, -4978962, -5315317, 29416931, 1847569, -20654173, -16484855, 4714547, -9600655},
+ },
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+ FieldElement{-15924766, -5229880, -17443532, 6410664, 3622847, 10243618, 20615400, 12405433, -23753030, -8436416},
+ },
+ {
+ FieldElement{-7091295, 12556208, -20191352, 9025187, -17072479, 4333801, 4378436, 2432030, 23097949, -566018},
+ FieldElement{4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264, 10103221, -18512313, 2424778},
+ FieldElement{366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678, 1344109, -3642553, 12412659},
+ },
+ {
+ FieldElement{-24001791, 7690286, 14929416, -168257, -32210835, -13412986, 24162697, -15326504, -3141501, 11179385},
+ FieldElement{18289522, -14724954, 8056945, 16430056, -21729724, 7842514, -6001441, -1486897, -18684645, -11443503},
+ FieldElement{476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959, 13403813, 11052904, 5219329},
+ },
+ },
+ {
+ {
+ FieldElement{20678546, -8375738, -32671898, 8849123, -5009758, 14574752, 31186971, -3973730, 9014762, -8579056},
+ FieldElement{-13644050, -10350239, -15962508, 5075808, -1514661, -11534600, -33102500, 9160280, 8473550, -3256838},
+ FieldElement{24900749, 14435722, 17209120, -15292541, -22592275, 9878983, -7689309, -16335821, -24568481, 11788948},
+ },
+ {
+ FieldElement{-3118155, -11395194, -13802089, 14797441, 9652448, -6845904, -20037437, 10410733, -24568470, -1458691},
+ FieldElement{-15659161, 16736706, -22467150, 10215878, -9097177, 7563911, 11871841, -12505194, -18513325, 8464118},
+ FieldElement{-23400612, 8348507, -14585951, -861714, -3950205, -6373419, 14325289, 8628612, 33313881, -8370517},
+ },
+ {
+ FieldElement{-20186973, -4967935, 22367356, 5271547, -1097117, -4788838, -24805667, -10236854, -8940735, -5818269},
+ FieldElement{-6948785, -1795212, -32625683, -16021179, 32635414, -7374245, 15989197, -12838188, 28358192, -4253904},
+ FieldElement{-23561781, -2799059, -32351682, -1661963, -9147719, 10429267, -16637684, 4072016, -5351664, 5596589},
+ },
+ {
+ FieldElement{-28236598, -3390048, 12312896, 6213178, 3117142, 16078565, 29266239, 2557221, 1768301, 15373193},
+ FieldElement{-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902, -4504991, -24660491, 3442910},
+ FieldElement{-30210571, 5124043, 14181784, 8197961, 18964734, -11939093, 22597931, 7176455, -18585478, 13365930},
+ },
+ {
+ FieldElement{-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107, -8570186, -9689599, -3031667},
+ FieldElement{25008904, -10771599, -4305031, -9638010, 16265036, 15721635, 683793, -11823784, 15723479, -15163481},
+ FieldElement{-9660625, 12374379, -27006999, -7026148, -7724114, -12314514, 11879682, 5400171, 519526, -1235876},
+ },
+ {
+ FieldElement{22258397, -16332233, -7869817, 14613016, -22520255, -2950923, -20353881, 7315967, 16648397, 7605640},
+ FieldElement{-8081308, -8464597, -8223311, 9719710, 19259459, -15348212, 23994942, -5281555, -9468848, 4763278},
+ FieldElement{-21699244, 9220969, -15730624, 1084137, -25476107, -2852390, 31088447, -7764523, -11356529, 728112},
+ },
+ {
+ FieldElement{26047220, -11751471, -6900323, -16521798, 24092068, 9158119, -4273545, -12555558, -29365436, -5498272},
+ FieldElement{17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007, 12327945, 10750447, 10014012},
+ FieldElement{-10312768, 3936952, 9156313, -8897683, 16498692, -994647, -27481051, -666732, 3424691, 7540221},
+ },
+ {
+ FieldElement{30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422, -16317219, -9244265, 15258046},
+ FieldElement{13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406, 2711395, 1062915, -5136345},
+ FieldElement{-19240248, -11254599, -29509029, -7499965, -5835763, 13005411, -6066489, 12194497, 32960380, 1459310},
+ },
+ },
+ {
+ {
+ FieldElement{19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197, -6101885, 18638003, -11174937},
+ FieldElement{31395534, 15098109, 26581030, 8030562, -16527914, -5007134, 9012486, -7584354, -6643087, -5442636},
+ FieldElement{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 9677543, -32294889, -6456008},
+ },
+ {
+ FieldElement{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, -7839692, -7852844, -8138429},
+ FieldElement{-15236356, -15433509, 7766470, 746860, 26346930, -10221762, -27333451, 10754588, -9431476, 5203576},
+ FieldElement{31834314, 14135496, -770007, 5159118, 20917671, -16768096, -7467973, -7337524, 31809243, 7347066},
+ },
+ {
+ FieldElement{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 19797970, -12211255, 15192876, -2087490},
+ FieldElement{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 10609330, 12694420, 33473243, -13382104},
+ FieldElement{33184999, 11180355, 15832085, -11385430, -1633671, 225884, 15089336, -11023903, -6135662, 14480053},
+ },
+ {
+ FieldElement{31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 5496208, 13685227, 27595050, 8737275},
+ FieldElement{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022, -31008351, -12610604, 26498114, 66511},
+ FieldElement{22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 30540766, -4286747, -13327787, -7515095},
+ },
+ {
+ FieldElement{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 8205540, 13585437, -17127465, 15115439},
+ FieldElement{23711543, -672915, 31206561, -8362711, 6164647, -9709987, -33535882, -1426096, 8236921, 16492939},
+ FieldElement{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 19574902, 10071562, 6708380, -6222424},
+ },
+ {
+ FieldElement{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 9328700, 29955601, -11678310},
+ FieldElement{3096359, 9271816, -21620864, -15521844, -14847996, -7592937, -25892142, -12635595, -9917575, 6216608},
+ FieldElement{-32615849, 338663, -25195611, 2510422, -29213566, -13820213, 24822830, -6146567, -26767480, 7525079},
+ },
+ {
+ FieldElement{-23066649, -13985623, 16133487, -7896178, -3389565, 778788, -910336, -2782495, -19386633, 11994101},
+ FieldElement{21691500, -13624626, -641331, -14367021, 3285881, -3483596, -25064666, 9718258, -7477437, 13381418},
+ FieldElement{18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 23111648, -6375247, 28535282, 15779576},
+ },
+ {
+ FieldElement{30098053, 3089662, -9234387, 16662135, -21306940, 11308411, -14068454, 12021730, 9955285, -16303356},
+ FieldElement{9734894, -14576830, -7473633, -9138735, 2060392, 11313496, -18426029, 9924399, 20194861, 13380996},
+ FieldElement{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792, -1984914, 15707771, 26342023, 10146099},
+ },
+ },
+ {
+ {
+ FieldElement{-26016874, -219943, 21339191, -41388, 19745256, -2878700, -29637280, 2227040, 21612326, -545728},
+ FieldElement{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 25764461, 12243797, -20856566, 11649658},
+ FieldElement{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 6114064, 33514190, 2333242},
+ },
+ {
+ FieldElement{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, -6679750, -12670638, 24350578, -13450001},
+ FieldElement{-4116307, -11271533, -23886186, 4843615, -30088339, 690623, -31536088, -10406836, 8317860, 12352766},
+ FieldElement{18200138, -14475911, -33087759, -2696619, -23702521, -9102511, -23552096, -2287550, 20712163, 6719373},
+ },
+ {
+ FieldElement{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, -3763210, 26224235, -3297458},
+ FieldElement{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 21728352, 9493610, 18620611, -16428628},
+ FieldElement{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, -5269471, -9725556, -30701573, -16479657},
+ },
+ {
+ FieldElement{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 12248509, -5240639, 13735342, 1934062},
+ FieldElement{25089769, 6742589, 17081145, -13406266, 21909293, -16067981, -15136294, -3765346, -21277997, 5473616},
+ FieldElement{31883677, -7961101, 1083432, -11572403, 22828471, 13290673, -7125085, 12469656, 29111212, -5451014},
+ },
+ {
+ FieldElement{24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 24367466, 6388839, -10295587, 452383},
+ FieldElement{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, -24236251, -5915248, 15766062, 8407814},
+ FieldElement{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, -8917023, -4388953, -8067909, 2276718},
+ },
+ {
+ FieldElement{30157918, 12924066, -17712050, 9245753, 19895028, 3368142, -23827587, 5096219, 22740376, -7303417},
+ FieldElement{2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 24051124, 13742383, -15637599, 13295222},
+ FieldElement{33338237, -8505733, 12532113, 7977527, 9106186, -1715251, -17720195, -4612972, -4451357, -14669444},
+ },
+ {
+ FieldElement{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, -2469266, -4141880, 7770569, 9620597},
+ FieldElement{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, -1694323, -33502340, -14767970},
+ FieldElement{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 1220118, 30494170, -11440799},
+ },
+ {
+ FieldElement{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, -26739026, 926050, -1684339, -13333647},
+ FieldElement{13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 9021034, 9078865, 3353509, 4033511},
+ FieldElement{-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 23161163, 8839127, 27485041, 7356032},
+ },
+ },
+ {
+ {
+ FieldElement{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 2625015, 28431036, -16771834},
+ FieldElement{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, -22545972, 14150565, 15970762, 4099461},
+ FieldElement{29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 13617293, -9937143, 11465739, 8317062},
+ },
+ {
+ FieldElement{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 14898637, 3848455, 20969334, -5157516},
+ FieldElement{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, -21610826, -3649888, 11177095, 14989547},
+ FieldElement{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 13515641, 2581286, -28487508, 9930240},
+ },
+ {
+ FieldElement{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 18345767, -13403753, 16291481, -5314038},
+ FieldElement{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 6957617, 4368891, 9788741},
+ FieldElement{16660756, 7281060, -10830758, 12911820, 20108584, -8101676, -21722536, -8613148, 16250552, -11111103},
+ },
+ {
+ FieldElement{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 10604807, -30190403, 4782747},
+ FieldElement{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, -9981571, 4383045, 22546403, 437323},
+ FieldElement{31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 27343084, 2786261, -30633590, -14097016},
+ },
+ {
+ FieldElement{-14467279, -683715, -33374107, 7448552, 19294360, 14334329, -19690631, 2355319, -19284671, -6114373},
+ FieldElement{15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 18952177, 15496498, -29380133, 11754228},
+ FieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141},
+ },
+ {
+ FieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399},
+ FieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831},
+ FieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376},
+ },
+ {
+ FieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313},
+ FieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958},
+ FieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577},
+ },
+ {
+ FieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743},
+ FieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684},
+ FieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476},
+ },
+ },
+}
diff --git a/internal/edwards25519/edwards25519.go b/internal/edwards25519/edwards25519.go
new file mode 100644
index 0000000..9079818
--- /dev/null
+++ b/internal/edwards25519/edwards25519.go
@@ -0,0 +1,1773 @@
+// 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 edwards25519 implements operations in GF(2**255-19) and on an
+// Edwards curve that is isomorphic to curve25519. See
+// http://ed25519.cr.yp.to/.
+package edwards25519
+
+// 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.
+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)
+}
diff --git a/internal/extra25519/extra25519.go b/internal/extra25519/extra25519.go
new file mode 100644
index 0000000..7e1d227
--- /dev/null
+++ b/internal/extra25519/extra25519.go
@@ -0,0 +1,291 @@
+// 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 "gitlab.com/yawning/obfs4.git/internal/edwards25519"
+
+// 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)
+}
+
+// UnsafeBrokenScalarBaseMult 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 UnsafeBrokenScalarBaseMult(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
+}
+
+// UnsafeBrokenRepresentativeToPublicKey converts a uniform representative
+// value for a curve25519 public key, as produced by UnsafeBrokenScalarBaseMult,
+// to a curve25519 public key.
+func UnsafeBrokenRepresentativeToPublicKey(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)
+}
diff --git a/internal/extra25519/extra25519_test.go b/internal/extra25519/extra25519_test.go
new file mode 100644
index 0000000..7cf1e97
--- /dev/null
+++ b/internal/extra25519/extra25519_test.go
@@ -0,0 +1,61 @@
+// 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 (
+ "bytes"
+ "crypto/rand"
+ "testing"
+
+ "golang.org/x/crypto/curve25519"
+)
+
+func TestElligator(t *testing.T) {
+ var publicKey, publicKey2, publicKey3, representative, privateKey [32]byte
+
+ for i := 0; i < 1000; i++ {
+ rand.Reader.Read(privateKey[:])
+
+ if !UnsafeBrokenScalarBaseMult(&publicKey, &representative, &privateKey) {
+ continue
+ }
+ UnsafeBrokenRepresentativeToPublicKey(&publicKey2, &representative)
+ if !bytes.Equal(publicKey[:], publicKey2[:]) {
+ t.Fatal("The resulting public key doesn't match the initial one.")
+ }
+
+ curve25519.ScalarBaseMult(&publicKey3, &privateKey)
+ if !bytes.Equal(publicKey[:], publicKey3[:]) {
+ t.Fatal("The public key doesn't match the value that curve25519 produced.")
+ }
+ }
+}
+
+func BenchmarkKeyGeneration(b *testing.B) {
+ var publicKey, representative, privateKey [32]byte
+
+ // Find the private key that results in a point that's in the image of the map.
+ for {
+ rand.Reader.Read(privateKey[:])
+ if UnsafeBrokenScalarBaseMult(&publicKey, &representative, &privateKey) {
+ break
+ }
+ }
+
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ UnsafeBrokenScalarBaseMult(&publicKey, &representative, &privateKey)
+ }
+}
+
+func BenchmarkMap(b *testing.B) {
+ var publicKey, representative [32]byte
+ rand.Reader.Read(representative[:])
+
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ UnsafeBrokenRepresentativeToPublicKey(&publicKey, &representative)
+ }
+}
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