pax_global_header00006660000000000000000000000064145663747550014540gustar00rootroot0000000000000052 comment=f465e9d95e7ea513d9ad85d085d837806377c990 golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/000077500000000000000000000000001456637475500214665ustar00rootroot00000000000000golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/LICENSE000066400000000000000000000020471456637475500224760ustar00rootroot00000000000000MIT License Copyright (c) 2021 Travis Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/README.md000066400000000000000000000022211456637475500227420ustar00rootroot00000000000000# siec Super-Isolated Elliptic Curve Implementation in Go This package exports a super-isolated elliptic curve. Over the base field 𝔽ₚ, the curve E does not admit any isogenies to other curves. We can verify the curve properties in Sage. ```python K. = QuadraticField(-3) pi = 2^127 + 2^25 + 2^12 + 2^6 + (1 - isqrt3)/2 p = ZZ(pi.norm()) N = ZZ((pi-1).norm()) E = EllipticCurve(GF(p),[0,19]) # E: y^2 = x^3 + 19 G = E([5,12]) # p is a 255 bit prime with hamming weight 14 assert p.is_prime() assert len(p.bits()) == 255 assert sum(p.bits()) == 14 # N is a 255 bit prime assert N.is_prime() assert len(N.bits()) == 255 # E has N points assert E.count_points() == N # The Frobenius endomorphism on E satisfies the same minimal polynomial as pi assert E.frobenius_polynomial() == pi.minpoly() # pi generates a maximal order assert K.order([pi]).is_maximal() # K has class number 1 assert K.class_number() == 1 # Examples of order 6 endomorphisms assert E([28948022309329048855892746252183396360433790236562615305258360005404201062400*G[0],-G[1]]) == 170141183460469231731687303715917664320*G assert 170141183460469231731687303715917664320^3 % N == N-1 ``` golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/bench_test.go000066400000000000000000000043561456637475500241430ustar00rootroot00000000000000package siec import ( "crypto/elliptic" "crypto/sha256" "testing" "github.com/tscholl2/siec/edwards25519" ) func BenchmarkDouble(b *testing.B) { curve := SIEC255() x, y := curve.ScalarBaseMult(hash(1)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Double(x, y) } } func BenchmarkDoubleP256(b *testing.B) { curve := elliptic.P256() x, y := curve.ScalarBaseMult(hash(1)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Double(x, y) } } func BenchmarkDoubleP224(b *testing.B) { curve := elliptic.P224() x, y := curve.ScalarBaseMult(hash(1)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Double(x, y) } } func BenchmarkAdd(b *testing.B) { curve := SIEC255() x1, y1 := curve.ScalarBaseMult(hash(1)) x2, y2 := curve.ScalarBaseMult(hash(2)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Add(x1, y1, x2, y2) } } func BenchmarkAddP256(b *testing.B) { curve := elliptic.P256() x1, y1 := curve.ScalarBaseMult(hash(1)) x2, y2 := curve.ScalarBaseMult(hash(2)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Add(x1, y1, x2, y2) } } func BenchmarkAddP224(b *testing.B) { curve := elliptic.P224() x1, y1 := curve.ScalarBaseMult(hash(1)) x2, y2 := curve.ScalarBaseMult(hash(2)) b.ResetTimer() for i := 0; i < b.N; i++ { curve.Add(x1, y1, x2, y2) } } func BenchmarkScale(b *testing.B) { curve := SIEC255() k := hash(1) b.ResetTimer() for i := 0; i < b.N; i++ { curve.ScalarBaseMult(k) } } func BenchmarkScale2(b *testing.B) { curve := SIEC255() k := hash(1) b.ResetTimer() for i := 0; i < b.N; i++ { curve.scalarMult2(curve.Gx, curve.Gy, k) } } func BenchmarkScaleP256(b *testing.B) { curve := elliptic.P256() k := hash(1) b.ResetTimer() for i := 0; i < b.N; i++ { curve.ScalarBaseMult(k) } } func BenchmarkScaleP224(b *testing.B) { curve := elliptic.P224() k := hash(1) b.ResetTimer() for i := 0; i < b.N; i++ { curve.ScalarBaseMult(k) } } func BenchmarkScaleEd25519(b *testing.B) { arr := hash(1) arr[0] &= 248 arr[31] &= 127 arr[31] |= 64 var A edwards25519.ExtendedGroupElement var hBytes [32]byte copy(hBytes[:], arr) b.ResetTimer() for i := 0; i < b.N; i++ { edwards25519.GeScalarMultBase(&A, &hBytes) } } func hash(i int) []byte { arr := sha256.Sum256([]byte{byte(i)}) return arr[:] } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/curve_test.go000066400000000000000000000201741456637475500242040ustar00rootroot00000000000000package siec import ( "crypto/elliptic" "math/big" "reflect" "testing" ) func TestDouble(t *testing.T) { curve := SIEC255() x1, y1 := curve.Gx, curve.Gy x, y := curve.Double(x1, y1) u, _ := new(big.Int).SetString("6784692728748995825599862402855483522016546426567910438357042338075027826575", 10) v, _ := new(big.Int).SetString("14982863109320699114866362806305859444453206692004135551371801829915686450358", 10) if x.Cmp(u) != 0 { t.Errorf("curve.Double() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.Double() gotY = %v, want %v", y, v) } x, y = curve.Double(x, y) u, _ = new(big.Int).SetString("12318642006867402687195826566147291859634823582672295191656499276835526033145", 10) v, _ = new(big.Int).SetString("9343467693237486709905252998911952863134805995110526737200728195882424275543", 10) if x.Cmp(u) != 0 { t.Errorf("curve.Double() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.Double() gotY = %v, want %v", y, v) } } func TestAdd(t *testing.T) { curve := SIEC255() x1, _ := new(big.Int).SetString("5", 10) y1, _ := new(big.Int).SetString("12", 10) x2, _ := new(big.Int).SetString("6784692728748995825599862402855483522016546426567910438357042338075027826575", 10) y2, _ := new(big.Int).SetString("14982863109320699114866362806305859444453206692004135551371801829915686450358", 10) x, y := curve.Add(x1, y1, x2, y2) u, _ := new(big.Int).SetString("15508762693928266726769396085241920452527964114034020358226683933290603861217", 10) v, _ := new(big.Int).SetString("14902066091748681388520350407103675753252739566337731413744921722214701412361", 10) if x.Cmp(u) != 0 { t.Errorf("curve.Add() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.Add() gotY = %v, want %v", y, v) } } func TestIsOnCurve(t *testing.T) { curve := SIEC255() x, _ := new(big.Int).SetString("5", 10) y, _ := new(big.Int).SetString("12", 10) if !curve.IsOnCurve(x, y) { t.Errorf("curve.IsOnCurve() got = %v, want %v", false, true) } y.Add(y, big.NewInt(1)) if curve.IsOnCurve(x, y) { t.Errorf("curve.IsOnCurve() got = %v, want %v", true, false) } x, _ = new(big.Int).SetString("6784692728748995825599862402855483522016546426567910438357042338075027826575", 10) y, _ = new(big.Int).SetString("14982863109320699114866362806305859444453206692004135551371801829915686450358", 10) if !curve.IsOnCurve(x, y) { t.Errorf("curve.IsOnCurve() got = %v, want %v", false, true) } x, _ = new(big.Int).SetString("15508762693928266726769396085241920452527964114034020358226683933290603861217", 10) y, _ = new(big.Int).SetString("14902066091748681388520350407103675753252739566337731413744921722214701412361", 10) if !curve.IsOnCurve(x, y) { t.Errorf("curve.IsOnCurve() got = %v, want %v", false, true) } } func TestScale(t *testing.T) { curve := SIEC255() x, y := curve.ScalarBaseMult([]byte{0x40}) u, _ := new(big.Int).SetString("22784956368772284587014129354783824301122808969944262244690262244567645543628", 10) v, _ := new(big.Int).SetString("15122902039027963115899162435007755480175091927763482548168795824636596047001", 10) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } } func TestScaleP256(t *testing.T) { // Sage: // P256 = EllipticCurve(GF(115792089210356248762697446949407573530086143415290314195533631308867097853951),[-3,0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b]) // G = P256([0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296,0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5]) // 0x4001*G curve := elliptic.P256() x, y := curve.ScalarBaseMult([]byte{0x40, 0x1}) u, _ := new(big.Int).SetString("72695894326801153147216665794252886088922051068954702128373097767796982305416", 10) v, _ := new(big.Int).SetString("115052900738981503025783750032562471330247412874189631362047844261308430634379", 10) if x.Cmp(u) != 0 { t.Errorf("P256.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("P256.ScalarBaseMult() gotY = %v, want %v", y, v) } } func TestLiftX(t *testing.T) { curve := SIEC255() type args struct { X *big.Int } tests := []struct { name string args args wantX *big.Int wantY *big.Int }{ {"generator lifts correctly", args{curve.Gx}, curve.Gx, curve.Gy}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { gotX, gotY := curve.liftX(tt.args.X) if !reflect.DeepEqual(gotX, tt.wantX) { t.Errorf("LiftX() gotX = %v, want %v", gotX, tt.wantX) } if !reflect.DeepEqual(gotY, tt.wantY) { t.Errorf("LiftX() gotY = %v, want %v", gotY, tt.wantY) } }) } } func TestCompress(t *testing.T) { curve := SIEC255() var x1, y1, x2, y2 *big.Int x1, _ = new(big.Int).SetString("5", 10) y1, _ = new(big.Int).SetString("12", 10) x2, y2 = curve.Decompress(curve.Compress(x1, y1)) if x2.Cmp(x1) != 0 || y2.Cmp(y1) != 0 { t.Errorf("(%v,%v) did not compress/decompress correctly, got (%v,%v)", x1, y1, x2, y2) } x1, _ = new(big.Int).SetString("6784692728748995825599862402855483522016546426567910438357042338075027826575", 10) y1, _ = new(big.Int).SetString("14982863109320699114866362806305859444453206692004135551371801829915686450358", 10) x2, y2 = curve.Decompress(curve.Compress(x1, y1)) if x2.Cmp(x1) != 0 || y2.Cmp(y1) != 0 { t.Errorf("(%v,%v) did not compress/decompress correctly, got (%v,%v)", x1, y1, x2, y2) } } func TestCompressRandom(t *testing.T) { curve := SIEC255() for i := 0; i < 10; i++ { x1, y1 := curve.ScalarBaseMult(hash(i)) x2, y2 := curve.Decompress(curve.Compress(x1, y1)) if x2.Cmp(x1) != 0 || y2.Cmp(y1) != 0 { t.Errorf("(%v,%v) did not compress/decompress correctly, got (%v,%v)", x1, y1, x2, y2) } } } func TestScalarMult2Random(t *testing.T) { curve := SIEC255() for i := 0; i < 10; i++ { x, y := curve.ScalarBaseMult(hash(i)) u, v := curve.scalarMult2(curve.Gx, curve.Gy, hash(i)) if x.Cmp(u) != 0 || y.Cmp(v) != 0 { t.Errorf("wanted (%v,%v), got (%v,%v)", x, y, u, v) } } } func TestEndomorphism(t *testing.T) { curve := SIEC255() x, y := curve.Gx, curve.Gy u, _ := new(big.Int).SetString("28948022309329048855892746252183396359753225502720738378331610790540530405116", 10) v, _ := new(big.Int).SetString("28948022309329048855892746252183396360603931420023084536990047309120118726709", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } u, _ = new(big.Int).SetString("850705917302346158658436518579588321600", 10) v, _ = new(big.Int).SetString("12", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } u, _ = new(big.Int).SetString("5", 10) v, _ = new(big.Int).SetString("28948022309329048855892746252183396360603931420023084536990047309120118726709", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } u, _ = new(big.Int).SetString("28948022309329048855892746252183396359753225502720738378331610790540530405116", 10) v, _ = new(big.Int).SetString("12", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } u, _ = new(big.Int).SetString("850705917302346158658436518579588321600", 10) v, _ = new(big.Int).SetString("28948022309329048855892746252183396360603931420023084536990047309120118726709", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } u, _ = new(big.Int).SetString("5", 10) v, _ = new(big.Int).SetString("12", 10) x, y = curve.phi(x, y) if x.Cmp(u) != 0 { t.Errorf("curve.ScalarBaseMult() gotX = %v, want %v", x, u) } if y.Cmp(v) != 0 { t.Errorf("curve.ScalarBaseMult() gotY = %v, want %v", y, v) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/edwards25519/000077500000000000000000000000001456637475500235255ustar00rootroot00000000000000golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/edwards25519/const.go000066400000000000000000002743421456637475500252160ustar00rootroot00000000000000// Copyright 2016 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package edwards25519 // These values are from the public domain, “ref10” implementation of ed25519 // from SUPERCOP. // d is a constant in the Edwards curve equation. var d = FieldElement{ -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, } // d2 is 2*d. var d2 = FieldElement{ -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, } // SqrtM1 is the square-root of -1 in the field. var SqrtM1 = FieldElement{ -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482, } // A is a constant in the Montgomery-form of curve25519. var A = FieldElement{ 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, } // bi contains precomputed multiples of the base-point. See the Ed25519 paper // for a discussion about how these values are used. 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}, }, } // base contains precomputed multiples of the base-point. See the Ed25519 paper // for a discussion about how these values are used. 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}, }, { FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777}, 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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}, }, }, } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/edwards25519/edwards25519.go000066400000000000000000001233451456637475500261230ustar00rootroot00000000000000// Copyright 2016 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package edwards25519 import "encoding/binary" // This code is a port of the public domain, “ref10” implementation of ed25519 // from SUPERCOP. // FieldElement represents an element of the field GF(2^255 - 19). An element // t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 // t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on // context. type FieldElement [10]int32 var zero FieldElement func FeZero(fe *FieldElement) { copy(fe[:], zero[:]) } func FeOne(fe *FieldElement) { FeZero(fe) fe[0] = 1 } func FeAdd(dst, a, b *FieldElement) { dst[0] = a[0] + b[0] dst[1] = a[1] + b[1] dst[2] = a[2] + b[2] dst[3] = a[3] + b[3] dst[4] = a[4] + b[4] dst[5] = a[5] + b[5] dst[6] = a[6] + b[6] dst[7] = a[7] + b[7] dst[8] = a[8] + b[8] dst[9] = a[9] + b[9] } func FeSub(dst, a, b *FieldElement) { dst[0] = a[0] - b[0] dst[1] = a[1] - b[1] dst[2] = a[2] - b[2] dst[3] = a[3] - b[3] dst[4] = a[4] - b[4] dst[5] = a[5] - b[5] dst[6] = a[6] - b[6] dst[7] = a[7] - b[7] dst[8] = a[8] - b[8] dst[9] = a[9] - b[9] } func FeCopy(dst, src *FieldElement) { copy(dst[:], src[:]) } // Replace (f,g) with (g,g) if b == 1; // replace (f,g) with (f,g) if b == 0. // // Preconditions: b in {0,1}. func FeCMove(f, g *FieldElement, b int32) { b = -b f[0] ^= b & (f[0] ^ g[0]) f[1] ^= b & (f[1] ^ g[1]) f[2] ^= b & (f[2] ^ g[2]) f[3] ^= b & (f[3] ^ g[3]) f[4] ^= b & (f[4] ^ g[4]) f[5] ^= b & (f[5] ^ g[5]) f[6] ^= b & (f[6] ^ g[6]) f[7] ^= b & (f[7] ^ g[7]) f[8] ^= b & (f[8] ^ g[8]) f[9] ^= b & (f[9] ^ g[9]) } func load3(in []byte) int64 { var r int64 r = int64(in[0]) r |= int64(in[1]) << 8 r |= int64(in[2]) << 16 return r } func load4(in []byte) int64 { var r int64 r = int64(in[0]) r |= int64(in[1]) << 8 r |= int64(in[2]) << 16 r |= int64(in[3]) << 24 return r } func FeFromBytes(dst *FieldElement, src *[32]byte) { h0 := load4(src[:]) h1 := load3(src[4:]) << 6 h2 := load3(src[7:]) << 5 h3 := load3(src[10:]) << 3 h4 := load3(src[13:]) << 2 h5 := load4(src[16:]) h6 := load3(src[20:]) << 7 h7 := load3(src[23:]) << 5 h8 := load3(src[26:]) << 4 h9 := (load3(src[29:]) & 8388607) << 2 FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) } // FeToBytes marshals h to s. // Preconditions: // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. // // Write p=2^255-19; q=floor(h/p). // Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). // // Proof: // Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. // Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. // // Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). // Then 0> 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]<= -15 { r[i] -= r[i+b] << uint(b) for k := i + b; k < 256; k++ { if r[k] == 0 { r[k] = 1 break } r[k] = 0 } } else { break } } } } } } // GeDoubleScalarMultVartime sets r = a*A + b*B // where a = a[0]+256*a[1]+...+256^31 a[31]. // and b = b[0]+256*b[1]+...+256^31 b[31]. // B is the Ed25519 base point (x,4/5) with x positive. func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { var aSlide, bSlide [256]int8 var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A var t CompletedGroupElement var u, A2 ExtendedGroupElement var i int slide(&aSlide, a) slide(&bSlide, b) A.ToCached(&Ai[0]) A.Double(&t) t.ToExtended(&A2) for i := 0; i < 7; i++ { geAdd(&t, &A2, &Ai[i]) t.ToExtended(&u) u.ToCached(&Ai[i+1]) } r.Zero() for i = 255; i >= 0; i-- { if aSlide[i] != 0 || bSlide[i] != 0 { break } } for ; i >= 0; i-- { r.Double(&t) if aSlide[i] > 0 { t.ToExtended(&u) geAdd(&t, &u, &Ai[aSlide[i]/2]) } else if aSlide[i] < 0 { t.ToExtended(&u) geSub(&t, &u, &Ai[(-aSlide[i])/2]) } if bSlide[i] > 0 { t.ToExtended(&u) geMixedAdd(&t, &u, &bi[bSlide[i]/2]) } else if bSlide[i] < 0 { t.ToExtended(&u) geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) } t.ToProjective(r) } } // equal returns 1 if b == c and 0 otherwise, assuming that b and c are // non-negative. func equal(b, c int32) int32 { x := uint32(b ^ c) x-- return int32(x >> 31) } // negative returns 1 if b < 0 and 0 otherwise. func negative(b int32) int32 { return (b >> 31) & 1 } func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { FeCMove(&t.yPlusX, &u.yPlusX, b) FeCMove(&t.yMinusX, &u.yMinusX, b) FeCMove(&t.xy2d, &u.xy2d, b) } func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { var minusT PreComputedGroupElement bNegative := negative(b) bAbs := b - (((-bNegative) & b) << 1) t.Zero() for i := int32(0); i < 8; i++ { PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) } FeCopy(&minusT.yPlusX, &t.yMinusX) FeCopy(&minusT.yMinusX, &t.yPlusX) FeNeg(&minusT.xy2d, &t.xy2d) PreComputedGroupElementCMove(t, &minusT, bNegative) } // GeScalarMultBase computes h = a*B, where // a = a[0]+256*a[1]+...+256^31 a[31] // B is the Ed25519 base point (x,4/5) with x positive. // // Preconditions: // a[31] <= 127 func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { var e [64]int8 for i, v := range a { e[2*i] = int8(v & 15) e[2*i+1] = int8((v >> 4) & 15) } // each e[i] is between 0 and 15 and e[63] is between 0 and 7. carry := int8(0) for i := 0; i < 63; i++ { e[i] += carry carry = (e[i] + 8) >> 4 e[i] -= carry << 4 } e[63] += carry // each e[i] is between -8 and 8. h.Zero() var t PreComputedGroupElement var r CompletedGroupElement for i := int32(1); i < 64; i += 2 { selectPoint(&t, i/2, int32(e[i])) geMixedAdd(&r, h, &t) r.ToExtended(h) } var s ProjectiveGroupElement h.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToExtended(h) for i := int32(0); i < 64; i += 2 { selectPoint(&t, i/2, int32(e[i])) geMixedAdd(&r, h, &t) r.ToExtended(h) } } // The scalars are GF(2^252 + 27742317777372353535851937790883648493). // Input: // a[0]+256*a[1]+...+256^31*a[31] = a // b[0]+256*b[1]+...+256^31*b[31] = b // c[0]+256*c[1]+...+256^31*c[31] = c // // Output: // s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l // where l = 2^252 + 27742317777372353535851937790883648493. func ScMulAdd(s, a, b, c *[32]byte) { a0 := 2097151 & load3(a[:]) a1 := 2097151 & (load4(a[2:]) >> 5) a2 := 2097151 & (load3(a[5:]) >> 2) a3 := 2097151 & (load4(a[7:]) >> 7) a4 := 2097151 & (load4(a[10:]) >> 4) a5 := 2097151 & (load3(a[13:]) >> 1) a6 := 2097151 & (load4(a[15:]) >> 6) a7 := 2097151 & (load3(a[18:]) >> 3) a8 := 2097151 & load3(a[21:]) a9 := 2097151 & (load4(a[23:]) >> 5) a10 := 2097151 & (load3(a[26:]) >> 2) a11 := (load4(a[28:]) >> 7) b0 := 2097151 & load3(b[:]) b1 := 2097151 & (load4(b[2:]) >> 5) b2 := 2097151 & (load3(b[5:]) >> 2) b3 := 2097151 & (load4(b[7:]) >> 7) b4 := 2097151 & (load4(b[10:]) >> 4) b5 := 2097151 & (load3(b[13:]) >> 1) b6 := 2097151 & (load4(b[15:]) >> 6) b7 := 2097151 & (load3(b[18:]) >> 3) b8 := 2097151 & load3(b[21:]) b9 := 2097151 & (load4(b[23:]) >> 5) b10 := 2097151 & (load3(b[26:]) >> 2) b11 := (load4(b[28:]) >> 7) c0 := 2097151 & load3(c[:]) c1 := 2097151 & (load4(c[2:]) >> 5) c2 := 2097151 & (load3(c[5:]) >> 2) c3 := 2097151 & (load4(c[7:]) >> 7) c4 := 2097151 & (load4(c[10:]) >> 4) c5 := 2097151 & (load3(c[13:]) >> 1) c6 := 2097151 & (load4(c[15:]) >> 6) c7 := 2097151 & (load3(c[18:]) >> 3) c8 := 2097151 & load3(c[21:]) c9 := 2097151 & (load4(c[23:]) >> 5) c10 := 2097151 & (load3(c[26:]) >> 2) c11 := (load4(c[28:]) >> 7) var carry [23]int64 s0 := c0 + a0*b0 s1 := c1 + a0*b1 + a1*b0 s2 := c2 + a0*b2 + a1*b1 + a2*b0 s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0 s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 s20 := a9*b11 + a10*b10 + a11*b9 s21 := a10*b11 + a11*b10 s22 := a11 * b11 s23 := int64(0) carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[18] = (s18 + (1 << 20)) >> 21 s19 += carry[18] s18 -= carry[18] << 21 carry[20] = (s20 + (1 << 20)) >> 21 s21 += carry[20] s20 -= carry[20] << 21 carry[22] = (s22 + (1 << 20)) >> 21 s23 += carry[22] s22 -= carry[22] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 carry[17] = (s17 + (1 << 20)) >> 21 s18 += carry[17] s17 -= carry[17] << 21 carry[19] = (s19 + (1 << 20)) >> 21 s20 += carry[19] s19 -= carry[19] << 21 carry[21] = (s21 + (1 << 20)) >> 21 s22 += carry[21] s21 -= carry[21] << 21 s11 += s23 * 666643 s12 += s23 * 470296 s13 += s23 * 654183 s14 -= s23 * 997805 s15 += s23 * 136657 s16 -= s23 * 683901 s23 = 0 s10 += s22 * 666643 s11 += s22 * 470296 s12 += s22 * 654183 s13 -= s22 * 997805 s14 += s22 * 136657 s15 -= s22 * 683901 s22 = 0 s9 += s21 * 666643 s10 += s21 * 470296 s11 += s21 * 654183 s12 -= s21 * 997805 s13 += s21 * 136657 s14 -= s21 * 683901 s21 = 0 s8 += s20 * 666643 s9 += s20 * 470296 s10 += s20 * 654183 s11 -= s20 * 997805 s12 += s20 * 136657 s13 -= s20 * 683901 s20 = 0 s7 += s19 * 666643 s8 += s19 * 470296 s9 += s19 * 654183 s10 -= s19 * 997805 s11 += s19 * 136657 s12 -= s19 * 683901 s19 = 0 s6 += s18 * 666643 s7 += s18 * 470296 s8 += s18 * 654183 s9 -= s18 * 997805 s10 += s18 * 136657 s11 -= s18 * 683901 s18 = 0 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 s5 += s17 * 666643 s6 += s17 * 470296 s7 += s17 * 654183 s8 -= s17 * 997805 s9 += s17 * 136657 s10 -= s17 * 683901 s17 = 0 s4 += s16 * 666643 s5 += s16 * 470296 s6 += s16 * 654183 s7 -= s16 * 997805 s8 += s16 * 136657 s9 -= s16 * 683901 s16 = 0 s3 += s15 * 666643 s4 += s15 * 470296 s5 += s15 * 654183 s6 -= s15 * 997805 s7 += s15 * 136657 s8 -= s15 * 683901 s15 = 0 s2 += s14 * 666643 s3 += s14 * 470296 s4 += s14 * 654183 s5 -= s14 * 997805 s6 += s14 * 136657 s7 -= s14 * 683901 s14 = 0 s1 += s13 * 666643 s2 += s13 * 470296 s3 += s13 * 654183 s4 -= s13 * 997805 s5 += s13 * 136657 s6 -= s13 * 683901 s13 = 0 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[11] = s11 >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 s[0] = byte(s0 >> 0) s[1] = byte(s0 >> 8) s[2] = byte((s0 >> 16) | (s1 << 5)) s[3] = byte(s1 >> 3) s[4] = byte(s1 >> 11) s[5] = byte((s1 >> 19) | (s2 << 2)) s[6] = byte(s2 >> 6) s[7] = byte((s2 >> 14) | (s3 << 7)) s[8] = byte(s3 >> 1) s[9] = byte(s3 >> 9) s[10] = byte((s3 >> 17) | (s4 << 4)) s[11] = byte(s4 >> 4) s[12] = byte(s4 >> 12) s[13] = byte((s4 >> 20) | (s5 << 1)) s[14] = byte(s5 >> 7) s[15] = byte((s5 >> 15) | (s6 << 6)) s[16] = byte(s6 >> 2) s[17] = byte(s6 >> 10) s[18] = byte((s6 >> 18) | (s7 << 3)) s[19] = byte(s7 >> 5) s[20] = byte(s7 >> 13) s[21] = byte(s8 >> 0) s[22] = byte(s8 >> 8) s[23] = byte((s8 >> 16) | (s9 << 5)) s[24] = byte(s9 >> 3) s[25] = byte(s9 >> 11) s[26] = byte((s9 >> 19) | (s10 << 2)) s[27] = byte(s10 >> 6) s[28] = byte((s10 >> 14) | (s11 << 7)) s[29] = byte(s11 >> 1) s[30] = byte(s11 >> 9) s[31] = byte(s11 >> 17) } // Input: // s[0]+256*s[1]+...+256^63*s[63] = s // // Output: // s[0]+256*s[1]+...+256^31*s[31] = s mod l // where l = 2^252 + 27742317777372353535851937790883648493. func ScReduce(out *[32]byte, s *[64]byte) { s0 := 2097151 & load3(s[:]) s1 := 2097151 & (load4(s[2:]) >> 5) s2 := 2097151 & (load3(s[5:]) >> 2) s3 := 2097151 & (load4(s[7:]) >> 7) s4 := 2097151 & (load4(s[10:]) >> 4) s5 := 2097151 & (load3(s[13:]) >> 1) s6 := 2097151 & (load4(s[15:]) >> 6) s7 := 2097151 & (load3(s[18:]) >> 3) s8 := 2097151 & load3(s[21:]) s9 := 2097151 & (load4(s[23:]) >> 5) s10 := 2097151 & (load3(s[26:]) >> 2) s11 := 2097151 & (load4(s[28:]) >> 7) s12 := 2097151 & (load4(s[31:]) >> 4) s13 := 2097151 & (load3(s[34:]) >> 1) s14 := 2097151 & (load4(s[36:]) >> 6) s15 := 2097151 & (load3(s[39:]) >> 3) s16 := 2097151 & load3(s[42:]) s17 := 2097151 & (load4(s[44:]) >> 5) s18 := 2097151 & (load3(s[47:]) >> 2) s19 := 2097151 & (load4(s[49:]) >> 7) s20 := 2097151 & (load4(s[52:]) >> 4) s21 := 2097151 & (load3(s[55:]) >> 1) s22 := 2097151 & (load4(s[57:]) >> 6) s23 := (load4(s[60:]) >> 3) s11 += s23 * 666643 s12 += s23 * 470296 s13 += s23 * 654183 s14 -= s23 * 997805 s15 += s23 * 136657 s16 -= s23 * 683901 s23 = 0 s10 += s22 * 666643 s11 += s22 * 470296 s12 += s22 * 654183 s13 -= s22 * 997805 s14 += s22 * 136657 s15 -= s22 * 683901 s22 = 0 s9 += s21 * 666643 s10 += s21 * 470296 s11 += s21 * 654183 s12 -= s21 * 997805 s13 += s21 * 136657 s14 -= s21 * 683901 s21 = 0 s8 += s20 * 666643 s9 += s20 * 470296 s10 += s20 * 654183 s11 -= s20 * 997805 s12 += s20 * 136657 s13 -= s20 * 683901 s20 = 0 s7 += s19 * 666643 s8 += s19 * 470296 s9 += s19 * 654183 s10 -= s19 * 997805 s11 += s19 * 136657 s12 -= s19 * 683901 s19 = 0 s6 += s18 * 666643 s7 += s18 * 470296 s8 += s18 * 654183 s9 -= s18 * 997805 s10 += s18 * 136657 s11 -= s18 * 683901 s18 = 0 var carry [17]int64 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 s5 += s17 * 666643 s6 += s17 * 470296 s7 += s17 * 654183 s8 -= s17 * 997805 s9 += s17 * 136657 s10 -= s17 * 683901 s17 = 0 s4 += s16 * 666643 s5 += s16 * 470296 s6 += s16 * 654183 s7 -= s16 * 997805 s8 += s16 * 136657 s9 -= s16 * 683901 s16 = 0 s3 += s15 * 666643 s4 += s15 * 470296 s5 += s15 * 654183 s6 -= s15 * 997805 s7 += s15 * 136657 s8 -= s15 * 683901 s15 = 0 s2 += s14 * 666643 s3 += s14 * 470296 s4 += s14 * 654183 s5 -= s14 * 997805 s6 += s14 * 136657 s7 -= s14 * 683901 s14 = 0 s1 += s13 * 666643 s2 += s13 * 470296 s3 += s13 * 654183 s4 -= s13 * 997805 s5 += s13 * 136657 s6 -= s13 * 683901 s13 = 0 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[11] = s11 >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 out[0] = byte(s0 >> 0) out[1] = byte(s0 >> 8) out[2] = byte((s0 >> 16) | (s1 << 5)) out[3] = byte(s1 >> 3) out[4] = byte(s1 >> 11) out[5] = byte((s1 >> 19) | (s2 << 2)) out[6] = byte(s2 >> 6) out[7] = byte((s2 >> 14) | (s3 << 7)) out[8] = byte(s3 >> 1) out[9] = byte(s3 >> 9) out[10] = byte((s3 >> 17) | (s4 << 4)) out[11] = byte(s4 >> 4) out[12] = byte(s4 >> 12) out[13] = byte((s4 >> 20) | (s5 << 1)) out[14] = byte(s5 >> 7) out[15] = byte((s5 >> 15) | (s6 << 6)) out[16] = byte(s6 >> 2) out[17] = byte(s6 >> 10) out[18] = byte((s6 >> 18) | (s7 << 3)) out[19] = byte(s7 >> 5) out[20] = byte(s7 >> 13) out[21] = byte(s8 >> 0) out[22] = byte(s8 >> 8) out[23] = byte((s8 >> 16) | (s9 << 5)) out[24] = byte(s9 >> 3) out[25] = byte(s9 >> 11) out[26] = byte((s9 >> 19) | (s10 << 2)) out[27] = byte(s10 >> 6) out[28] = byte((s10 >> 14) | (s11 << 7)) out[29] = byte(s11 >> 1) out[30] = byte(s11 >> 9) out[31] = byte(s11 >> 17) } // order is the order of Curve25519 in little-endian form. var order = [4]uint64{0x5812631a5cf5d3ed, 0x14def9dea2f79cd6, 0, 0x1000000000000000} // ScMinimal returns true if the given scalar is less than the order of the // curve. func ScMinimal(scalar *[32]byte) bool { for i := 3; ; i-- { v := binary.LittleEndian.Uint64(scalar[i*8:]) if v > order[i] { return false } else if v < order[i] { break } else if i == 0 { return false } } return true } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/000077500000000000000000000000001456637475500220615ustar00rootroot00000000000000golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/add.go000066400000000000000000000004501456637475500231370ustar00rootroot00000000000000package ff func add(x, y Element) (z Element) { // x + y var k uint64 for i := 0; i < 4; i++ { z[i] = x[i] + y[i] + k // if x[i] + y[i]+ k > 0xffffffffffffffff if (x[i] == 0xffffffffffffffff && k == 1) || x[i]+k > 0xffffffffffffffff-y[i] { k = 1 } else { k = 0 } } return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/add_test.go000066400000000000000000000016001456637475500241740ustar00rootroot00000000000000package ff import ( "math/big" "testing" ) func Test_add_random(t *testing.T) { for i := 0; i < 100; i++ { x := randomElement(2 * i) y := randomElement(2*i + 1) y[3] = y[3] & 0x7fffffffffffffff // clear last bit so garunteed it will fit in 256 bits got := add(x, y) want := BigIntToElement(new(big.Int).Add(ElementToBigInt(x), ElementToBigInt(y))) if got != want { t.Errorf("/%d add(%v,%v) = %v, want %v", i, x, y, got, want) t.FailNow() } } } func Benchmark_add(b *testing.B) { x := randomElement(1) y := randomElement(2) y[3] = y[3] & 0x7fffffffffffffff b.ResetTimer() for i := 0; i < b.N; i++ { add(x, y) } } func Benchmark_add_BI(b *testing.B) { x := randomElement(1) y := randomElement(2) y[3] = y[3] & 0x7fffffffffffffff A, B := ElementToBigInt(x), ElementToBigInt(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Add(A, B) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/cmp.go000066400000000000000000000003411456637475500231650ustar00rootroot00000000000000package ff // cmp returns 1 if a > b, 0 if a = b, and -1 if a < b. func cmp(a [4]uint64, b [4]uint64) int { for i := 3; i >= 0; i-- { if a[i] > b[i] { return 1 } else if a[i] < b[i] { return -1 } } return 0 } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/cmp_test.go000066400000000000000000000022631456637475500242310ustar00rootroot00000000000000package ff import "testing" func Test_cmp(t *testing.T) { type args struct { a [4]uint64 b [4]uint64 } tests := []struct { name string args args want int }{ {"1=1", args{[4]uint64{1, 0, 0, 0}, [4]uint64{1, 0, 0, 0}}, 0}, {"1<2", args{[4]uint64{1, 0, 0, 0}, [4]uint64{2, 0, 0, 0}}, -1}, {"2>1", args{[4]uint64{2, 0, 0, 0}, [4]uint64{1, 0, 0, 0}}, 1}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if got := cmp(tt.args.a, tt.args.b); got != tt.want { t.Errorf("cmp() = %v, want %v", got, tt.want) } }) } } func Test_cmp_random(t *testing.T) { for i := 0; i < 100; i++ { x := randomElement(2 * i) y := randomElement(2*i + 1) A, B := ElementToBigInt(x), ElementToBigInt(y) got := cmp(x, y) want := A.Cmp(B) if got != want { t.Errorf("/%d add(%v,%v) = %v, want %v", i, x, y, got, want) t.FailNow() } } } func Benchmark_cmp(b *testing.B) { x := randomElement(1) y := randomElement(2) b.ResetTimer() for i := 0; i < b.N; i++ { cmp(x, y) } } func Benchmark_cmp_BI(b *testing.B) { x := ElementToBigInt(randomElement(1)) y := ElementToBigInt(randomElement(2)) b.ResetTimer() for i := 0; i < b.N; i++ { x.Cmp(y) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/double.go000066400000000000000000000007141456637475500236640ustar00rootroot00000000000000package ff var ( doublep = Element{0xfff3ff3cf6e79f3d, 0x7fffffffffffffff, 0xfffffffff9ffcf3e, 0x3fffffffffffffff} ) // Note: only works if 2*a < 2^256. One way to garuntee this works is: // // b := double(a) // if a[3]>>63 == 0 { // b = add(doublep, normalize(a)) // } // func double(a Element) (b Element) { b[0] = a[0] << 1 b[1] = (a[1] << 1) | (a[0] >> 63) b[2] = (a[2] << 1) | (a[1] >> 63) b[3] = (a[3] << 1) | (a[2] >> 63) return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/double_test.go000066400000000000000000000024761456637475500247320ustar00rootroot00000000000000package ff import ( "math/big" "reflect" "testing" ) func Test_double(t *testing.T) { type args struct { a Element } tests := []struct { name string args args wantB Element }{ {"1", args{Element{1, 0, 0, 0}}, Element{2, 0, 0, 0}}, {"2^64", args{Element{0, 1, 0, 0}}, Element{0, 2, 0, 0}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotB := double(tt.args.a); !reflect.DeepEqual(gotB, tt.wantB) { t.Errorf("double() = %v, want %v", gotB, tt.wantB) } }) } } func Test_double_random(t *testing.T) { for i := 0; i < 100; i++ { a := randomElement(i) a[3] = a[3] & 0x7fffffffffffffff // clear last bit so garunteed it will fit in 256 bits got := double(a) want := BigIntToElement(new(big.Int).Lsh(ElementToBigInt(a), 1)) if got != want { t.Errorf("/%d double(%v) = %v, want %v", i, a, got, want) t.FailNow() } } } func Benchmark_double(b *testing.B) { a := randomElement(1) a[3] = a[3] & 0x7fffffffffffffff b.ResetTimer() for i := 0; i < b.N; i++ { double(a) } } func Benchmark_double_BI(b *testing.B) { a := ElementToBigInt(randomElement(1)) z := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { z.Lsh(a, 1) } } func Benchmark_double_add(b *testing.B) { a := randomElement(1) b.ResetTimer() for i := 0; i < b.N; i++ { add(a, a) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/ff.go000066400000000000000000000033111456637475500230010ustar00rootroot00000000000000package ff import ( "math/big" ) var ( // p as a *big.Int. pBI, _ = new(big.Int).SetString("28948022309329048855892746252183396360603931420023084536990047309120118726721", 10) // p as an Element. p = Element{1126179130581057, 9223372036854775808, 33558592, 4611686018427387904} ) // Element represents a single element a ∈ 𝔽ₚ, where // // p = 28948022309329048855892746252183396360603931420023084536990047309120118726721. // // Elements are represnted by an unsigned 256 bit integer. // This is represented by an array of four 64 bit unsigned integers. // Specifically, // // element = [4]uint64{a,b,c,d} = (a + b*2^64 + c*2^128 + d*2^192) // // A representation is "normalized" if the it represents an integer in the interval [0,p). type Element [4]uint64 // ElementToBigInt converts an element to a *big.Int. func ElementToBigInt(a Element) (z *big.Int) { return uint64ArrayToBigInt(a[:]) } // BigIntToElement converts a big.Int to an element. func BigIntToElement(z *big.Int) (a Element) { arr := bigIntToUint64Array(z) for len(arr) < 4 { arr = append(arr, 0) } return Element{arr[0], arr[1], arr[2], arr[3]} } // Transform a *big.Int to []uint64 in little-endian. func bigIntToUint64Array(z *big.Int) (arr []uint64) { z = new(big.Int).Set(z) // Use a copy to avoid overwriting anything. low64 := new(big.Int).SetUint64(0xffffffffffffffff) for z.BitLen() > 0 { arr = append(arr, new(big.Int).And(z, low64).Uint64()) z.Rsh(z, 64) } return } func uint64ArrayToBigInt(arr []uint64) (z *big.Int) { if len(arr) == 0 { return new(big.Int) } z = new(big.Int).SetUint64(arr[len(arr)-1]) for i := len(arr) - 2; i >= 0; i-- { z.Add(z.Lsh(z, 64), new(big.Int).SetUint64(arr[i])) } return z } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/ff_test.go000066400000000000000000000035331456637475500240460ustar00rootroot00000000000000package ff import ( "math/big" "math/rand" "reflect" "testing" ) func TestElementToBigInt(t *testing.T) { type args struct { a Element } tests := []struct { name string args args wantZ *big.Int }{ {"0", args{Element{0, 0, 0, 0}}, big.NewInt(0)}, {"1", args{Element{1, 0, 0, 0}}, big.NewInt(1)}, {"1", args{Element{1, 0, 0, 0}}, big.NewInt(1)}, {"2^64", args{Element{0, 1, 0, 0}}, new(big.Int).Lsh(big.NewInt(1), 64)}, {"p", args{p}, pBI}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotZ := ElementToBigInt(tt.args.a); !reflect.DeepEqual(gotZ, tt.wantZ) { t.Errorf("ElementToBigInt() = %v, want %v", gotZ, tt.wantZ) } }) } } func TestBigIntToElement(t *testing.T) { type args struct { z *big.Int } tests := []struct { name string args args wantA Element }{ {"0", args{big.NewInt(0)}, Element{0, 0, 0, 0}}, {"1", args{big.NewInt(1)}, Element{1, 0, 0, 0}}, {"2^64", args{new(big.Int).Lsh(big.NewInt(1), 64)}, Element{0, 1, 0, 0}}, {"p", args{pBI}, p}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotA := BigIntToElement(tt.args.z); !reflect.DeepEqual(gotA, tt.wantA) { t.Errorf("BigIntToElement() = %v, want %v", gotA, tt.wantA) } }) } } func Test_BigIntToElement_ElementToBigInt_random(t *testing.T) { for i := 0; i < 10; i++ { a := randomElement(i) n := ElementToBigInt(a) aa := BigIntToElement(n) if a != aa { t.Errorf("/%d BigIntToElement(ElementToBigInt(%v)) = %v, want %v", i, a, aa, a) t.FailNow() } nn := ElementToBigInt(aa) if n.Cmp(nn) != 0 { t.Errorf("/%d ElementToBigInt(BigIntToElement(%v)) = %v, want %v", i, n, nn, n) t.FailNow() } } } func randomElement(seed int) Element { r := rand.NewSource(int64(seed)) return Element{uint64(r.Int63()), uint64(r.Int63()), uint64(r.Int63()), uint64(r.Int63())} } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul.go000066400000000000000000000115761456637475500232170ustar00rootroot00000000000000package ff var ( negative2ToThe256ModP = Element{0x1001040c208104, 0x0, 0x8004102, 0x0} ) func mul(x, y Element) (z Element) { /* x = x₁*2^B + x₀ y = y₁*2^B + y₀ z₂ = x₁y₁ z₁ = (x₀-x₁)(y₁-y₀) + x₁y₁ + x₀y₀ = x₀y₁ + x₁y₀ z₀ = x₀y₀ x*y = z₂*2^(2B) + z₁*2^B + z₀ */ x0 := [2]uint64{x[0], x[1]} x1 := [2]uint64{x[2], x[3]} y0 := [2]uint64{y[0], y[1]} y1 := [2]uint64{y[2], y[3]} // Compute z₀,z₂ z2 := to320(mul128(x1, y1)) z0 := to320(mul128(x0, y0)) // a = |(x₀-x₁)(y₁-y₀)| var a [5]uint64 x0IsLessThanx1 := cmp(Element{x0[0], x0[1], 0, 0}, Element{x1[0], x1[1], 0, 0}) == -1 y1IsLessThany0 := cmp(Element{y1[0], y1[1], 0, 0}, Element{y0[0], y0[1], 0, 0}) == -1 aIsNegative := x0IsLessThanx1 != y1IsLessThany0 if aIsNegative { if x0IsLessThanx1 { a = to320(mul128(sub128(x1, x0), sub128(y1, y0))) } else { a = to320(mul128(sub128(x0, x1), sub128(y0, y1))) } } else { if x0IsLessThanx1 { a = to320(mul128(sub128(x1, x0), sub128(y0, y1))) } else { a = to320(mul128(sub128(x0, x1), sub128(y1, y0))) } } // Now add z₀ and z₂. z0Plusz2 := add320(z0, z2) // Now compute z₁ = z₂ + z₀ +/- a. var z1 [5]uint64 if aIsNegative { // z₁ = z₀ + z₂ - a // Note that because z₁ = x₀y₁ + x₁y₀ ≥ 0, we know that this subtraction is ok. z1 = sub320(z0Plusz2, a) } else { // z₁ = a + z₀ + z₂ z1 = add320(z0Plusz2, a) } // z = z₂*2^(256) + z₁*2^(128) + z₀ // TODO: // z5 is z as a 320 bit number z5 := add320(z0, [5]uint64{0, 0, z1[0], z1[1], 0}) //z5 = add320(z5, mulBy256([5]uint64{z1[2], z1[3], 0, 0, 0})) //z5 = add320(z5, mulBy256(z2)) // TODO: turn z5 into z z[0] = z5[0] return } func mulBy256(a [5]uint64) [5]uint64 { // // (-2^256 % p) = {0x1001040c208104,0,0x8004102} // (2^384 % p) = {0x30c30d2479030608, 0x800c, 0x30030c24618300} // // -{a0,a1,a2,a3}*2^256 = {a0,a1}*(-2^256) - {a2,a3}*2^384 // = ~~~~~~~ζ~~~~~~~~ - ~~~~~~ω~~~~~~ // // ζ = {a0,a1}*{0x1001040c208104,0,0x8004102} // = a0*{0x1001040c208104,0,0x8004102} + {0,a1}*0x1001040c208104 + (a1*0x8004102)*2^192 // ~~~~~~~~~~~~~~~ζ₀~~~~~~~~~~~~~~~~ ~~~~~~~~~~ζ₁~~~~~~~~~~~ // = ζ₀ + ζ₁ + (a1*0x8004102={b0,b1})*2^192 // = ζ₀ + ζ₁ + {0,0,0,b0} + b1*2^256 // ~~~~ζ₂~~~~ // = ζ₀ + ζ₁ + ζ₂ + b1*2^256 // = ζ₀ + ζ₁ + ζ₂ - b1*{0x1001040c208104,0,0x8004102} // ~~~~~~~~~~~~~~~ζ₃~~~~~~~~~~~~~~~~ // = ζ₀ + ζ₁ + ζ₂ - ζ₃ // // ω = {a2,a3}*{0x30c30d2479030608,0x800c,0x30030c24618300} // = a2*{0x30c30d2479030608,0x800c,0x30030c24618300} + {0,a3}*{0x30c30d2479030608,0x800c} + {0,a3}*{0,0,0x30030c24618300} // ~~~~~~~~~~~~~~~~~~~~~~ω₀~~~~~~~~~~~~~~~~~~~~~~~ + ~~~~~~~~~~~~~~~~~~ω₁~~~~~~~~~~~~~~ // = ω₀ + ω₁ + {0,a3}*{0,0,0x30030c24618300} // = ω₀ + ω₁ + (a3*0x30030c24618300={c1,c2})*2^192 // = ω₀ + ω₁ + {0,0,0,c1} + c2*2^256 // = ω₀ + ω₁ + ~~~~ω₂~~~~ - c2*{0x1001040c208104,0,0x8004102} // = ω₀ + ω₁ + ω₂ - c2*{0x1001040c208104,0,0x8004102} // ~~~~~~~~~~~~~~ω₃~~~~~~~~~~~~~~~~~ // = ω₀ + ω₁ + ω₂ - ω₃ // var ζ [4]Element // ζ₀ ζ0a := mul64(a[0], 0x1001040c208104) ζ0b := mul64(a[0], 0x8004102) ζ[0][0], ζ[0][1], ζ[0][2], ζ[0][3] = ζ0a[0], ζ0a[1], ζ0b[0], ζ0b[1] // ζ₁ ζ1a := mul64(a[1], 0x1001040c208104) ζ[1][1], ζ[1][2] = ζ1a[0], ζ1a[1] // ζ₂ b := mul64(a[1], 0x8004102) ζ[2][3] = b[0] // ζ₃ ζ3a := mul64(b[1], 0x1001040c208104) ζ3b := mul64(b[1], 0x8004102) ζ[3][0], ζ[3][1], ζ[3][2], ζ[3][3] = ζ3a[0], ζ3a[1], ζ3b[0], ζ3b[1] // TODO: think about normalization and check overflow ζall := add(normalize(ζ[0]), normalize(ζ[1])) ζall = add(ζall, normalize(ζ[2])) ζall = sub(ζall, ζ[3]) if a[2] == 0 && a[3] == 0 { return [5]uint64{0, 0, 0, 0, 0} //TODO: ζall } return [5]uint64{0, 0, 0, 0, 0} } func sub128(x, y [2]uint64) (z [2]uint64) { // x - y var k uint64 for i := 0; i < 2; i++ { z[i] = x[i] - (y[i] + k) // if x[i] - (y[i]+k) < 0 if (x[i] == 0 && k == 1) || x[i]-k < y[i] { k = 1 } else { k = 0 } } return } func to320(x [4]uint64) (z [5]uint64) { z[0], z[1], z[2], z[3] = x[0], x[1], x[2], x[3] return } func add320(x, y [5]uint64) (z [5]uint64) { var k uint64 for i := 0; i < 5; i++ { z[i] = x[i] + y[i] + k if (x[i] == 0xffffffffffffffff && k == 1) || x[i]+k > 0xffffffffffffffff-y[i] { k = 1 } else { k = 0 } } return } func sub320(x, y [5]uint64) (z [5]uint64) { var k uint64 for i := 0; i < 5; i++ { z[i] = x[i] - (y[i] + k) if (x[i] == 0 && k == 1) || x[i]-k < y[i] { k = 1 } else { k = 0 } } return } func cmp320(a [5]uint64, b [5]uint64) int { for i := 4; i >= 0; i-- { if a[i] > b[i] { return 1 } else if a[i] < b[i] { return -1 } } return 0 } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul128.go000066400000000000000000000037461456637475500234520ustar00rootroot00000000000000package ff // mul128 multiplies 2 unsigned 128bit integers and returns a 256 bit unsigned integer. func mul128(x, y [2]uint64) (z [4]uint64) { /* x = x₁*2^B + x₀ y = y₁*2^B + y₀ z₂ = x₁y₁ z₁ = (x₀-x₁)(y₁-y₀) + x₁y₁ + x₀y₀ = x₀y₁ + x₁y₀ z₀ = x₀y₀ xy = z₂*2^(2B) + z₁*2^B + z₀ */ z2 := mul64(x[1], y[1]) z0 := mul64(x[0], y[0]) // a = |(x₀-x₁)(y₁-y₀)| var a [2]uint64 aIsNegative := (x[0] < x[1] && y[1] > y[0]) || (x[0] > x[1] && y[1] < y[0]) if aIsNegative { if x[0] < x[1] { a = mul64(x[1]-x[0], y[1]-y[0]) } else { a = mul64(x[0]-x[1], y[0]-y[1]) } } else { if x[0] < x[1] { a = mul64(x[1]-x[0], y[0]-y[1]) } else { a = mul64(x[0]-x[1], y[1]-y[0]) } } // Now add z₀ and z₂. var z0Plusz2 [4]uint64 var k uint64 z0Plusz2[0] = z0[0] + z2[0] if z0[0] > 0xffffffffffffffff-z2[0] { k = 1 } z0Plusz2[1] = z0[1] + z2[1] + k if (z0[1] == 0xffffffffffffffff && k > 0) || z0[1]+k > 0xffffffffffffffff-z2[1] { z0Plusz2[2] = 1 } k = 0 var z1 [4]uint64 // Now compute z₁. if aIsNegative { // z₁ = z₀ + z₂ - a for i := 0; i < 2; i++ { z1[i] = z0Plusz2[i] - (a[i] + k) if (z0Plusz2[i] == 0 && k == 1) || z0Plusz2[i]-k < a[i] { k = 1 } else { k = 0 } } z1[2] = z0Plusz2[2] - k } else { // z₁ = a + z₀ + z₂ for i := 0; i < 2; i++ { z1[i] = a[i] + z0Plusz2[i] + k if (a[i] == 0xffffffffffffffff && k == 1) || a[i]+k > 0xffffffffffffffff-z0Plusz2[i] { k = 1 } else { k = 0 } } z1[2] = z0Plusz2[2] + k } k = 0 // [ z0[0], z0[1], z2[0], z2[1] ] // + [ 0 , z1[0], z1[1], z1[2] ] // __________________________________ // z z[0] = z0[0] z[1] = z0[1] + z1[0] if z0[1] > 0xffffffffffffffff-z1[0] { k = 1 } else { k = 0 } z[2] = z2[0] + z1[1] + k if (z2[0] == 0xffffffffffffffff && k == 1) || z2[0]+k > 0xffffffffffffffff-z1[1] { k = 1 } else { k = 0 } z[3] = z2[1] + z1[2] + k return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul128_test.go000066400000000000000000000032361456637475500245030ustar00rootroot00000000000000package ff import ( "math/big" "reflect" "testing" ) func Test_mul128(t *testing.T) { type args struct { x [2]uint64 y [2]uint64 } tests := []struct { name string args args wantZ [4]uint64 }{ {"2*1", args{[2]uint64{2, 0}, [2]uint64{1, 0}}, [4]uint64{2, 0, 0, 0}}, {"(2^128 - 1)(2^128 - 1)", args{[2]uint64{0xffffffffffffffff, 0xffffffffffffffff}, [2]uint64{0xffffffffffffffff, 0xffffffffffffffff}}, [4]uint64{0x1, 0, 0xfffffffffffffffe, 0xffffffffffffffff}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotZ := mul128(tt.args.x, tt.args.y); !reflect.DeepEqual(gotZ, tt.wantZ) { t.Errorf("mul128() = %v, want %v", gotZ, tt.wantZ) } }) } } func Test_mul128_random(t *testing.T) { for i := 0; i < 100; i++ { x := [2]uint64{randomElement(2 * i)[0], randomElement(2 * i)[1]} y := [2]uint64{randomElement(2*i + 1)[0], randomElement(2*i + 1)[1]} got := mul128(x, y) want := BigIntToElement(new(big.Int).Mul(ElementToBigInt(Element{x[0], x[1], 0, 0}), ElementToBigInt(Element{y[0], y[1], 0, 0}))) if got != want { t.Errorf("/%d mul128(%v,%v) = %v, want %v", i, x, y, got, want) t.FailNow() } } } func Benchmark_mul128(b *testing.B) { x := [2]uint64{randomElement(1)[0], randomElement(1)[1]} y := [2]uint64{randomElement(2)[0], randomElement(2)[1]} b.ResetTimer() for i := 0; i < b.N; i++ { mul128(x, y) } } func Benchmark_mul128_BI(b *testing.B) { x := [4]uint64{randomElement(1)[0], randomElement(1)[1], 0, 0} y := [4]uint64{randomElement(2)[0], randomElement(2)[1], 0, 0} A, B := ElementToBigInt(x), ElementToBigInt(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Mul(A, B) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul256.go000066400000000000000000000022671456637475500234510ustar00rootroot00000000000000package ff func mul256(x, y [4]uint64) (z [8]uint64) { /* B = 128 x = x₁*2^B + x₀ y = y₁*2^B + y₀ z₂ = x₁y₁ z₁ = (x₀-x₁)(y₁-y₀) + x₁y₁ + x₀y₀ = x₀y₁ + x₁y₀ z₀ = x₀y₀ x*y = z₂*2^(2B) + z₁*2^B + z₀ */ x0 := [2]uint64{x[0], x[1]} x1 := [2]uint64{x[2], x[3]} y0 := [2]uint64{y[0], y[1]} y1 := [2]uint64{y[2], y[3]} z0 := mul128(x0, y0) z2 := mul128(x1, y1) x0y1 := mul128(x0, y1) x1y0 := mul128(x1, y0) var k1, k2 uint64 z[0] = z0[0] z[1] = z0[1] z[2], k1 = addThree64(z0[2], x0y1[0], k1) z[2], k2 = addTwo64(z[2], x1y0[0]) z[3], k1 = addThree64(z0[3], x0y1[1], k1+k2) z[3], k2 = addTwo64(z[3], x1y0[1]) z[4], k1 = addThree64(z2[0], x0y1[2], k1+k2) z[4], k2 = addTwo64(z[4], x1y0[2]) z[5], k1 = addThree64(z2[1], x0y1[3], k1+k2) z[5], k2 = addTwo64(z[5], x1y0[3]) z[6], k1 = addTwo64(z2[2], k1+k2) z[7], _ = addTwo64(z2[3], k1) return } func addThree64(x, y, z uint64) (sum, carry uint64) { sum = x + y + z if (y > 0xffffffffffffffff-x) || (z > 0xffffffffffffffff-(x+y)) { carry = 1 } return } func addTwo64(x, y uint64) (sum, carry uint64) { sum = x + y if y > 0xffffffffffffffff-x { carry = 1 } return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul256_test.go000066400000000000000000000030611456637475500245010ustar00rootroot00000000000000package ff import ( "math/big" "reflect" "testing" ) func Test_mul256(t *testing.T) { type args struct { x [4]uint64 y [4]uint64 } tests := []struct { name string args args wantZ [8]uint64 }{ {"2*1", args{[4]uint64{2, 0, 0, 0}, [4]uint64{1, 0, 0, 0}}, [8]uint64{2, 0, 0, 0, 0, 0, 0, 0}}, { "(2^128 - 1)(2^128 - 1)", args{ [4]uint64{0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}, [4]uint64{0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}, }, [8]uint64{1, 0, 0, 0, 0xfffffffffffffffe, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotZ := mul256(tt.args.x, tt.args.y); !reflect.DeepEqual(gotZ, tt.wantZ) { t.Errorf("mul256() = %v, want %v", gotZ, tt.wantZ) } }) } } func Test_mul256_random(t *testing.T) { for i := 0; i < 100; i++ { a := randomElement(i) A := ElementToBigInt(a) b := randomElement(i + 1) B := ElementToBigInt(b) c := mul256(a, b) C := new(big.Int).Mul(A, B) if C.Cmp(uint64ArrayToBigInt(c[:])) != 0 { t.Errorf("mul256() = %v, want %v", c, bigIntToUint64Array(C)) } } } func Benchmark_mul256(b *testing.B) { x := randomElement(1) y := randomElement(2) b.ResetTimer() for i := 0; i < b.N; i++ { mul256(x, y) } } func Benchmark_mul256_BI(b *testing.B) { x := randomElement(1) y := randomElement(2) A, B := ElementToBigInt(x), ElementToBigInt(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Mul(A, B) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul64.go000066400000000000000000000012741456637475500233630ustar00rootroot00000000000000package ff // mul64 multiplies 2 unsigned 64 bit integers and returns a 128 bit unsigned integer. func mul64(x, y uint64) (z [2]uint64) { /* B = 2^32 x = x₁*B + x₀ y = y₁*B + y₀ z₂ = x₁y₁ z₁ = x₁y₀ + x₀y₁ z₀ = x₀y₀ xy = z₂*2^(2B) + z₁*2^B + z₀ */ x1, x0 := x>>32, x&0xffffffff y1, y0 := y>>32, y&0xffffffff z[0], z[1] = x0*y0, x1*y1 a := x1 * y0 b := x0 * y1 c := a + b // overflow is checked below and stored in w[1] // w = z1*2^B w := [2]uint64{c << 32, c >> 32} if a > 0xffffffffffffffff-b { w[1] = w[1] | 0x100000000 } // z + w if z[0] > 0xffffffffffffffff-w[0] { w[1]++ } z[0] = z[0] + w[0] z[1] = z[1] + w[1] return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul64_test.go000066400000000000000000000026441456637475500244240ustar00rootroot00000000000000package ff import ( "math/big" "math/rand" "reflect" "testing" ) func Test_mul64(t *testing.T) { type args struct { x uint64 y uint64 } tests := []struct { name string args args wantZ [2]uint64 }{ {"2*1", args{2, 1}, [2]uint64{2, 0}}, {"(2^64 - 1)(2^64 - 1)", args{0xffffffffffffffff, 0xffffffffffffffff}, [2]uint64{1, 0xfffffffffffffffe}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotZ := mul64(tt.args.x, tt.args.y); !reflect.DeepEqual(gotZ, tt.wantZ) { t.Errorf("mul64() = %v, want %v", gotZ, tt.wantZ) } }) } } func Test_mul64_random(t *testing.T) { r := rand.NewSource(1) for i := 0; i < 100; i++ { a := uint64(r.Int63()) b := uint64(r.Int63()) A := new(big.Int).SetUint64(a) B := new(big.Int).SetUint64(b) got := mul64(a, b) C1 := new(big.Int).Mul(A, B) want := [2]uint64{C1.Uint64(), C1.Rsh(C1, 64).Uint64()} if got != want { t.Errorf("/%d mul64(%v,%v) = %v, want %v", i, a, b, got, want) t.FailNow() } } } func Benchmark_mul64(b *testing.B) { r := rand.NewSource(1) x := uint64(r.Int63()) y := uint64(r.Int63()) b.ResetTimer() for i := 0; i < b.N; i++ { mul64(x, y) } } func Benchmark_mul64_BI(b *testing.B) { r := rand.NewSource(1) x := uint64(r.Int63()) y := uint64(r.Int63()) A := new(big.Int).SetUint64(x) B := new(big.Int).SetUint64(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Mul(A, B) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/mul_test.go000066400000000000000000000015441456637475500242500ustar00rootroot00000000000000package ff import ( "math/big" "reflect" "testing" ) func Test_mul(t *testing.T) { type args struct { x Element y Element } tests := []struct { name string args args wantZ Element }{ {"1*1", args{Element{1, 0, 0, 0}, Element{1, 0, 0, 0}}, Element{1, 0, 0, 0}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if gotZ := mul(tt.args.x, tt.args.y); !reflect.DeepEqual(gotZ, tt.wantZ) { t.Errorf("mul() = %v, want %v", gotZ, tt.wantZ) } }) } } func Benchmark_mul(b *testing.B) { x := randomElement(1) y := randomElement(2) b.ResetTimer() for i := 0; i < b.N; i++ { mul(x, y) } } func Benchmark_mul_BI(b *testing.B) { x := randomElement(1) y := randomElement(2) A, B := ElementToBigInt(x), ElementToBigInt(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Mod(C.Mul(A, B), pBI) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/normalize.go000066400000000000000000000002501456637475500244050ustar00rootroot00000000000000package ff func normalize(a Element) Element { c := cmp(a, p) if c == -1 { return a } if c == 0 { return Element{0, 0, 0, 0} } return normalize(sub(a, p)) } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/normalize_test.go000066400000000000000000000015471456637475500254560ustar00rootroot00000000000000package ff import ( "reflect" "testing" ) func Test_normalize(t *testing.T) { type args struct { a Element } tests := []struct { name string args args want Element }{ {"1", args{Element{1, 0, 0, 0}}, Element{1, 0, 0, 0}}, {"0", args{Element{0, 0, 0, 0}}, Element{0, 0, 0, 0}}, {"p", args{p}, Element{0, 0, 0, 0}}, {"2p", args{Element{0x8008206104082, 0, 0x4002081, 0x8000000000000000}}, Element{0, 0, 0, 0}}, {"2p+1", args{Element{0x8008206104083, 0, 0x4002081, 0x8000000000000000}}, Element{1, 0, 0, 0}}, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { if got := normalize(tt.args.a); !reflect.DeepEqual(got, tt.want) { t.Errorf("normalize() = %v, want %v", got, tt.want) } }) } } func Benchmark_normalize(b *testing.B) { a := randomElement(1) b.ResetTimer() for i := 0; i < b.N; i++ { normalize(a) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/sub.go000066400000000000000000000003661456637475500232060ustar00rootroot00000000000000package ff func sub(x, y Element) (z Element) { // x - y var k uint64 for i := 0; i < 4; i++ { z[i] = x[i] - (y[i] + k) // if x[i] - (y[i]+k) < 0 if (x[i] == 0 && k == 1) || x[i]-k < y[i] { k = 1 } else { k = 0 } } return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/ff/sub_test.go000066400000000000000000000016371456637475500242470ustar00rootroot00000000000000package ff import ( "math/big" "testing" ) func Test_sub_random(t *testing.T) { for i := 0; i < 100; i++ { x := randomElement(2 * i) y := randomElement(2*i + 1) x[3] = x[3] | 0x8000000000000000 // set top bit so x > y y[3] = y[3] & 0x7fffffffffffffff // clear last bit so y < x got := sub(x, y) want := BigIntToElement(new(big.Int).Sub(ElementToBigInt(x), ElementToBigInt(y))) if got != want { t.Errorf("/%d sub(%v,%v) = %v, want %v", i, x, y, got, want) t.FailNow() } } } func Benchmark_sub(b *testing.B) { x := randomElement(1) y := randomElement(2) y[3] = y[3] & 0x7fffffffffffffff b.ResetTimer() for i := 0; i < b.N; i++ { sub(x, y) } } func Benchmark_sub_BI(b *testing.B) { x := randomElement(1) y := randomElement(2) y[3] = y[3] & 0x7fffffffffffffff A, B := ElementToBigInt(x), ElementToBigInt(y) C := new(big.Int) b.ResetTimer() for i := 0; i < b.N; i++ { C.Sub(A, B) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/projective.go000066400000000000000000000100701456637475500241650ustar00rootroot00000000000000package siec import ( "math/big" ) func (curve *SIEC255Params) affineToProjective(x, y *big.Int) (X, Y, Z *big.Int) { X, Y, Z = new(big.Int), new(big.Int), new(big.Int) X.Set(x) X.Mod(X, curve.P) Y.Set(y) Y.Mod(Y, curve.P) Z.SetInt64(1) return } func (curve *SIEC255Params) projectiveToAffine(X, Y, Z *big.Int) (x, y *big.Int) { x, y = new(big.Int), new(big.Int) if Z.Sign() == 0 { return new(big.Int), new(big.Int) } Zinv := new(big.Int).ModInverse(Z, curve.P) Zinvsq := new(big.Int).Mul(Zinv, Zinv) x.Mul(X, Zinvsq) x.Mod(x, curve.P) y.Mul(Y, Zinvsq.Mul(Zinvsq, Zinv)) y.Mod(y, curve.P) return } // assumes Z1 = Z2 = 1. func (curve *SIEC255Params) mmadd2007bl(X1, Y1, X2, Y2 *big.Int) (X3, Y3, Z3 *big.Int) { w := new(big.Int) ww := new(big.Int) // H = X2-X1 H := new(big.Int).Sub(X2, X1) // HH = H^2 HH := new(big.Int).Mul(H, H) // I = 4*HH I := new(big.Int).Lsh(HH, 2) // J = H*I J := new(big.Int).Mul(H, I) // r = 2*(Y2-Y1) w.Sub(Y2, Y1) r := new(big.Int).Lsh(w, 1) // V = X1*I V := new(big.Int).Mul(X1, I) // X3 = r^2-J-2*V w.Mul(r, r) ww.Add(J, ww.Lsh(V, 1)) X3 = new(big.Int).Sub(w, ww) // Y3 = r*(V-X3)-2*Y1*J w.Mul(r, w.Sub(V, X3)) ww.Lsh(ww.Mul(Y1, J), 1) Y3 = new(big.Int).Sub(w, ww) // Z3 = 2*H Z3 = new(big.Int).Lsh(H, 1) return } // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2001-b func (curve *SIEC255Params) add2007bl(X1, Y1, Z1, X2, Y2, Z2 *big.Int) (X3, Y3, Z3 *big.Int) { if X1.BitLen() == 0 && Y1.BitLen() == 0 && Z1.BitLen() == 0 { return X2, Y2, Z2 } if X2.BitLen() == 0 && Y2.BitLen() == 0 && Z2.BitLen() == 0 { return X1, Y1, Z1 } w := new(big.Int) ww := new(big.Int) // Z1Z1 = Z1^2 Z1Z1 := new(big.Int).Mul(Z1, Z1) // Z2Z2 = Z2^2 Z2Z2 := new(big.Int).Mul(Z2, Z2) // U1 = X1*Z2Z2 U1 := new(big.Int).Mul(X1, Z2Z2) // U2 = X2*Z1Z1 U2 := new(big.Int).Mul(X2, Z1Z1) // S1 = Y1*Z2*Z2Z2 w.Mul(Z2, Z2Z2) S1 := new(big.Int).Mul(Y1, w) // S2 = Y2*Z1*Z1Z1 w.Mul(Z1, Z1Z1) S2 := new(big.Int).Mul(Y2, w) // H = U2-U1 H := new(big.Int).Sub(U2, U1) // I = (2*H)^2 ww.Lsh(H, 1) I := new(big.Int).Mul(ww, ww) // J = H*I J := new(big.Int).Mul(H, I) // r = 2*(S2-S1) w.Sub(S2, S1) r := new(big.Int).Lsh(w, 1) // V = U1*I V := new(big.Int).Mul(U1, I) // X3 = r^2-J-2*V w.Mul(r, r) ww.Lsh(V, 1) ww.Add(J, ww) X3 = new(big.Int).Sub(w, ww) X3.Mod(X3, curve.P) // Y3 = r*(V-X3)-2*S1*J w.Sub(V, X3) w.Mul(r, w) ww.Mul(S1, J) ww.Lsh(ww, 1) Y3 = new(big.Int).Sub(w, ww) Y3.Mod(Y3, curve.P) // Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H w.Add(Z1, Z2) w.Mul(w, w) ww.Add(Z1Z1, Z2Z2) w.Sub(w, ww) Z3 = new(big.Int).Mul(w, H) Z3.Mod(Z3, curve.P) return } // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l func (curve *SIEC255Params) dbl2009l(X1, Y1, Z1 *big.Int) (X3, Y3, Z3 *big.Int) { w := new(big.Int) m := new(big.Int) // A = X1^2 A := new(big.Int).Mul(X1, X1) A.Mod(A, curve.P) // B = Y1^2 B := new(big.Int).Mul(Y1, Y1) B.Mod(B, curve.P) // C = B^2 C := new(big.Int).Mul(B, B) C.Mod(C, curve.P) // D = 2*((X1+B)^2-A-C) w.Add(X1, B) D := new(big.Int).Lsh(w.Sub(w.Mul(w, w), m.Add(A, C)), 1) D.Mod(D, curve.P) // E = 3*A E := new(big.Int).Mul(three, A) E.Mod(E, curve.P) // F = E^2 F := new(big.Int).Mul(E, E) F.Mod(F, curve.P) // X3 = F-2*D X3 = new(big.Int).Sub(F, w.Lsh(D, 1)) X3.Mod(X3, curve.P) // Y3 = E*(D-X3)-8*C Y3 = new(big.Int).Sub(w.Mul(E, w.Sub(D, X3)), m.Mul(eight, C)) Y3.Mod(Y3, curve.P) // Z3 = 2*Y1*Z1 Z3 = w.Lsh(w.Mul(Z1, Y1), 1) Z3.Mod(Z3, curve.P) return } func (curve *SIEC255Params) projectiveScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) { Bz := new(big.Int).SetInt64(1) x, y, z := new(big.Int), new(big.Int), new(big.Int) for _, byte := range k { for bitNum := 0; bitNum < 8; bitNum++ { x, y, z = curve.dbl2009l(x, y, z) if byte&0x80 == 0x80 { x, y, z = curve.add2007bl(Bx, By, Bz, x, y, z) } byte <<= 1 } } return curve.projectiveToAffine(x, y, z) } func (curve *SIEC255Params) projectiveScalarBaseMult(k []byte) (*big.Int, *big.Int) { return curve.projectiveScalarMult(curve.Gx, curve.Gy, k) } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/projective_test.go000066400000000000000000000070441456637475500252330ustar00rootroot00000000000000package siec import ( "math/big" "testing" ) func Test_projectiveToAffine(t *testing.T) { curve := SIEC255() var x, y, X, Y, Z, u, v *big.Int X, Y, Z = big.NewInt(5), big.NewInt(12), big.NewInt(1) x, y = curve.projectiveToAffine(X, Y, Z) u, v = big.NewInt(5), big.NewInt(12) if x.Cmp(u) != 0 || y.Cmp(v) != 0 { t.Errorf("got: (0x%x,0x%x) wanted (0x%x,0x%x)", x, y, u, v) } } func Test_affineToProjective(t *testing.T) { curve := SIEC255() var x, y, X, Y, Z, U, V, W *big.Int x, y = big.NewInt(5), big.NewInt(12) X, Y, Z = curve.affineToProjective(x, y) U, V, W = big.NewInt(5), big.NewInt(12), big.NewInt(1) if X.Cmp(U) != 0 || Y.Cmp(V) != 0 || Z.Cmp(W) != 0 { t.Errorf("got: (0x%x,0x%x,0x%x) wanted (0x%x,0x%x,0x%x)", X, Y, Z, U, V, W) } } func Test_add2007bl(t *testing.T) { curve := SIEC255() var X1, Y1, Z1, X2, Y2, Z2, X3, Y3, Z3, U, V, W *big.Int X1 = big.NewInt(5) Y1 = big.NewInt(12) Z1 = big.NewInt(1) X2, _ = new(big.Int).SetString("f0000000000000000000000007803cf1e000000000000000000f00f3cb5e78f", 16) Y2, _ = new(big.Int).SetString("21200000000000000000000001090869624000000000000000021221a611b4b6", 16) Z2, _ = new(big.Int).SetString("1", 16) X3, Y3, Z3 = curve.add2007bl(X1, Y1, Z1, X2, Y2, Z2) X3.Mod(X3, curve.P) Y3.Mod(Y3, curve.P) Z3.Mod(Z3, curve.P) U, _ = new(big.Int).SetString("3db00000000000000000000001ed8faa2b600000000000000003db3ea9ac13f6", 16) V, _ = new(big.Int).SetString("334e00000000000000000000019a7d07349c000000000000000335141da62f6f", 16) W, _ = new(big.Int).SetString("1e000000000000000000000000f0079e3c000000000000000001e01e796bcf14", 16) if X3.Cmp(U) != 0 || Y3.Cmp(V) != 0 || Z3.Cmp(W) != 0 { t.Errorf("got: (0x%x,0x%x,0x%x) wanted (0x%x,0x%x,0x%x)", X3, Y3, Z3, U, V, W) } } func Test_dbl2009l(t *testing.T) { curve := SIEC255() var X1, Y1, Z1, X3, Y3, Z3, U, V, W *big.Int X1, _ = new(big.Int).SetString("f0000000000000000000000007803cf1e000000000000000000f00f3cb5e78f", 16) Y1, _ = new(big.Int).SetString("21200000000000000000000001090869624000000000000000021221a611b4b6", 16) Z1, _ = new(big.Int).SetString("1", 16) X3, Y3, Z3 = curve.dbl2009l(X1, Y1, Z1) X3.Mod(X3, curve.P) Y3.Mod(Y3, curve.P) Z3.Mod(Z3, curve.P) U, _ = new(big.Int).SetString("3a2e9fc0000000000000000001d183c44aecff80000000000003a325161bd1c7", 16) V, _ = new(big.Int).SetString("1e40efa67c0000000000000000f20f2bed3b24c8f80000000001e42db5c9fc62", 16) W, _ = new(big.Int).SetString("2400000000000000000000000120092448000000000000000002402491b492b", 16) if X3.Cmp(U) != 0 || Y3.Cmp(V) != 0 || Z3.Cmp(W) != 0 { t.Errorf("got: (0x%x,0x%x,0x%x) wanted (0x%x,0x%x,0x%x)", X3, Y3, Z3, U, V, W) } x, y := curve.projectiveToAffine(X3, Y3, Z3) u, v := new(big.Int), new(big.Int) u.SetString("1b3c1ab2fff8cc396d2c1528fd5e7229b58c06888decb98657312b2a0abc3ef9", 16) v.SetString("14a8371e5d745fe915613984af6451eb655cb3150a91caa9bafc5781f18d1257", 16) if x.Cmp(u) != 0 || y.Cmp(v) != 0 { t.Errorf("got: (0x%x,0x%x) wanted (0x%x,0x%x)", x, y, u, v) } } func Test_projectiveScalarBaseMult(t *testing.T) { curve := SIEC255() x, y := curve.projectiveScalarBaseMult([]byte{0x4}) u, _ := new(big.Int).SetString("1b3c1ab2fff8cc396d2c1528fd5e7229b58c06888decb98657312b2a0abc3ef9", 16) v, _ := new(big.Int).SetString("14a8371e5d745fe915613984af6451eb655cb3150a91caa9bafc5781f18d1257", 16) if x.Cmp(u) != 0 || y.Cmp(v) != 0 { t.Errorf("got: (0x%x,0x%x) wanted (0x%x,0x%x)", x, y, u, v) } } func BenchmarkScale_projective(b *testing.B) { curve := SIEC255() b.ResetTimer() k := hash(1) for i := 0; i < b.N; i++ { curve.projectiveScalarBaseMult(k) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/search/000077500000000000000000000000001456637475500227335ustar00rootroot00000000000000golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/search/main.go000066400000000000000000000023621456637475500242110ustar00rootroot00000000000000package main import ( "fmt" "math" "math/big" "sort" "sync" ) func main() { // build results var results [][2]*big.Int M := new(big.Int).Lsh(one, 256) var mutex = &sync.Mutex{} var waitGroup sync.WaitGroup waitGroup.Add(255) for j := uint(0); j < 255; j++ { go func(j uint) { defer waitGroup.Done() M2 := new(big.Int).Set(M) M2.Sub(M2, new(big.Int).Lsh(one, j)) var q, t *big.Int q = M2 for { q, t = nextSiec(q) N := new(big.Int).Add(q, big.NewInt(1)) N.Sub(N, t) N2 := new(big.Int).Add(q, big.NewInt(1)) N2.Add(N2, t) if true || N.ProbablyPrime(25) || N2.ProbablyPrime(25) { mutex.Lock() results = append(results, [2]*big.Int{q, t}) mutex.Unlock() return } } }(j) } waitGroup.Wait() // rank results sort.Slice(results, func(i int, j int) bool { n1 := results[i][0].BitLen() c1 := popCount(results[i][0]) n2 := results[j][0].BitLen() c2 := popCount(results[j][0]) return math.Abs(float64(n1-2*int(c1))) < math.Abs(float64(n2-2*int(c2))) }) // print for _, r := range results { fmt.Printf("q = %x\ns = %d\nt = %x\n", r[0], popCount(r[0]), r[1]) } } func popCount(a *big.Int) (c uint) { n := a.BitLen() for i := 0; i <= n; i++ { c += a.Bit(i) } return } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/search/search.go000066400000000000000000000016461456637475500245360ustar00rootroot00000000000000package main import ( "math/big" ) var ( _D = []*big.Int{big.NewInt(3), big.NewInt(4), big.NewInt(7), big.NewInt(8), big.NewInt(11), big.NewInt(19), big.NewInt(43), big.NewInt(67), big.NewInt(163)} one = big.NewInt(1) two = big.NewInt(2) ) func test(t *big.Int, q *big.Int) bool { for _, d := range _D { q.Exp(t, two, nil) q.Add(q, d) if q.Bits()[0]&3 != 0 { continue } q.Rsh(q, 2) // TODO: check if q is a prime power if q.ProbablyPrime(20) { return true } } return false } func nextSiec(M *big.Int) (q, t *big.Int) { t = new(big.Int) q = new(big.Int) t.Lsh(M, 2) t.Sub(t, _D[8]) t.Sqrt(t) t.Sub(t, one) for { if test(t, q) && q.Cmp(M) > 0 { return } t.Add(t, one) } } func prevSiec(M *big.Int) (q, t *big.Int) { t = new(big.Int) q = new(big.Int) t.Lsh(M, 2) t.Sub(t, _D[0]) t.Sqrt(t) t.Add(t, one) for { if test(t, q) && q.Cmp(M) < 0 { return } t.Sub(t, one) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/search/search_test.go000066400000000000000000000017031456637475500255670ustar00rootroot00000000000000package main import ( "math/big" "reflect" "testing" ) func BenchmarkNextSiec(b *testing.B) { M := new(big.Int) M.SetString("100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", 16) b.ResetTimer() for i := 1; i < b.N; i++ { nextSiec(M) } } func Test_nextSiec(t *testing.T) { type args struct { M *big.Int } tests := []struct { name string args args wantT *big.Int wantQ *big.Int }{ { name: "10^10", args: args{M: big.NewInt(10000000000)}, wantT: big.NewInt(200001), wantQ: big.NewInt(10000100003), }, } for _, tt := range tests { t.Run(tt.name, func(t *testing.T) { gotT, gotQ := nextSiec(tt.args.M) if !reflect.DeepEqual(gotT, tt.wantT) { t.Errorf("nextSiec() gotT = %v, want %v", gotT, tt.wantT) } if !reflect.DeepEqual(gotQ, tt.wantQ) { t.Errorf("nextSiec() gotQ = %v, want %v", gotQ, tt.wantQ) } }) } } golang-github-tscholl2-siec-0.0~git20210707.9bdfc48/siec.go000066400000000000000000000171601456637475500227450ustar00rootroot00000000000000package siec import ( "fmt" "io" "math" "math/big" "sync" ) var ( initonce sync.Once zero = big.NewInt(0) two = big.NewInt(2) three = big.NewInt(3) eight = big.NewInt(8) nineteen = big.NewInt(19) ) // SIEC255Params contains the parameters of an elliptic curve and also provides // a generic, non-constant time implementation of Curve. type SIEC255Params struct { P *big.Int // the order of the underlying field N *big.Int // the order of the base point A *big.Int // the other constant of the curve equation B *big.Int // the constant of the curve equation Gx, Gy *big.Int // (x,y) of the base point BitSize int // the size of the underlying field Name string // the canonical name of the curve } // Params returns the parameters for the curve. func (curve *SIEC255Params) Params() *SIEC255Params { return curve } // IsOnCurve reports whether the given (x,y) lies on the curve. func (curve *SIEC255Params) IsOnCurve(x, y *big.Int) bool { // y² = x³ + 19 LHS := new(big.Int).Exp(y, two, curve.P) LHS.Mod(LHS, curve.P) RHS := new(big.Int).Exp(x, three, curve.P) RHS.Add(RHS, nineteen) RHS.Mod(RHS, curve.P) return LHS.Cmp(RHS) == 0 } // Add returns the sum of (x1,y1) and (x2,y2) func (curve *SIEC255Params) Add(x1, y1, x2, y2 *big.Int) (x3, y3 *big.Int) { if x1.BitLen() == 0 && y1.BitLen() == 0 { return x2, y2 } if x2.BitLen() == 0 && y2.BitLen() == 0 { return x1, y1 } if x1.Cmp(x2) == 0 && y1.Cmp(y2) == 0 { return curve.Double(x1, y1) } // TODO: optimize // λ = (y2 - y1)/(x2 - x1) z := new(big.Int).Sub(x2, x1) lambda := new(big.Int).Sub(y2, y1) if z.BitLen() == 0 { return z.Set(zero), lambda.Set(zero) } z.ModInverse(z, curve.P) lambda.Mul(lambda, z) lambda.Mod(lambda, curve.P) // x3 = λ² - x1 - x2 x3 = new(big.Int).Exp(lambda, two, curve.P) x3.Sub(x3, z.Add(x1, x2)) x3.Mod(x3, curve.P) // y = λ(x1 - x3) - y1 y3 = new(big.Int).Mul(lambda, z.Sub(x1, x3)) y3.Mod(y3, curve.P) y3.Sub(y3, y1) y3.Mod(y3, curve.P) return } // Double returns 2*(x,y) func (curve *SIEC255Params) Double(x1, y1 *big.Int) (x3, y3 *big.Int) { x3 = new(big.Int) y3 = new(big.Int) // TODO: optimize // λ = (3x1^2)/(2y1) lambda := new(big.Int).Mul(three, x3.Exp(x1, two, curve.P)) if y1.BitLen() == 0 { return x3.Set(zero), y3.Set(zero) } x3.Mul(two, y1) x3.ModInverse(x3, curve.P) lambda.Mul(lambda, x3) // x3 = λ² - x1 - x2 x3.Exp(lambda, two, curve.P) x3.Sub(x3, y3.Add(x1, x1)) x3.Mod(x3, curve.P) // y = λ(x1 - x3) - y1 y3.Mul(lambda, new(big.Int).Sub(x1, x3)) y3.Mod(y3, curve.P) y3.Sub(y3, y1) y3.Mod(y3, curve.P) return } var ( phiX, _ = new(big.Int).SetString("4000000000000000000000000200104000000000000000000004004101081000", 16) ) // Order 6 endomorphism. func (curve *SIEC255Params) phi(x1, y1 *big.Int) (x2, y2 *big.Int) { x2 = new(big.Int).Mul(phiX, x1) x2.Mod(x2, curve.P) y2 = new(big.Int).Neg(y1) y2.Mod(y2, curve.P) return } var ( endK, _ = new(big.Int).SetString("80000000000000000000000002001040", 16) ) func (curve *SIEC255Params) scalarMult2(x1, y1 *big.Int, k []byte) (x, y *big.Int) { z := new(big.Int).SetBytes(k) r := new(big.Int) z.QuoRem(z, endK, r) zb := z.Bytes() rb := r.Bytes() for len(zb) != len(rb) { if len(zb) < len(rb) { zb = append([]byte{0}, zb...) } else { rb = append([]byte{0}, rb...) } } x2, y2 := curve.phi(x1, y1) x3, y3 := curve.Add(x1, y1, x2, y2) x, y = new(big.Int), new(big.Int) for i := range zb { b1, b2 := zb[i], rb[i] for bitNum := 0; bitNum < 8; bitNum++ { x, y = curve.Double(x, y) if b1&0x80 == 0x80 && b2&0x80 == 0x80 { // if top bit set x, y = curve.Add(x, y, x3, y3) } else if b1&0x80 == 0x80 && b2&0x80 != 0x80 { x, y = curve.Add(x, y, x2, y2) } else if b1&0x80 != 0x80 && b2&0x80 == 0x80 { x, y = curve.Add(x, y, x1, y1) } else if b1&0x80 != 0x80 && b2&0x80 != 0x80 { // pass } b1 <<= 1 b2 <<= 1 } } return } // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. func (curve *SIEC255Params) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) { x, y = new(big.Int), new(big.Int) for _, b := range k { for bitNum := 0; bitNum < 8; bitNum++ { x, y = curve.Double(x, y) if b&0x80 == 0x80 { // if top bit set x, y = curve.Add(x1, y1, x, y) } b <<= 1 } } return x, y } // ScalarBaseMult returns k*G, where G is the base point of the group // and k is an integer in big-endian form. func (curve *SIEC255Params) ScalarBaseMult(k []byte) (x, y *big.Int) { return curve.ScalarMult(curve.Gx, curve.Gy, k) } // liftX returns a point on the curve (x,y) with the given x-value. // If there is more than one, it returns the one whose y-value // is smaller in the interval [0,p). If no such point exists, // then this function panics. func (curve *SIEC255Params) liftX(X *big.Int) (x, y *big.Int) { // y² = x³ + Ax + B x = new(big.Int).Set(X) y = new(big.Int) y.Exp(x, three, curve.P) y.Add(y, new(big.Int).Mul(x, curve.A)) y.Mod(y, curve.P) y.Add(y, curve.B) y.Mod(y, curve.P) y = y.ModSqrt(y, curve.P) if y == nil { panic(fmt.Sprintf("%d is not a point on the curve", X)) } if y.Cmp(new(big.Int).Sub(curve.P, y)) > 0 { y.Sub(curve.P, y) } return x, y } // Compress compresses a point down to 32 bytes. // The first 31 bytes (index 0,1,2, and most of 3) // represent the x coordinate (little endian). // The last bit of the fourth byte is the parity of the y coordinate. func (curve *SIEC255Params) Compress(x, y *big.Int) (c [32]byte) { b := reverse(x.Bytes()) for i := 0; i < int(math.Min(32, float64(len(b)))); i++ { c[i] = b[i] } c[31] = (c[31] & 0x7f) | (byte(y.Bit(0)) << 7) return } // Decompress uncompresses a point and returns an // actual (x,y) pair. func (curve *SIEC255Params) Decompress(c [32]byte) (x, y *big.Int) { s := c[31] >> 7 c[31] = c[31] & 0x7f x, y = curve.liftX(new(big.Int).SetBytes(reverse(c[:]))) if y.Bit(0) != uint(s) { y.Sub(curve.P, y) } return } var siec255 *SIEC255Params func initSIEC255() { siec255 = &SIEC255Params{Name: "SIEC255"} siec255.Gx = big.NewInt(5) siec255.Gy = big.NewInt(12) siec255.A = big.NewInt(0) siec255.B = big.NewInt(19) siec255.P, _ = new(big.Int).SetString("28948022309329048855892746252183396360603931420023084536990047309120118726721", 10) siec255.N, _ = new(big.Int).SetString("28948022309329048855892746252183396360263649053102146073526672701688283398081", 10) siec255.BitSize = 255 } // SIEC255 returns a Curve which implements SIEC255. func SIEC255() *SIEC255Params { initonce.Do(initSIEC255) return siec255 } var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f} // GenerateKey returns a public/private key pair. The private key is // generated using the given reader, which must return random data. // This is copied from https://golang.org/src/crypto/elliptic/elliptic.go?s=7368:7453#L266 func (curve *SIEC255Params) GenerateKey(rand io.Reader) (k []byte, x, y *big.Int, err error) { N := curve.N bitSize := N.BitLen() byteLen := (bitSize + 7) >> 3 k = make([]byte, byteLen) for x == nil { _, err = io.ReadFull(rand, k) if err != nil { return } // We have to mask off any excess bits in the case that the size of the // underlying field is not a whole number of bytes. k[0] &= mask[bitSize%8] // This is because, in tests, rand will return all zeros and we don't // want to get the point at infinity and loop forever. k[1] ^= 0x42 // If the scalar is out of range, sample another random number. if new(big.Int).SetBytes(k).Cmp(N) >= 0 { continue } x, y = curve.ScalarBaseMult(k) } return } func reverse(s []byte) []byte { for i, j := 0, len(s)-1; i < j; i, j = i+1, j-1 { s[i], s[j] = s[j], s[i] } return s }