├── README.md ├── .gitignore ├── LICENSE ├── vrf.go ├── edwards25519 ├── edwards25519_test.go ├── edwards25519.go └── const.go ├── vxeddsa_test.go ├── vxeddsa.go └── extra25519 ├── extra25519_test.go └── extra25519.go /README.md: -------------------------------------------------------------------------------- 1 | # vxeddsa 2 | VXEDDSA implementation in Go. 3 | 4 | This is a fork of [Coniks](https://github.com/coniks-sys/coniks-go/pull/167) implementation 5 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Binaries for programs and plugins 2 | *.exe 3 | *.dll 4 | *.so 5 | *.dylib 6 | 7 | # Test binary, build with `go test -c` 8 | *.test 9 | 10 | # Output of the go coverage tool, specifically when used with LiteIDE 11 | *.out 12 | 13 | # Project-local glide cache, RE: https://github.com/Masterminds/glide/issues/736 14 | .glide/ 15 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2018 Alexey Ermishkin 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /vrf.go: -------------------------------------------------------------------------------- 1 | // Package vrf implements a verifiable random function using the Edwards form 2 | // of Curve25519, SHA3 and the Elligator map. 3 | // 4 | // E is Curve25519 (in Edwards coordinates), h is SHA512. 5 | // f is the elligator map (bytes->E) that covers half of E. 6 | // 8 is the cofactor of E, the group order is 8*l for prime l. 7 | // Setup : the prover publicly commits to a public key (P : E) 8 | // H : names -> E 9 | // H(n) = f(h(n))^8 10 | // VRF : keys -> names -> vrfs 11 | // VRF_x(n) = h(n, H(n)^x)) 12 | // Prove : keys -> names -> proofs 13 | // Prove_x(n) = tuple(c=h(n, g^r, H(n)^r), t=r-c*x, ii=H(n)^x) 14 | // where r = h(x, n) is used as a source of randomness 15 | // Check : E -> names -> vrfs -> proofs -> bool 16 | // Check(P, n, vrf, (c,t,ii)) = vrf == h(n, ii) 17 | // && c == h(n, g^t*P^c, H(n)^t*ii^c) 18 | package vxeddsa 19 | 20 | import ( 21 | "bytes" 22 | "crypto/rand" 23 | "io" 24 | 25 | "github.com/Scratch-net/vxeddsa/edwards25519" 26 | ) 27 | 28 | const ( 29 | PublicKeySize = 32 30 | PrivateKeySize = 64 31 | Size = 32 32 | intermediateSize = 32 33 | ProofSize = 32 + 32 + intermediateSize 34 | ) 35 | 36 | // PrivateKey represents a Curve25519 private key. 37 | type PrivateKey [PrivateKeySize]byte 38 | 39 | // PublicKey represents a Curve25519 private key. 40 | type PublicKey [PublicKeySize]byte 41 | 42 | // GenerateKey creates a Curve25519 public/private key pair using rnd for 43 | // randomness. 44 | // Only the private key sk is returned (call sk.Public() the get the 45 | // corresponding public key). 46 | // If rnd is nil, crypto/rand is used. 47 | func GenerateKey(rnd io.Reader) (sk PrivateKey, err error) { 48 | if rnd == nil { 49 | rnd = rand.Reader 50 | } 51 | _, err = io.ReadFull(rnd, sk[:32]) 52 | if err != nil { 53 | return 54 | } 55 | sk[0] &= 248 56 | sk[31] &= 127 57 | sk[31] |= 64 58 | 59 | var ed edwards25519.ExtendedGroupElement 60 | var u edwards25519.FieldElement 61 | var x [32]byte 62 | copy(x[:], sk[:]) 63 | 64 | // cache the public-key: 65 | edwards25519.GeScalarMultBase(&ed, &x) 66 | edwards25519.GeToMontX(&u, &ed) 67 | var pkBytes [PublicKeySize]byte 68 | edwards25519.FeToBytes(&pkBytes, &u) 69 | copy(sk[32:], pkBytes[:]) 70 | 71 | return 72 | } 73 | 74 | // Public extracts the public VRF key from the underlying private-key 75 | func (sk PrivateKey) Public() (publicKey PublicKey) { 76 | publicKeyB := new([PublicKeySize]byte) 77 | copy(publicKeyB[:], sk[32:]) 78 | publicKey = PublicKey(*publicKeyB) 79 | return 80 | } 81 | 82 | // Compute generates the vrf value for the byte slice m using the 83 | // underlying private key sk. 84 | func (sk PrivateKey) Compute(m []byte) []byte { 85 | var a, aNeg, A [32]byte 86 | copy(a[:], sk[:32]) 87 | // copy(uB[:], sk[32:64]) 88 | 89 | // XXX use the cached public key instead: 90 | var x [32]byte 91 | copy(x[:], sk[:]) 92 | var edPubKey edwards25519.ExtendedGroupElement 93 | edwards25519.GeScalarMultBase(&edPubKey, &x) 94 | edPubKey.ToBytes(&A) 95 | 96 | // Force Edwards sign bit to zero 97 | //copy(A[:], []byte(pkBytes)) 98 | signBit := (A[31] & 0x80) >> 7 99 | copy(a[:], sk[:32]) 100 | edwards25519.ScNeg(&aNeg, &a) 101 | edwards25519.ScCMove(&a, &aNeg, int32(signBit)) 102 | A[31] &= 0x7F 103 | 104 | _, Vbytes := calculateBvAndV(a, A, m) 105 | vrfB := computeVrfFromV(*Vbytes) 106 | return vrfB[:] 107 | } 108 | 109 | // Sign returns the vrf value and a proof such that 110 | // Verify(m, vrf, proof) == true. The vrf value is the 111 | // same as returned by Compute(m). 112 | func (sk PrivateKey) Sign(m []byte) (signature []byte) { 113 | signature = sk.signInternal(m, nil) 114 | return 115 | } 116 | 117 | // Prove returns the vrf value and a proof such that Verify(pk, m, vrf, proof) 118 | // == true. The vrf value is the same as returned by Compute(m, sk). 119 | func (sk PrivateKey) Prove(m []byte) (vrf, proof []byte) { 120 | proof = sk.Sign(m) 121 | var V [32]byte 122 | copy(V[:], proof[:32]) 123 | v := computeVrfFromV(V) 124 | vrf = make([]byte, 32) 125 | copy(vrf, v[:]) 126 | return 127 | } 128 | 129 | // Verify returns true iff vrf=Compute(m) for the sk that 130 | // corresponds to pk. 131 | func (pk PublicKey) Verify(m, vrfBytes, signature []byte) bool { 132 | if len(vrfBytes) != Size { 133 | return false 134 | } 135 | if ok, vrf := pk.verifyInteral(m, signature); ok && 136 | bytes.Equal(vrfBytes, vrf[:]) { 137 | return true 138 | 139 | } 140 | return false 141 | } 142 | -------------------------------------------------------------------------------- /edwards25519/edwards25519_test.go: -------------------------------------------------------------------------------- 1 | // Copyright 2013 The Go Authors. All rights reserved. 2 | // Use of this source code is governed by a BSD-style 3 | // license that can be found in the LICENSE file. 4 | 5 | package edwards25519 6 | 7 | import ( 8 | "crypto/rand" 9 | "testing" 10 | ) 11 | 12 | func TestScNegMulAddIdentity(t *testing.T) { 13 | one := [32]byte{1} 14 | var seed [64]byte 15 | var a, minusA, x [32]byte 16 | 17 | rand.Reader.Read(seed[:]) 18 | ScReduce(&a, &seed) 19 | copy(minusA[:], a[:]) 20 | ScNeg(&minusA, &minusA) 21 | ScMulAdd(&x, &one, &a, &minusA) 22 | if x != [32]byte{} { 23 | t.Errorf("%x --neg--> %x --sum--> %x", a, minusA, x) 24 | } 25 | } 26 | 27 | func TestGeAddAgainstgeAdd(t *testing.T) { 28 | var x, y [32]byte 29 | rand.Reader.Read(x[:]) 30 | rand.Reader.Read(y[:]) 31 | var X, Y ExtendedGroupElement 32 | x[31] &= 127 33 | y[31] &= 127 34 | GeScalarMultBase(&X, &x) 35 | GeScalarMultBase(&Y, &y) 36 | 37 | var SBytesRef [32]byte 38 | var SRef ExtendedGroupElement 39 | var Ycached CachedGroupElement 40 | var SCompletedRef CompletedGroupElement 41 | Y.ToCached(&Ycached) 42 | geAdd(&SCompletedRef, &X, &Ycached) 43 | SCompletedRef.ToExtended(&SRef) 44 | SRef.ToBytes(&SBytesRef) 45 | 46 | var SBytes [32]byte 47 | var S ExtendedGroupElement 48 | GeAdd(&S, &X, &Y) 49 | S.ToBytes(&SBytes) 50 | 51 | if SBytes != SBytesRef { 52 | t.Errorf("GeAdd does not match geAdd: %x != %x", SBytes, SBytesRef) 53 | } 54 | } 55 | 56 | func TestGeScalarMultAgainstDoubleScalarMult(t *testing.T) { 57 | var zero [32]byte 58 | 59 | var x [32]byte 60 | rand.Reader.Read(x[:]) 61 | var X ExtendedGroupElement 62 | x[31] &= 127 63 | GeScalarMultBase(&X, &x) 64 | 65 | var a [32]byte 66 | rand.Reader.Read(a[:]) 67 | a[31] &= 127 68 | 69 | var ABytesRef [32]byte 70 | var Aref ProjectiveGroupElement 71 | GeDoubleScalarMultVartime(&Aref, &a, &X, &zero) 72 | Aref.ToBytes(&ABytesRef) 73 | 74 | var ABytes [32]byte 75 | var A ExtendedGroupElement 76 | GeScalarMult(&A, &a, &X) 77 | A.ToBytes(&ABytes) 78 | 79 | if ABytes != ABytesRef { 80 | t.Errorf("GeScalarMult does not match GeDoubleScalarMultVartime: %x != %x", ABytes, ABytesRef) 81 | } 82 | } 83 | 84 | func inc(b *[32]byte) { 85 | acc := uint(1) 86 | for i := 0; i < 32; i++ { 87 | acc += uint(b[i]) 88 | acc, b[i] = uint(acc>>8), byte(acc) 89 | } 90 | } 91 | 92 | func TestGeAddAgainstScalarMult(t *testing.T) { 93 | var x, s [32]byte 94 | rand.Reader.Read(x[:]) 95 | x[31] &= 127 96 | var X ExtendedGroupElement 97 | GeScalarMultBase(&X, &x) 98 | var A ExtendedGroupElement 99 | A.Zero() 100 | var ABytes, ABytesMult [32]byte 101 | A.ToBytes(&ABytes) 102 | var AMult ExtendedGroupElement 103 | GeScalarMult(&AMult, &s, &X) 104 | AMult.ToBytes(&ABytesMult) 105 | if ABytes != ABytesMult { 106 | t.Errorf("addition does not match multiplication (%x) -> %x != %x", s, ABytes, ABytesMult) 107 | } 108 | 109 | GeAdd(&A, &A, &X) 110 | inc(&s) 111 | } 112 | 113 | func TestGeAddAgainstDoubleScalarMult(t *testing.T) { 114 | var zero, x, s [32]byte 115 | rand.Reader.Read(x[:]) 116 | x[31] &= 127 117 | var X ExtendedGroupElement 118 | GeScalarMultBase(&X, &x) 119 | var A ExtendedGroupElement 120 | A.Zero() 121 | var ABytes, ABytesMult [32]byte 122 | A.ToBytes(&ABytes) 123 | var AMult ProjectiveGroupElement 124 | GeDoubleScalarMultVartime(&AMult, &s, &X, &zero) 125 | AMult.ToBytes(&ABytesMult) 126 | if ABytes != ABytesMult { 127 | t.Errorf("addition does not match multiplication (%x)", s) 128 | } 129 | 130 | GeAdd(&A, &A, &X) 131 | inc(&s) 132 | } 133 | 134 | func TestGeScalarMultiBaseScalarMultDH(t *testing.T) { 135 | var a, b [32]byte 136 | rand.Reader.Read(a[:]) 137 | rand.Reader.Read(b[:]) 138 | 139 | var A, B ExtendedGroupElement 140 | a[31] &= 127 141 | b[31] &= 127 142 | GeScalarMultBase(&A, &a) 143 | GeScalarMultBase(&B, &b) 144 | 145 | var kA, kB ExtendedGroupElement 146 | GeScalarMult(&kA, &a, &B) 147 | GeScalarMult(&kB, &b, &A) 148 | 149 | var ka, kb [32]byte 150 | kA.ToBytes(&ka) 151 | kB.ToBytes(&kb) 152 | 153 | if ka != kb { 154 | t.Fatal("DH shared secrets do not match") 155 | } 156 | } 157 | 158 | func TestGeScalarMultDH(t *testing.T) { 159 | var one [32]byte 160 | one[0] = 1 161 | var base ExtendedGroupElement 162 | GeScalarMultBase(&base, &one) 163 | 164 | var a, b [32]byte 165 | rand.Reader.Read(a[:]) 166 | rand.Reader.Read(b[:]) 167 | 168 | var A, B ExtendedGroupElement 169 | GeScalarMult(&A, &a, &base) 170 | GeScalarMult(&B, &b, &base) 171 | 172 | var kA, kB ExtendedGroupElement 173 | GeScalarMult(&kA, &a, &B) 174 | GeScalarMult(&kB, &b, &A) 175 | 176 | var ka, kb [32]byte 177 | kA.ToBytes(&ka) 178 | kB.ToBytes(&kb) 179 | 180 | if ka != kb { 181 | t.Fatal("DH shared secrets do not match") 182 | } 183 | 184 | var bytesA, bytesB [32]byte 185 | A.ToBytes(&bytesA) 186 | B.ToBytes(&bytesB) 187 | 188 | if bytesA == bytesB { 189 | t.Fatalf("DH public key collision: g^%x = g^%x = %x", a, b, A) 190 | } 191 | } 192 | 193 | func BenchmarkGeScalarMultBase(b *testing.B) { 194 | var s [32]byte 195 | rand.Reader.Read(s[:]) 196 | var P ExtendedGroupElement 197 | 198 | b.ResetTimer() 199 | for i := 0; i < b.N; i++ { 200 | GeScalarMultBase(&P, &s) 201 | } 202 | } 203 | 204 | func BenchmarkGeScalarMult(b *testing.B) { 205 | var s [32]byte 206 | rand.Reader.Read(s[:]) 207 | 208 | var P ExtendedGroupElement 209 | s[31] &= 127 210 | GeScalarMultBase(&P, &s) 211 | 212 | b.ResetTimer() 213 | for i := 0; i < b.N; i++ { 214 | GeScalarMult(&P, &s, &P) 215 | } 216 | } 217 | 218 | func BenchmarkGeDoubleScalarMultVartime(b *testing.B) { 219 | var s [32]byte 220 | rand.Reader.Read(s[:]) 221 | 222 | var P, Pout ExtendedGroupElement 223 | s[31] &= 127 224 | GeScalarMultBase(&P, &s) 225 | 226 | var out ProjectiveGroupElement 227 | 228 | b.ResetTimer() 229 | for i := 0; i < b.N; i++ { 230 | GeDoubleScalarMultVartime(&out, &s, &P, &[32]byte{}) 231 | out.ToExtended(&Pout) 232 | } 233 | } 234 | 235 | func BenchmarkGeAdd(b *testing.B) { 236 | var s [32]byte 237 | rand.Reader.Read(s[:]) 238 | 239 | var R, P ExtendedGroupElement 240 | s[31] &= 127 241 | GeScalarMultBase(&P, &s) 242 | R = P 243 | 244 | b.ResetTimer() 245 | for i := 0; i < b.N; i++ { 246 | GeAdd(&R, &R, &P) 247 | } 248 | } 249 | 250 | func BenchmarkGeDouble(b *testing.B) { 251 | var s [32]byte 252 | rand.Reader.Read(s[:]) 253 | 254 | var R, P ExtendedGroupElement 255 | s[31] &= 127 256 | GeScalarMultBase(&P, &s) 257 | R = P 258 | 259 | b.ResetTimer() 260 | for i := 0; i < b.N; i++ { 261 | GeDouble(&R, &P) 262 | } 263 | } 264 | -------------------------------------------------------------------------------- /vxeddsa_test.go: -------------------------------------------------------------------------------- 1 | package vxeddsa 2 | 3 | import ( 4 | "bytes" 5 | "crypto/sha512" 6 | "testing" 7 | 8 | "github.com/Scratch-net/vxeddsa/edwards25519" 9 | ) 10 | 11 | func TestVxed25519Vectors(t *testing.T) { 12 | signature_correct := [96]byte{ 13 | 0x23, 0xc6, 0xe5, 0x93, 0x3f, 0xcd, 0x56, 0x47, 14 | 0x7a, 0x86, 0xc9, 0x9b, 0x76, 0x2c, 0xb5, 0x24, 15 | 0xc3, 0xd6, 0x05, 0x55, 0x38, 0x83, 0x4d, 0x4f, 16 | 0x8d, 0xb8, 0xf0, 0x31, 0x07, 0xec, 0xeb, 0xa0, 17 | 0xa0, 0x01, 0x50, 0xb8, 0x4c, 0xbb, 0x8c, 0xcd, 18 | 0x23, 0xdc, 0x65, 0xfd, 0x0e, 0x81, 0xb2, 0x86, 19 | 0x06, 0xa5, 0x6b, 0x0c, 0x4f, 0x53, 0x6d, 0xc8, 20 | 0x8b, 0x8d, 0xc9, 0x04, 0x6e, 0x4a, 0xeb, 0x08, 21 | 0xce, 0x08, 0x71, 0xfc, 0xc7, 0x00, 0x09, 0xa4, 22 | 0xd6, 0xc0, 0xfd, 0x2d, 0x1a, 0xe5, 0xb6, 0xc0, 23 | 0x7c, 0xc7, 0x22, 0x3b, 0x69, 0x59, 0xa8, 0x26, 24 | 0x2b, 0x57, 0x78, 0xd5, 0x46, 0x0e, 0x0f, 0x05, 25 | } 26 | var privKey [32]byte 27 | var vrfOutPrev, vrfOut [32]byte 28 | privKey[8] = 189 29 | edwards25519.ScClamp(&privKey) 30 | msgLen := 200 31 | msg := make([]byte, msgLen) 32 | r := bytes.NewReader(privKey[:]) 33 | sk, _ := GenerateKey(r) 34 | 35 | random := make([]byte, 64) 36 | signature := sk.signInternal(msg, bytes.NewReader(random)) 37 | if !bytes.Equal(signature, signature_correct[:]) { 38 | t.Fatal("VXEdDSA sign failed") 39 | } 40 | var ok bool 41 | if ok, vrfOut = sk.Public().verifyInteral(msg, signature); !ok { 42 | t.Fatal("VXEdDSA verify #1 failed") 43 | } 44 | 45 | copy(vrfOutPrev[:], vrfOut[:]) 46 | signature[0] ^= 1 47 | if ok, _ := sk.Public().verifyInteral(msg, signature); ok { 48 | t.Fatal("VXEdDSA verify #2 should have failed!") 49 | } 50 | var sigPrev [96]byte 51 | copy(sigPrev[:], signature[:]) 52 | sigPrev[0] ^= 1 // undo prev disturbance 53 | 54 | random[0] ^= 1 55 | signature = sk.signInternal(msg, bytes.NewReader(random)) 56 | if ok, vrfOut = sk.Public().verifyInteral(msg, signature); !ok { 57 | t.Fatal("VXEdDSA verify #3 failed") 58 | } 59 | if !bytes.Equal(vrfOut[:], vrfOutPrev[:]) { 60 | t.Fatal("VXEdDSA VRF value has changed") 61 | } 62 | if bytes.Equal(signature[32:96], sigPrev[32:96]) { 63 | t.Fatal("VXEdDSA (h, s) changed") 64 | } 65 | } 66 | 67 | func TestVxed25519VectorsSlow(t *testing.T) { 68 | if testing.Short() { 69 | t.Skip("Skipped slow test in short mode") 70 | } 71 | signature_10k_correct := [96]byte{ 72 | 0xa1, 0x96, 0x96, 0xe5, 0x87, 0x3f, 0x6e, 0x5c, 73 | 0x2e, 0xd3, 0x73, 0xab, 0x04, 0x0c, 0x1f, 0x26, 74 | 0x3c, 0xca, 0x52, 0xc4, 0x7e, 0x49, 0xaa, 0xce, 75 | 0xb5, 0xd6, 0xa2, 0x29, 0x46, 0x3f, 0x1b, 0x54, 76 | 0x45, 0x94, 0x9b, 0x6c, 0x27, 0xf9, 0x2a, 0xed, 77 | 0x17, 0xa4, 0x72, 0xbf, 0x35, 0x37, 0xc1, 0x90, 78 | 0xac, 0xb3, 0xfd, 0x2d, 0xf1, 0x01, 0x05, 0xbe, 79 | 0x56, 0x5c, 0xaf, 0x63, 0x65, 0xad, 0x38, 0x04, 80 | 0x70, 0x53, 0xdf, 0x2b, 0xc1, 0x45, 0xc8, 0xee, 81 | 0x02, 0x0d, 0x2b, 0x22, 0x23, 0x7a, 0xbf, 0xfa, 82 | 0x43, 0x31, 0xb3, 0xac, 0x26, 0xd9, 0x76, 0xfc, 83 | 0xfe, 0x30, 0xa1, 0x7c, 0xce, 0x10, 0x67, 0x0e, 84 | } 85 | 86 | signature_100k_correct := [96]byte{ 87 | 0xc9, 0x11, 0x2b, 0x55, 0xfa, 0xc4, 0xb2, 0xfe, 88 | 0x00, 0x7d, 0xf6, 0x45, 0xcb, 0xd2, 0x73, 0xc9, 89 | 0x43, 0xba, 0x20, 0xf6, 0x9c, 0x18, 0x84, 0xef, 90 | 0x6c, 0x65, 0x7a, 0xdb, 0x49, 0xfc, 0x1e, 0xbe, 91 | 0x31, 0xb3, 0xe6, 0xa4, 0x68, 0x2f, 0xd0, 0x30, 92 | 0x81, 0xfc, 0x0d, 0xcd, 0x2d, 0x00, 0xab, 0xae, 93 | 0x9f, 0x08, 0xf0, 0x99, 0xff, 0x9f, 0xdc, 0x2d, 94 | 0x68, 0xd6, 0xe7, 0xe8, 0x44, 0x2a, 0x5b, 0x0e, 95 | 0x48, 0x67, 0xe2, 0x41, 0x4a, 0xd9, 0x0c, 0x2a, 96 | 0x2b, 0x4e, 0x66, 0x09, 0x87, 0xa0, 0x6b, 0x3b, 97 | 0xd1, 0xd9, 0xa3, 0xe3, 0xa5, 0x69, 0xed, 0xc1, 98 | 0x42, 0x03, 0x93, 0x0d, 0xbc, 0x7e, 0xe9, 0x08, 99 | } 100 | 101 | signature_1m_correct := [96]byte{ 102 | 0xf8, 0xb1, 0x20, 0xf2, 0x1e, 0x5c, 0xbf, 0x5f, 103 | 0xea, 0x07, 0xcb, 0xb5, 0x77, 0xb8, 0x03, 0xbc, 104 | 0xcb, 0x6d, 0xf1, 0xc1, 0xa5, 0x03, 0x05, 0x7b, 105 | 0x01, 0x63, 0x9b, 0xf9, 0xed, 0x3e, 0x57, 0x47, 106 | 0xd2, 0x5b, 0xf4, 0x7e, 0x7c, 0x45, 0xce, 0xfc, 107 | 0x06, 0xb3, 0xf4, 0x05, 0x81, 0x9f, 0x53, 0xb0, 108 | 0x18, 0xe3, 0xfa, 0xcb, 0xb2, 0x52, 0x3e, 0x57, 109 | 0xcb, 0x34, 0xcc, 0x81, 0x60, 0xb9, 0x0b, 0x04, 110 | 0x07, 0x79, 0xc0, 0x53, 0xad, 0xc4, 0x4b, 0xd0, 111 | 0xb5, 0x7d, 0x95, 0x4e, 0xbe, 0xa5, 0x75, 0x0c, 112 | 0xd4, 0xbf, 0xa7, 0xc0, 0xcf, 0xba, 0xe7, 0x7c, 113 | 0xe2, 0x90, 0xef, 0x61, 0xa9, 0x29, 0x66, 0x0d, 114 | } 115 | 116 | signature_10m_correct := [96]byte{ 117 | 0xf5, 0xa4, 0xbc, 0xec, 0xc3, 0x3d, 0xd0, 0x43, 118 | 0xd2, 0x81, 0x27, 0x9e, 0xf0, 0x4c, 0xbe, 0xf3, 119 | 0x77, 0x01, 0x56, 0x41, 0x0e, 0xff, 0x0c, 0xb9, 120 | 0x66, 0xec, 0x4d, 0xe0, 0xb7, 0x25, 0x63, 0x6b, 121 | 0x5c, 0x08, 0x39, 0x80, 0x4e, 0x37, 0x1b, 0x2c, 122 | 0x46, 0x6f, 0x86, 0x99, 0x1c, 0x4e, 0x31, 0x60, 123 | 0xdb, 0x4c, 0xfe, 0xc5, 0xa2, 0x4d, 0x71, 0x2b, 124 | 0xd6, 0xd0, 0xc3, 0x98, 0x88, 0xdb, 0x0e, 0x0c, 125 | 0x68, 0x4a, 0xd3, 0xc7, 0x56, 0xac, 0x8d, 0x95, 126 | 0x7b, 0xbd, 0x99, 0x50, 0xe8, 0xd3, 0xea, 0xf3, 127 | 0x7b, 0x26, 0xf2, 0xa2, 0x2b, 0x02, 0x58, 0xca, 128 | 0xbd, 0x2c, 0x2b, 0xf7, 0x77, 0x58, 0xfe, 0x09, 129 | } 130 | 131 | const msgLen = 200 132 | var privateKey [32]byte 133 | msg := make([]byte, msgLen) 134 | var random [64]byte 135 | signature := bytes.Repeat([]byte{3}, 96) 136 | 137 | t.Log("Pseudorandom VXEdDSA...\n") 138 | 139 | // up to 10000000 iterations (super slow ... ) 140 | const iterations = 100000 141 | for count := 1; count <= iterations; count++ { 142 | b := sha512.Sum512(signature[:96]) 143 | copy(privateKey[:], b[:32]) 144 | b = sha512.Sum512(privateKey[:32]) 145 | copy(random[:], b[:64]) 146 | 147 | edwards25519.ScClamp(&privateKey) 148 | sk, err := GenerateKey(bytes.NewReader(privateKey[:])) 149 | if err != nil { 150 | t.Fatalf("Couldn't generate key in %d\n", count) 151 | } 152 | pk := sk.Public() 153 | signature = sk.signInternal(msg[:], bytes.NewReader(random[:])) 154 | if ok, _ := pk.verifyInteral(msg[:], signature); !ok { 155 | t.Fatalf("VXEdDSA verify failure #1 %d\n", count) 156 | } 157 | if (b[63] & 1) == 1 { 158 | signature[count%96] ^= 1 159 | } else { 160 | msg[count%msgLen] ^= 1 161 | } 162 | 163 | if count == 10000 { 164 | if !bytes.Equal(signature, signature_10k_correct[:]) { 165 | t.Errorf("VXEDDSA 10K doesn't match %d\n", count) 166 | } 167 | } 168 | if count == 100000 { 169 | if !bytes.Equal(signature, signature_100k_correct[:]) { 170 | t.Errorf("VXEDDSA 100K doesn't match %d\n", count) 171 | } 172 | } 173 | if count == 1000000 { 174 | if !bytes.Equal(signature, signature_1m_correct[:]) { 175 | t.Errorf("VXEDDSA 1m doesn't match %d\n", count) 176 | } 177 | } 178 | if count == 10000000 { 179 | if !bytes.Equal(signature, signature_10m_correct[:]) { 180 | t.Errorf("VXEDDSA 10m doesn't match %d\n", count) 181 | } 182 | } 183 | } 184 | } 185 | -------------------------------------------------------------------------------- /vxeddsa.go: -------------------------------------------------------------------------------- 1 | package vxeddsa 2 | 3 | import ( 4 | "bytes" 5 | "crypto/rand" 6 | "crypto/sha512" 7 | "io" 8 | 9 | "github.com/Scratch-net/vxeddsa/edwards25519" 10 | "github.com/Scratch-net/vxeddsa/extra25519" 11 | ) 12 | 13 | // This code is a port of the public domain, VXEdDSA C implementation by Trevor 14 | // Perrin / Open Whisper Systems. Specification: 15 | // https://whispersystems.org/docs/specifications/xeddsa/#vxeddsa 16 | 17 | func calculateBv(A [32]byte, msg []byte) (Bv *edwards25519.ExtendedGroupElement) { 18 | /* Calculate SHA512(label(2) || A || msg) */ 19 | buf := make([]byte, 32) 20 | buf[0] = 0xFD 21 | for count := 1; count < 32; count++ { 22 | buf[count] = 0xFF 23 | } 24 | buffer := bytes.NewBuffer(buf) 25 | buffer.Write(A[:]) 26 | buffer.Write(msg) 27 | Bv = extra25519.HashToPoint(buffer.Bytes()) 28 | 29 | return 30 | } 31 | 32 | func (sk PrivateKey) signInternal(m []byte, randr io.Reader) (signature []byte) { 33 | if randr == nil { 34 | randr = rand.Reader 35 | } 36 | 37 | // TODO check for maxMsgLen 38 | var a, aNeg, rB [32]byte 39 | var R edwards25519.ExtendedGroupElement 40 | var x [32]byte 41 | copy(x[:], sk[:]) 42 | var edPubKey edwards25519.ExtendedGroupElement 43 | edwards25519.GeScalarMultBase(&edPubKey, &x) 44 | var A [32]byte 45 | edPubKey.ToBytes(&A) 46 | 47 | // Force Edwards sign bit to zero 48 | signBit := (A[31] & 0x80) >> 7 49 | copy(a[:], sk[:32]) 50 | edwards25519.ScNeg(&aNeg, &a) 51 | edwards25519.ScCMove(&a, &aNeg, int32(signBit)) 52 | 53 | A[31] &= 0x7F 54 | Bv, V := calculateBvAndV(a, A, m) 55 | 56 | /* r = SHA512(label(3) || a || V || random(64)) */ 57 | var Rv edwards25519.ExtendedGroupElement 58 | buf := make([]byte, 160) 59 | buf[0] = 0xFC 60 | for i := 1; i < 32; i++ { 61 | buf[i] = 0xFF 62 | } 63 | copy(buf[32:], a[:]) 64 | copy(buf[64:], V[:]) 65 | random := make([]byte, 64) 66 | if _, err := randr.Read(random); err != nil { 67 | panic("Couldn't read from random") 68 | } 69 | copy(buf[96:], random) 70 | 71 | rH := sha512.Sum512(buf[:160]) 72 | 73 | edwards25519.ScReduce(&rB, &rH) 74 | edwards25519.GeScalarMultBase(&R, &rB) 75 | edwards25519.GeScalarMult(&Rv, &rB, Bv) 76 | 77 | /* h = SHA512(label(4) || A || V || R || Rv || M) */ 78 | var Rb [32]byte 79 | R.ToBytes(&Rb) 80 | var RvB [32]byte 81 | Rv.ToBytes(&RvB) 82 | buf = append(buf, m...) 83 | buf[0] = 0xFB 84 | for i := 1; i < 32; i++ { 85 | buf[i] = 0xFF 86 | } 87 | copy(buf[32:], A[:]) 88 | copy(buf[64:], V[:]) 89 | copy(buf[96:], Rb[:]) 90 | copy(buf[128:], RvB[:]) 91 | //copy(buf[:160], m) 92 | hB := sha512.Sum512(buf[:160+len(m)]) 93 | var h [32]byte 94 | edwards25519.ScReduce(&h, &hB) 95 | var s [32]byte 96 | edwards25519.ScMulAdd(&s, &h, &a, &rB) 97 | 98 | signature = make([]byte, ProofSize) 99 | copy(signature[:32], V[:]) 100 | copy(signature[32:64], h[:]) 101 | copy(signature[64:96], s[:]) 102 | 103 | return 104 | } 105 | 106 | func (pkB PublicKey) verifyInteral(m, signature []byte) (bool, [32]byte) { 107 | var vrf [32]byte 108 | if len(signature) != ProofSize || len(pkB) != PublicKeySize { 109 | return false, vrf 110 | } 111 | // TODO check for max. message length 112 | var u edwards25519.FieldElement 113 | pubKey := [32]byte(pkB) 114 | edwards25519.FeFromBytes(&u, &pubKey) 115 | var strict [32]byte 116 | edwards25519.FeToBytes(&strict, &u) 117 | if !(edwards25519.FeCompare(strict, pubKey) == 0) { 118 | return false, vrf 119 | } 120 | var y edwards25519.FieldElement 121 | extra25519.FeMontgomeryXToEdwardsY(&y, &u) 122 | var edPubKey [32]byte 123 | edwards25519.FeToBytes(&edPubKey, &y) 124 | Bv := calculateBv(edPubKey, m) 125 | 126 | // verifybuf = V || h || s || m 127 | verifBuf := make([]byte, len(m)+160) 128 | copy(verifBuf, signature[:96]) 129 | copy(verifBuf[96:], m) 130 | 131 | if verifBuf[63]&224 == 1 { 132 | return false, vrf 133 | } 134 | if verifBuf[95]&224 == 1 { 135 | return false, vrf 136 | } 137 | 138 | // Load -A: 139 | var minusA edwards25519.ExtendedGroupElement 140 | edwards25519.FeFromBytes(&minusA.Y, &edPubKey) 141 | if !minusA.FromParityAndY((edPubKey[31]>>7)^0x01, &minusA.Y) { 142 | return false, vrf 143 | } 144 | 145 | // Load -V 146 | var minusV edwards25519.ExtendedGroupElement 147 | var Vb [32]byte 148 | copy(Vb[:], signature[:32]) 149 | edwards25519.FeFromBytes(&minusV.Y, &Vb) 150 | if !minusV.FromParityAndY((Vb[31]>>7)^0x01, &minusV.Y) { 151 | return false, vrf 152 | } 153 | 154 | // Load h, s 155 | var h, s [32]byte 156 | copy(h[:], verifBuf[32:64]) 157 | copy(s[:], verifBuf[64:96]) 158 | if h[31]&224 == 1 { 159 | return false, vrf 160 | } /* strict parsing of h */ 161 | if s[31]&224 == 1 { 162 | return false, vrf 163 | } /* strict parsing of s */ 164 | 165 | var A, cA, V, cV edwards25519.ExtendedGroupElement 166 | edwards25519.GeNeg(&A, minusA) 167 | edwards25519.GeNeg(&V, minusV) 168 | 169 | edwards25519.GeDouble(&cA, &A) 170 | edwards25519.GeDouble(&cA, &cA) 171 | edwards25519.GeDouble(&cA, &cA) 172 | 173 | edwards25519.GeDouble(&cV, &V) 174 | edwards25519.GeDouble(&cV, &cV) 175 | edwards25519.GeDouble(&cV, &cV) 176 | 177 | if edwards25519.GeIsNeutral(&cA) || edwards25519.GeIsNeutral(&cV) || 178 | edwards25519.GeIsNeutral(Bv) { 179 | return false, vrf 180 | } 181 | 182 | // R = (s*B) + (h * -A)) 183 | var R edwards25519.ProjectiveGroupElement 184 | edwards25519.GeDoubleScalarMultVartime(&R, &h, &minusA, &s) 185 | 186 | // s * Bv 187 | var sBv edwards25519.ExtendedGroupElement 188 | edwards25519.GeScalarMult(&sBv, &s, Bv) 189 | 190 | // h * -V 191 | var hMinusV edwards25519.ExtendedGroupElement 192 | edwards25519.GeScalarMult(&hMinusV, &h, &minusV) 193 | 194 | // Rv = (sc * Bv) + (hc * (-V)) 195 | var Rv edwards25519.ExtendedGroupElement 196 | edwards25519.GeAdd(&Rv, &sBv, &hMinusV) 197 | 198 | // Check h == SHA512(label(4) || A || V || R || Rv || M) 199 | var VBytes, RBytes, RVBytes [32]byte 200 | V.ToBytes(&VBytes) 201 | R.ToBytes(&RBytes) 202 | Rv.ToBytes(&RVBytes) 203 | 204 | vrfBuf := make([]byte, 160+len(m)) 205 | vrfBuf[0] = 0xFB // label 4 206 | for count := 1; count < 32; count++ { 207 | vrfBuf[count] = 0xFF 208 | } 209 | 210 | //copy(vrfBuf, vrfBuf)) 211 | copy(vrfBuf[32:], edPubKey[:]) 212 | copy(vrfBuf[64:], VBytes[:]) 213 | copy(vrfBuf[96:], RBytes[:]) 214 | copy(vrfBuf[128:], RVBytes[:]) 215 | copy(vrfBuf[160:], verifBuf[96:96+len(m)]) 216 | 217 | hCheck := sha512.Sum512(vrfBuf[:160+len(m)]) 218 | 219 | var hCheckReduced [32]byte 220 | edwards25519.ScReduce(&hCheckReduced, &hCheck) 221 | 222 | if edwards25519.FeCompare(hCheckReduced, h) == 0 { 223 | // compute VRF from cV: 224 | var cVBytes [32]byte 225 | cV.ToBytes(&cVBytes) 226 | vrfBuf[0] = 0xFA // label 5 227 | copy(vrfBuf[32:], cVBytes[:]) 228 | vrfOutput := sha512.Sum512(vrfBuf[:64]) 229 | copy(vrf[:], vrfOutput[:32]) 230 | // vrf = cV || hash_5(cV) (mod 2^b) 231 | return true, vrf 232 | } 233 | 234 | return false, vrf 235 | } 236 | 237 | func calculateBvAndV(a, Abytes [32]byte, 238 | msg []byte) (Bv *edwards25519.ExtendedGroupElement, V *[32]byte) { 239 | p3 := edwards25519.ExtendedGroupElement{} 240 | 241 | Bv = calculateBv(Abytes, msg) 242 | edwards25519.GeScalarMult(&p3, &a, Bv) 243 | V = new([32]byte) 244 | p3.ToBytes(V) 245 | return 246 | } 247 | 248 | func computeVrfFromV(Vbytes [32]byte) (vrf [32]byte) { 249 | var V, cV edwards25519.ExtendedGroupElement 250 | V.FromBytes(&Vbytes) 251 | 252 | edwards25519.GeDouble(&cV, &V) 253 | edwards25519.GeDouble(&cV, &cV) 254 | edwards25519.GeDouble(&cV, &cV) 255 | buffer := make([]byte, 32) 256 | buffer[0] = 0xFA // label 5 257 | for count := 1; count < 32; count++ { 258 | buffer[count] = 0xFF 259 | } 260 | var cVBytes [32]byte 261 | cV.ToBytes(&cVBytes) 262 | buf := bytes.NewBuffer(buffer) 263 | buf.Write(cVBytes[:]) 264 | 265 | hash := sha512.Sum512(buf.Bytes()) 266 | copy(vrf[:], hash[:32]) 267 | return 268 | } 269 | -------------------------------------------------------------------------------- /extra25519/extra25519_test.go: -------------------------------------------------------------------------------- 1 | // Copyright 2013 The Go Authors. All rights reserved. 2 | // Use of this source code is governed by a BSD-style 3 | // license that can be found in the LICENSE file. 4 | 5 | package extra25519 6 | 7 | import ( 8 | "bytes" 9 | "crypto/rand" 10 | "crypto/sha512" 11 | "encoding/hex" 12 | "fmt" 13 | "testing" 14 | 15 | "github.com/Scratch-net/vxeddsa/edwards25519" 16 | "golang.org/x/crypto/curve25519" 17 | "golang.org/x/crypto/ed25519" 18 | ) 19 | 20 | func TestCurve25519Conversion(t *testing.T) { 21 | public, private, _ := ed25519.GenerateKey(rand.Reader) 22 | var pubBytes [32]byte 23 | copy(pubBytes[:], public) 24 | var privBytes [64]byte 25 | copy(privBytes[:], private) 26 | 27 | var curve25519Public, curve25519Public2, curve25519Private [32]byte 28 | PrivateKeyToCurve25519(&curve25519Private, &privBytes) 29 | curve25519.ScalarBaseMult(&curve25519Public, &curve25519Private) 30 | 31 | if !PublicKeyToCurve25519(&curve25519Public2, &pubBytes) { 32 | t.Fatalf("PublicKeyToCurve25519 failed") 33 | } 34 | 35 | if !bytes.Equal(curve25519Public[:], curve25519Public2[:]) { 36 | t.Errorf("Values didn't match: curve25519 produced %x, conversion produced %x", curve25519Public[:], curve25519Public2[:]) 37 | } 38 | } 39 | 40 | func TestHashNoCollisions(t *testing.T) { 41 | type intpair struct { 42 | i int 43 | j uint 44 | } 45 | rainbow := make(map[[32]byte]intpair) 46 | N := 25 47 | if testing.Short() { 48 | N = 3 49 | } 50 | var h [32]byte 51 | // NOTE: hash values 0b100000000000... and 0b00000000000... both map to 52 | // the identity. this is a core part of the elligator function and not a 53 | // collision we need to worry about because an attacker would need to find 54 | // the preimages of these hashes to exploit it. 55 | h[0] = 1 56 | for i := 0; i < N; i++ { 57 | for j := uint(0); j < 257; j++ { 58 | if j < 256 { 59 | h[j>>3] ^= byte(1) << (j & 7) 60 | } 61 | 62 | var P edwards25519.ExtendedGroupElement 63 | HashToEdwards(&P, &h) 64 | var p [32]byte 65 | P.ToBytes(&p) 66 | if c, ok := rainbow[p]; ok { 67 | t.Fatalf("found collision: (%d, %d) and (%d, %d)", i, j, c.i, c.j) 68 | } 69 | rainbow[p] = intpair{i, j} 70 | 71 | if j < 256 { 72 | h[j>>3] ^= byte(1) << (j & 7) 73 | } 74 | } 75 | hh := sha512.Sum512(h[:]) // this package already imports sha512 76 | copy(h[:], hh[:]) 77 | } 78 | } 79 | 80 | func TestElligator(t *testing.T) { 81 | var publicKey, publicKey2, publicKey3, representative, privateKey [32]byte 82 | 83 | for i := 0; i < 1000; i++ { 84 | rand.Reader.Read(privateKey[:]) 85 | 86 | if !ScalarBaseMult(&publicKey, &representative, &privateKey) { 87 | continue 88 | } 89 | RepresentativeToPublicKey(&publicKey2, &representative) 90 | if !bytes.Equal(publicKey[:], publicKey2[:]) { 91 | t.Fatal("The resulting public key doesn't match the initial one.") 92 | } 93 | 94 | curve25519.ScalarBaseMult(&publicKey3, &privateKey) 95 | if !bytes.Equal(publicKey[:], publicKey3[:]) { 96 | t.Fatal("The public key doesn't match the value that curve25519 produced.") 97 | } 98 | } 99 | } 100 | 101 | func BenchmarkKeyGeneration(b *testing.B) { 102 | var publicKey, representative, privateKey [32]byte 103 | 104 | // Find the private key that results in a point that's in the image of the map. 105 | for { 106 | rand.Reader.Read(privateKey[:]) 107 | if ScalarBaseMult(&publicKey, &representative, &privateKey) { 108 | break 109 | } 110 | } 111 | 112 | b.ResetTimer() 113 | for i := 0; i < b.N; i++ { 114 | ScalarBaseMult(&publicKey, &representative, &privateKey) 115 | } 116 | } 117 | 118 | func BenchmarkMap(b *testing.B) { 119 | var publicKey, representative [32]byte 120 | rand.Reader.Read(representative[:]) 121 | 122 | b.ResetTimer() 123 | for i := 0; i < b.N; i++ { 124 | RepresentativeToPublicKey(&publicKey, &representative) 125 | } 126 | } 127 | 128 | // copied test-vectors from: 129 | // https://github.com/WhisperSystems/curve25519-java/blob/master/android/jni/ed25519/tests/tests.c 130 | var sha512_correct_output = [64]byte{ 131 | 0x8E, 0x95, 0x9B, 0x75, 0xDA, 0xE3, 0x13, 0xDA, 132 | 0x8C, 0xF4, 0xF7, 0x28, 0x14, 0xFC, 0x14, 0x3F, 133 | 0x8F, 0x77, 0x79, 0xC6, 0xEB, 0x9F, 0x7F, 0xA1, 134 | 0x72, 0x99, 0xAE, 0xAD, 0xB6, 0x88, 0x90, 0x18, 135 | 0x50, 0x1D, 0x28, 0x9E, 0x49, 0x00, 0xF7, 0xE4, 136 | 0x33, 0x1B, 0x99, 0xDE, 0xC4, 0xB5, 0x43, 0x3A, 137 | 0xC7, 0xD3, 0x29, 0xEE, 0xB6, 0xDD, 0x26, 0x54, 138 | 0x5E, 0x96, 0xE5, 0x5B, 0x87, 0x4B, 0xE9, 0x09, 139 | } 140 | 141 | var elligator_correct_output = [32]byte{ 142 | 0x5f, 0x35, 0x20, 0x00, 0x1c, 0x6c, 0x99, 0x36, 143 | 0xa3, 0x12, 0x06, 0xaf, 0xe7, 0xc7, 0xac, 0x22, 144 | 0x4e, 0x88, 0x61, 0x61, 0x9b, 0xf9, 0x88, 0x72, 145 | 0x44, 0x49, 0x15, 0x89, 0x9d, 0x95, 0xf4, 0x6e, 146 | } 147 | 148 | var hashtopoint_correct_output1 = [32]byte{ 149 | 0xce, 0x89, 0x9f, 0xb2, 0x8f, 0xf7, 0x20, 0x91, 150 | 0x5e, 0x14, 0xf5, 0xb7, 0x99, 0x08, 0xab, 0x17, 151 | 0xaa, 0x2e, 0xe2, 0x45, 0xb4, 0xfc, 0x2b, 0xf6, 152 | 0x06, 0x36, 0x29, 0x40, 0xed, 0x7d, 0xe7, 0xed, 153 | } 154 | 155 | var hashtopoint_correct_output2 = [32]byte{ 156 | 0xa0, 0x35, 0xbb, 0xa9, 0x4d, 0x30, 0x55, 0x33, 157 | 0x0d, 0xce, 0xc2, 0x7f, 0x83, 0xde, 0x79, 0xd0, 158 | 0x89, 0x67, 0x72, 0x4c, 0x07, 0x8d, 0x68, 0x9d, 159 | 0x61, 0x52, 0x1d, 0xf9, 0x2c, 0x5c, 0xba, 0x77, 160 | } 161 | 162 | var calculatev_correct_output = [32]byte{ 163 | 0x1b, 0x77, 0xb5, 0xa0, 0x44, 0x84, 0x7e, 0xb9, 164 | 0x23, 0xd7, 0x93, 0x18, 0xce, 0xc2, 0xc5, 0xe2, 165 | 0x84, 0xd5, 0x79, 0x6f, 0x65, 0x63, 0x1b, 0x60, 166 | 0x9b, 0xf1, 0xf8, 0xce, 0x88, 0x0b, 0x50, 0x9c, 167 | } 168 | 169 | func TestElligatorFast(t *testing.T) { 170 | var b [32]byte 171 | for count := 0; count < 32; count++ { 172 | b[count] = byte(count) 173 | } 174 | var in edwards25519.FieldElement 175 | var out edwards25519.FieldElement 176 | edwards25519.FeFromBytes(&in, &b) 177 | 178 | elligator(&out, in) 179 | 180 | var b2 [32]byte 181 | edwards25519.FeToBytes(&b2, &out) 182 | //TEST("Elligator vector", memcmp(bytes, elligator_correct_output, 32) == 0); 183 | if !bytes.Equal(b2[:], elligator_correct_output[:]) { 184 | t.Fatal("Elligator test vector faile") 185 | } 186 | 187 | /* Elligator(0) == 0 test */ 188 | edwards25519.FeZero(&in) 189 | elligator(&out, in) 190 | 191 | var bi, bo [32]byte 192 | edwards25519.FeToBytes(&bi, &in) 193 | edwards25519.FeToBytes(&bo, &out) 194 | 195 | if !bytes.Equal(bi[:], bo[:]) { 196 | t.Error("Elligator(0) != 0") 197 | } 198 | 199 | /* ge_montx_to_p3(0) -> order2 point test */ 200 | var one, negone, zero edwards25519.FieldElement 201 | edwards25519.FeOne(&one) 202 | edwards25519.FeZero(&zero) 203 | edwards25519.FeSub(&negone, &zero, &one) 204 | var p3 edwards25519.ExtendedGroupElement 205 | GeMontXtoExtendedGroupElement(&p3, zero, 0) 206 | if !(edwards25519.FeIsequal(p3.X, zero) == 1 && 207 | edwards25519.FeIsequal(p3.Y, negone) == 1 && 208 | edwards25519.FeIsequal(p3.Z, one) == 1 && 209 | edwards25519.FeIsequal(p3.T, zero) == 1) { 210 | t.Fatal("ge_montx_to_p3(0) isn't a order 2 point") 211 | } 212 | 213 | /* Hash to point vector test */ 214 | var htp [32]byte 215 | for count := 0; count < 32; count++ { 216 | htp[count] = byte(count) 217 | } 218 | 219 | p3 = *HashToPoint(htp[:]) 220 | 221 | p3.ToBytes(&htp) 222 | if !bytes.Equal(htp[:], hashtopoint_correct_output1[:]) { 223 | fmt.Println(hex.Dump(htp[:])) 224 | fmt.Println(hex.Dump(hashtopoint_correct_output1[:])) 225 | t.Fatal("hash_to_point #1 failed") 226 | } 227 | 228 | for count := 0; count < 32; count++ { 229 | htp[count] = byte(count + 1) 230 | } 231 | 232 | p3 = *HashToPoint(htp[:]) 233 | p3.ToBytes(&htp) 234 | //TEST("hash_to_point #2", memcmp(htp, hashtopoint_correct_output2, 32) == 0); 235 | if !bytes.Equal(htp[:], hashtopoint_correct_output2[:]) { 236 | fmt.Println(hex.Dump(htp[:])) 237 | fmt.Println(hex.Dump(hashtopoint_correct_output2[:])) 238 | t.Fatal("hash_to_point #2 failed") 239 | } 240 | 241 | /* TODO move to VRF package: calculate_V vector test */ 242 | //Bv := edwards25519.ExtendedGroupElement{} 243 | //var V [32]byte 244 | //Vbuf := make([]byte, 200) 245 | //var a [32]byte 246 | //var A [32]byte 247 | //Vmsg := []byte{0x00, 0x01, 0x02} 248 | // 249 | //for count:=0; count < 32; count++ { 250 | // a[count] = byte(8 + count) 251 | // A[count] = byte(9 + count) 252 | //} 253 | //// sc_clamp(a): 254 | //a[0] &= 248; a[31] &= 127; a[31] |= 64 255 | //_ = calculateBvAndV(&V, Vbuf, a, A, Vmsg) 256 | //if !bytes.Equal(V[:], calculatev_correct_output[:]) { 257 | // fmt.Println(hex.Dump(V[:])) 258 | // fmt.Println(hex.Dump(calculatev_correct_output[:])) 259 | // t.Fatal("calculate_Bv_and_V vector failed") 260 | //} 261 | } 262 | -------------------------------------------------------------------------------- /extra25519/extra25519.go: -------------------------------------------------------------------------------- 1 | // Copyright 2013 The Go Authors. All rights reserved. 2 | // Use of this source code is governed by a BSD-style 3 | // license that can be found in the LICENSE file. 4 | 5 | package extra25519 6 | 7 | import ( 8 | "crypto/sha512" 9 | 10 | "github.com/Scratch-net/vxeddsa/edwards25519" 11 | ) 12 | 13 | // PrivateKeyToCurve25519 converts an ed25519 private key into a corresponding 14 | // curve25519 private key such that the resulting curve25519 public key will 15 | // equal the result from PublicKeyToCurve25519. 16 | func PrivateKeyToCurve25519(curve25519Private *[32]byte, privateKey *[64]byte) { 17 | h := sha512.New() 18 | h.Write(privateKey[:32]) 19 | digest := h.Sum(nil) 20 | 21 | digest[0] &= 248 22 | digest[31] &= 127 23 | digest[31] |= 64 24 | 25 | copy(curve25519Private[:], digest) 26 | } 27 | 28 | func edwardsToMontgomeryX(outX, y *edwards25519.FieldElement) { 29 | // We only need the x-coordinate of the curve25519 point, which I'll 30 | // call u. The isomorphism is u=(y+1)/(1-y), since y=Y/Z, this gives 31 | // u=(Y+Z)/(Z-Y). We know that Z=1, thus u=(Y+1)/(1-Y). 32 | var oneMinusY edwards25519.FieldElement 33 | edwards25519.FeOne(&oneMinusY) 34 | edwards25519.FeSub(&oneMinusY, &oneMinusY, y) 35 | edwards25519.FeInvert(&oneMinusY, &oneMinusY) 36 | 37 | edwards25519.FeOne(outX) 38 | edwards25519.FeAdd(outX, outX, y) 39 | 40 | edwards25519.FeMul(outX, outX, &oneMinusY) 41 | } 42 | 43 | // PublicKeyToCurve25519 converts an Ed25519 public key into the curve25519 44 | // public key that would be generated from the same private key. 45 | func PublicKeyToCurve25519(curve25519Public *[32]byte, publicKey *[32]byte) bool { 46 | var A edwards25519.ExtendedGroupElement 47 | if !A.FromBytes(publicKey) { 48 | return false 49 | } 50 | 51 | // A.Z = 1 as a postcondition of FromBytes. 52 | var x edwards25519.FieldElement 53 | edwardsToMontgomeryX(&x, &A.Y) 54 | edwards25519.FeToBytes(curve25519Public, &x) 55 | return true 56 | } 57 | 58 | // sqrtMinusA is sqrt(-486662) 59 | var sqrtMinusA = edwards25519.FieldElement{ 60 | 12222970, 8312128, 11511410, -9067497, 15300785, 241793, -25456130, -14121551, 12187136, -3972024, 61 | } 62 | 63 | // sqrtMinusHalf is sqrt(-1/2) 64 | var sqrtMinusHalf = edwards25519.FieldElement{ 65 | -17256545, 3971863, 28865457, -1750208, 27359696, -16640980, 12573105, 1002827, -163343, 11073975, 66 | } 67 | 68 | // halfQMinus1Bytes is (2^255-20)/2 expressed in little endian form. 69 | var halfQMinus1Bytes = [32]byte{ 70 | 0xf6, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x3f, 71 | } 72 | 73 | // feBytesLess returns one if a <= b and zero otherwise. 74 | func feBytesLE(a, b *[32]byte) int32 { 75 | equalSoFar := int32(-1) 76 | greater := int32(0) 77 | 78 | for i := uint(31); i < 32; i-- { 79 | x := int32(a[i]) 80 | y := int32(b[i]) 81 | 82 | greater = (^equalSoFar & greater) | (equalSoFar & ((x - y) >> 31)) 83 | equalSoFar = equalSoFar & (((x ^ y) - 1) >> 31) 84 | } 85 | 86 | return int32(^equalSoFar & 1 & greater) 87 | } 88 | 89 | // ScalarBaseMult computes a curve25519 public key from a private key and also 90 | // a uniform representative for that public key. Note that this function will 91 | // fail and return false for about half of private keys. 92 | // See http://elligator.cr.yp.to/elligator-20130828.pdf. 93 | func ScalarBaseMult(publicKey, representative, privateKey *[32]byte) bool { 94 | var maskedPrivateKey [32]byte 95 | copy(maskedPrivateKey[:], privateKey[:]) 96 | 97 | maskedPrivateKey[0] &= 248 98 | maskedPrivateKey[31] &= 127 99 | maskedPrivateKey[31] |= 64 100 | 101 | var A edwards25519.ExtendedGroupElement 102 | edwards25519.GeScalarMultBase(&A, &maskedPrivateKey) 103 | 104 | var inv1 edwards25519.FieldElement 105 | edwards25519.FeSub(&inv1, &A.Z, &A.Y) 106 | edwards25519.FeMul(&inv1, &inv1, &A.X) 107 | edwards25519.FeInvert(&inv1, &inv1) 108 | 109 | var t0, u edwards25519.FieldElement 110 | edwards25519.FeMul(&u, &inv1, &A.X) 111 | edwards25519.FeAdd(&t0, &A.Y, &A.Z) 112 | edwards25519.FeMul(&u, &u, &t0) 113 | 114 | var v edwards25519.FieldElement 115 | edwards25519.FeMul(&v, &t0, &inv1) 116 | edwards25519.FeMul(&v, &v, &A.Z) 117 | edwards25519.FeMul(&v, &v, &sqrtMinusA) 118 | 119 | var b edwards25519.FieldElement 120 | edwards25519.FeAdd(&b, &u, &edwards25519.A) 121 | 122 | var c, b3, b8 edwards25519.FieldElement 123 | edwards25519.FeSquare(&b3, &b) // 2 124 | edwards25519.FeMul(&b3, &b3, &b) // 3 125 | edwards25519.FeSquare(&c, &b3) // 6 126 | edwards25519.FeMul(&c, &c, &b) // 7 127 | edwards25519.FeMul(&b8, &c, &b) // 8 128 | edwards25519.FeMul(&c, &c, &u) 129 | q58(&c, &c) 130 | 131 | var chi edwards25519.FieldElement 132 | edwards25519.FeSquare(&chi, &c) 133 | edwards25519.FeSquare(&chi, &chi) 134 | 135 | edwards25519.FeSquare(&t0, &u) 136 | edwards25519.FeMul(&chi, &chi, &t0) 137 | 138 | edwards25519.FeSquare(&t0, &b) // 2 139 | edwards25519.FeMul(&t0, &t0, &b) // 3 140 | edwards25519.FeSquare(&t0, &t0) // 6 141 | edwards25519.FeMul(&t0, &t0, &b) // 7 142 | edwards25519.FeSquare(&t0, &t0) // 14 143 | edwards25519.FeMul(&chi, &chi, &t0) 144 | edwards25519.FeNeg(&chi, &chi) 145 | 146 | var chiBytes [32]byte 147 | edwards25519.FeToBytes(&chiBytes, &chi) 148 | // chi[1] is either 0 or 0xff 149 | if chiBytes[1] == 0xff { 150 | return false 151 | } 152 | 153 | // Calculate r1 = sqrt(-u/(2*(u+A))) 154 | var r1 edwards25519.FieldElement 155 | edwards25519.FeMul(&r1, &c, &u) 156 | edwards25519.FeMul(&r1, &r1, &b3) 157 | edwards25519.FeMul(&r1, &r1, &sqrtMinusHalf) 158 | 159 | var maybeSqrtM1 edwards25519.FieldElement 160 | edwards25519.FeSquare(&t0, &r1) 161 | edwards25519.FeMul(&t0, &t0, &b) 162 | edwards25519.FeAdd(&t0, &t0, &t0) 163 | edwards25519.FeAdd(&t0, &t0, &u) 164 | 165 | edwards25519.FeOne(&maybeSqrtM1) 166 | edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0)) 167 | edwards25519.FeMul(&r1, &r1, &maybeSqrtM1) 168 | 169 | // Calculate r = sqrt(-(u+A)/(2u)) 170 | var r edwards25519.FieldElement 171 | edwards25519.FeSquare(&t0, &c) // 2 172 | edwards25519.FeMul(&t0, &t0, &c) // 3 173 | edwards25519.FeSquare(&t0, &t0) // 6 174 | edwards25519.FeMul(&r, &t0, &c) // 7 175 | 176 | edwards25519.FeSquare(&t0, &u) // 2 177 | edwards25519.FeMul(&t0, &t0, &u) // 3 178 | edwards25519.FeMul(&r, &r, &t0) 179 | 180 | edwards25519.FeSquare(&t0, &b8) // 16 181 | edwards25519.FeMul(&t0, &t0, &b8) // 24 182 | edwards25519.FeMul(&t0, &t0, &b) // 25 183 | edwards25519.FeMul(&r, &r, &t0) 184 | edwards25519.FeMul(&r, &r, &sqrtMinusHalf) 185 | 186 | edwards25519.FeSquare(&t0, &r) 187 | edwards25519.FeMul(&t0, &t0, &u) 188 | edwards25519.FeAdd(&t0, &t0, &t0) 189 | edwards25519.FeAdd(&t0, &t0, &b) 190 | edwards25519.FeOne(&maybeSqrtM1) 191 | edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0)) 192 | edwards25519.FeMul(&r, &r, &maybeSqrtM1) 193 | 194 | var vBytes [32]byte 195 | edwards25519.FeToBytes(&vBytes, &v) 196 | vInSquareRootImage := feBytesLE(&vBytes, &halfQMinus1Bytes) 197 | edwards25519.FeCMove(&r, &r1, vInSquareRootImage) 198 | 199 | edwards25519.FeToBytes(publicKey, &u) 200 | edwards25519.FeToBytes(representative, &r) 201 | return true 202 | } 203 | 204 | // q58 calculates out = z^((p-5)/8). 205 | func q58(out, z *edwards25519.FieldElement) { 206 | var t1, t2, t3 edwards25519.FieldElement 207 | var i int 208 | 209 | edwards25519.FeSquare(&t1, z) // 2^1 210 | edwards25519.FeMul(&t1, &t1, z) // 2^1 + 2^0 211 | edwards25519.FeSquare(&t1, &t1) // 2^2 + 2^1 212 | edwards25519.FeSquare(&t2, &t1) // 2^3 + 2^2 213 | edwards25519.FeSquare(&t2, &t2) // 2^4 + 2^3 214 | edwards25519.FeMul(&t2, &t2, &t1) // 4,3,2,1 215 | edwards25519.FeMul(&t1, &t2, z) // 4..0 216 | edwards25519.FeSquare(&t2, &t1) // 5..1 217 | for i = 1; i < 5; i++ { // 9,8,7,6,5 218 | edwards25519.FeSquare(&t2, &t2) 219 | } 220 | edwards25519.FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 221 | edwards25519.FeSquare(&t2, &t1) // 10..1 222 | for i = 1; i < 10; i++ { // 19..10 223 | edwards25519.FeSquare(&t2, &t2) 224 | } 225 | edwards25519.FeMul(&t2, &t2, &t1) // 19..0 226 | edwards25519.FeSquare(&t3, &t2) // 20..1 227 | for i = 1; i < 20; i++ { // 39..20 228 | edwards25519.FeSquare(&t3, &t3) 229 | } 230 | edwards25519.FeMul(&t2, &t3, &t2) // 39..0 231 | edwards25519.FeSquare(&t2, &t2) // 40..1 232 | for i = 1; i < 10; i++ { // 49..10 233 | edwards25519.FeSquare(&t2, &t2) 234 | } 235 | edwards25519.FeMul(&t1, &t2, &t1) // 49..0 236 | edwards25519.FeSquare(&t2, &t1) // 50..1 237 | for i = 1; i < 50; i++ { // 99..50 238 | edwards25519.FeSquare(&t2, &t2) 239 | } 240 | edwards25519.FeMul(&t2, &t2, &t1) // 99..0 241 | edwards25519.FeSquare(&t3, &t2) // 100..1 242 | for i = 1; i < 100; i++ { // 199..100 243 | edwards25519.FeSquare(&t3, &t3) 244 | } 245 | edwards25519.FeMul(&t2, &t3, &t2) // 199..0 246 | edwards25519.FeSquare(&t2, &t2) // 200..1 247 | for i = 1; i < 50; i++ { // 249..50 248 | edwards25519.FeSquare(&t2, &t2) 249 | } 250 | edwards25519.FeMul(&t1, &t2, &t1) // 249..0 251 | edwards25519.FeSquare(&t1, &t1) // 250..1 252 | edwards25519.FeSquare(&t1, &t1) // 251..2 253 | edwards25519.FeMul(out, &t1, z) // 251..2,0 254 | } 255 | 256 | // chi calculates out = z^((p-1)/2). The result is either 1, 0, or -1 depending 257 | // on whether z is a non-zero square, zero, or a non-square. 258 | func chi(out, z *edwards25519.FieldElement) { 259 | var t0, t1, t2, t3 edwards25519.FieldElement 260 | var i int 261 | 262 | edwards25519.FeSquare(&t0, z) // 2^1 263 | edwards25519.FeMul(&t1, &t0, z) // 2^1 + 2^0 264 | edwards25519.FeSquare(&t0, &t1) // 2^2 + 2^1 265 | edwards25519.FeSquare(&t2, &t0) // 2^3 + 2^2 266 | edwards25519.FeSquare(&t2, &t2) // 4,3 267 | edwards25519.FeMul(&t2, &t2, &t0) // 4,3,2,1 268 | edwards25519.FeMul(&t1, &t2, z) // 4..0 269 | edwards25519.FeSquare(&t2, &t1) // 5..1 270 | for i = 1; i < 5; i++ { // 9,8,7,6,5 271 | edwards25519.FeSquare(&t2, &t2) 272 | } 273 | edwards25519.FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 274 | edwards25519.FeSquare(&t2, &t1) // 10..1 275 | for i = 1; i < 10; i++ { // 19..10 276 | edwards25519.FeSquare(&t2, &t2) 277 | } 278 | edwards25519.FeMul(&t2, &t2, &t1) // 19..0 279 | edwards25519.FeSquare(&t3, &t2) // 20..1 280 | for i = 1; i < 20; i++ { // 39..20 281 | edwards25519.FeSquare(&t3, &t3) 282 | } 283 | edwards25519.FeMul(&t2, &t3, &t2) // 39..0 284 | edwards25519.FeSquare(&t2, &t2) // 40..1 285 | for i = 1; i < 10; i++ { // 49..10 286 | edwards25519.FeSquare(&t2, &t2) 287 | } 288 | edwards25519.FeMul(&t1, &t2, &t1) // 49..0 289 | edwards25519.FeSquare(&t2, &t1) // 50..1 290 | for i = 1; i < 50; i++ { // 99..50 291 | edwards25519.FeSquare(&t2, &t2) 292 | } 293 | edwards25519.FeMul(&t2, &t2, &t1) // 99..0 294 | edwards25519.FeSquare(&t3, &t2) // 100..1 295 | for i = 1; i < 100; i++ { // 199..100 296 | edwards25519.FeSquare(&t3, &t3) 297 | } 298 | edwards25519.FeMul(&t2, &t3, &t2) // 199..0 299 | edwards25519.FeSquare(&t2, &t2) // 200..1 300 | for i = 1; i < 50; i++ { // 249..50 301 | edwards25519.FeSquare(&t2, &t2) 302 | } 303 | edwards25519.FeMul(&t1, &t2, &t1) // 249..0 304 | edwards25519.FeSquare(&t1, &t1) // 250..1 305 | for i = 1; i < 4; i++ { // 253..4 306 | edwards25519.FeSquare(&t1, &t1) 307 | } 308 | edwards25519.FeMul(out, &t1, &t0) // 253..4,2,1 309 | } 310 | 311 | // RepresentativeToPublicKey converts a uniform representative value for a 312 | // curve25519 public key, as produced by ScalarBaseMult, to a curve25519 public 313 | // key. 314 | func RepresentativeToPublicKey(publicKey, representative *[32]byte) { 315 | var rr2, v edwards25519.FieldElement 316 | edwards25519.FeFromBytes(&rr2, representative) 317 | representativeToMontgomeryX(&v, &rr2) 318 | edwards25519.FeToBytes(publicKey, &v) 319 | } 320 | 321 | // representativeToMontgomeryX consumes the rr2 input 322 | func representativeToMontgomeryX(v, rr2 *edwards25519.FieldElement) { 323 | var e edwards25519.FieldElement 324 | edwards25519.FeSquare2(rr2, rr2) 325 | rr2[0]++ 326 | edwards25519.FeInvert(rr2, rr2) 327 | edwards25519.FeMul(v, &edwards25519.A, rr2) 328 | edwards25519.FeNeg(v, v) 329 | 330 | var v2, v3 edwards25519.FieldElement 331 | edwards25519.FeSquare(&v2, v) 332 | edwards25519.FeMul(&v3, v, &v2) 333 | edwards25519.FeAdd(&e, &v3, v) 334 | edwards25519.FeMul(&v2, &v2, &edwards25519.A) 335 | edwards25519.FeAdd(&e, &v2, &e) 336 | chi(&e, &e) 337 | var eBytes [32]byte 338 | edwards25519.FeToBytes(&eBytes, &e) 339 | // eBytes[1] is either 0 (for e = 1) or 0xff (for e = -1) 340 | eIsMinus1 := int32(eBytes[1]) & 1 341 | var negV edwards25519.FieldElement 342 | edwards25519.FeNeg(&negV, v) 343 | edwards25519.FeCMove(v, &negV, eIsMinus1) 344 | 345 | edwards25519.FeZero(&v2) 346 | edwards25519.FeCMove(&v2, &edwards25519.A, eIsMinus1) 347 | edwards25519.FeSub(v, v, &v2) 348 | } 349 | 350 | func FeMontRhs(v2, u *edwards25519.FieldElement) { 351 | var u2, Au, inner edwards25519.FieldElement 352 | var one edwards25519.FieldElement 353 | edwards25519.FeOne(&one) 354 | 355 | edwards25519.FeSquare(&u2, u) /* u^2 */ 356 | edwards25519.FeMul(&Au, &edwards25519.A, u) /* Au */ 357 | edwards25519.FeAdd(&inner, &u2, &Au) /* u^2 + Au */ 358 | edwards25519.FeAdd(&inner, &inner, &one) /* u^2 + Au + 1 */ 359 | edwards25519.FeMul(v2, u, &inner) /* u(u^2 + Au + 1) */ 360 | } 361 | 362 | func legendreIsNonsquare(in edwards25519.FieldElement) int32 { 363 | var temp edwards25519.FieldElement 364 | edwards25519.FePow22523(&temp, &in) /* temp = in^((q-5)/8) */ 365 | edwards25519.FeSquare(&temp, &temp) /* in^((q-5)/4) */ 366 | edwards25519.FeSquare(&temp, &temp) /* in^((q-5)/2) */ 367 | edwards25519.FeMul(&temp, &temp, &in) /* in^((q-3)/2) */ 368 | edwards25519.FeMul(&temp, &temp, &in) /* in^((q-1)/2) */ 369 | 370 | /* temp is now the Legendre symbol: 371 | * 1 = square 372 | * 0 = input is zero 373 | * -1 = nonsquare 374 | */ 375 | var b [32]byte 376 | edwards25519.FeToBytes(&b, &temp) 377 | //fmt.Println(hex.Dump(b[:])) 378 | return int32(1 & b[31]) 379 | } 380 | 381 | // compare to fe_montx_to_edy 382 | func FeMontgomeryXToEdwardsY(out, x *edwards25519.FieldElement) { 383 | /* 384 | y = (u - 1) / (u + 1) 385 | 386 | NOTE: u=-1 is converted to y=0 since fe_invert is mod-exp 387 | */ 388 | var t, tt edwards25519.FieldElement 389 | edwards25519.FeOne(&t) 390 | edwards25519.FeAdd(&tt, x, &t) // u+1 391 | edwards25519.FeInvert(&tt, &tt) // 1/(u+1) 392 | edwards25519.FeSub(&t, x, &t) // u-1 393 | edwards25519.FeMul(out, &tt, &t) // (u-1)/(u+1) 394 | } 395 | 396 | func elligator(u *edwards25519.FieldElement, r edwards25519.FieldElement) { 397 | /* r = input 398 | * x = -A/(1+2r^2) # 2 is nonsquare 399 | * e = (x^3 + Ax^2 + x)^((q-1)/2) # legendre symbol 400 | * if e == 1 (square) or e == 0 (because x == 0 and 2r^2 + 1 == 0) 401 | * u = x 402 | * if e == -1 (nonsquare) 403 | * u = -x - A 404 | */ 405 | var A, one, twor2, twor2plus1, twor2plus1inv edwards25519.FieldElement 406 | var x, e, Atemp, uneg edwards25519.FieldElement 407 | A = edwards25519.A /* A = 486662 */ 408 | edwards25519.FeOne(&one) 409 | 410 | edwards25519.FeSquare2(&twor2, &r) /* 2r^2 */ 411 | edwards25519.FeAdd(&twor2plus1, &twor2, &one) /* 1+2r^2 */ 412 | edwards25519.FeInvert(&twor2plus1inv, &twor2plus1) /* 1/(1+2r^2) */ 413 | edwards25519.FeMul(&x, &twor2plus1inv, &A) /* A/(1+2r^2) */ 414 | edwards25519.FeNeg(&x, &x) /* x = -A/(1+2r^2) */ 415 | 416 | FeMontRhs(&e, &x) /* e = x^3 + Ax^2 + x */ 417 | 418 | nonsquare := legendreIsNonsquare(e) 419 | 420 | edwards25519.FeZero(&Atemp) 421 | 422 | edwards25519.FeCMove(&Atemp, &A, nonsquare) /* 0, or A if nonsquare */ 423 | edwards25519.FeAdd(u, &x, &Atemp) /* x, or x+A if nonsquare */ 424 | edwards25519.FeNeg(&uneg, u) /* -x, or -x-A if nonsquare */ 425 | edwards25519.FeCMove(u, &uneg, nonsquare) /* x, or -x-A if nonsquare */ 426 | } 427 | 428 | // HashToEdwards converts a 256-bit hash output into a point on the Edwards 429 | // curve isomorphic to Curve25519 in a manner that preserves 430 | // collision-resistance. The returned curve points are NOT indistinguishable 431 | // from random even if the hash value is. 432 | // Specifically, first one bit of the hash output is set aside for parity and 433 | // the rest is truncated and fed into the elligator bijection (which covers half 434 | // of the points on the elliptic curve). 435 | func HashToEdwards(out *edwards25519.ExtendedGroupElement, h *[32]byte) { 436 | hh := *h 437 | bit := hh[31] >> 7 438 | hh[31] &= 127 439 | edwards25519.FeFromBytes(&out.Y, &hh) 440 | representativeToMontgomeryX(&out.X, &out.Y) 441 | FeMontgomeryXToEdwardsY(&out.Y, &out.X) 442 | if ok := out.FromParityAndY(bit, &out.Y); !ok { 443 | panic("HashToEdwards: point not on curve") 444 | } 445 | } 446 | 447 | func HashToPoint(m []byte) *edwards25519.ExtendedGroupElement { 448 | // H(n) = (f(h(n))^8) 449 | var hmb [32]byte 450 | h64 := sha512.Sum512(m) 451 | copy(hmb[:], h64[:32]) 452 | var hm edwards25519.ExtendedGroupElement 453 | HashToEdwards(&hm, &hmb) 454 | edwards25519.GeDouble(&hm, &hm) 455 | edwards25519.GeDouble(&hm, &hm) 456 | edwards25519.GeDouble(&hm, &hm) 457 | return &hm 458 | } 459 | 460 | /* sqrt(-(A+2)) */ 461 | var a_bytes = [32]byte{ 462 | 0x06, 0x7e, 0x45, 0xff, 0xaa, 0x04, 0x6e, 0xcc, 463 | 0x82, 0x1a, 0x7d, 0x4b, 0xd1, 0xd3, 0xa1, 0xc5, 464 | 0x7e, 0x4f, 0xfc, 0x03, 0xdc, 0x08, 0x7b, 0xd2, 465 | 0xbb, 0x06, 0xa0, 0x60, 0xf4, 0xed, 0x26, 0x0f, 466 | } 467 | 468 | func GeMontXtoExtendedGroupElement(p *edwards25519.ExtendedGroupElement, u edwards25519.FieldElement, edSignBit byte) { 469 | var x, y, v, A, v2, iv, nx edwards25519.FieldElement 470 | 471 | edwards25519.FeFromBytes(&A, &a_bytes) 472 | 473 | /* given u, recover edwards y */ 474 | /* given u, recover v */ 475 | /* given u and v, recover edwards x */ 476 | 477 | FeMontgomeryXToEdwardsY(&y, &u) /* y = (u - 1) / (u + 1) */ 478 | 479 | FeMontRhs(&v2, &u) /* v^2 = u(u^2 + Au + 1) */ 480 | 481 | edwards25519.FeSqrt(&v, &v2) /* v = sqrt(v^2) */ 482 | 483 | edwards25519.FeMul(&x, &u, &A) /* x = u * sqrt(-(A+2)) */ 484 | edwards25519.FeInvert(&iv, &v) /* 1/v */ 485 | edwards25519.FeMul(&x, &x, &iv) /* x = (u/v) * sqrt(-(A+2)) */ 486 | 487 | edwards25519.FeNeg(&nx, &x) /* negate x to match sign bit */ 488 | edwards25519.FeCMove(&x, &nx, int32(edwards25519.FeIsNegative(&x)^edSignBit)) 489 | 490 | edwards25519.FeCopy(&(p.X), &x) 491 | edwards25519.FeCopy(&(p.Y), &y) 492 | edwards25519.FeOne(&(p.Z)) 493 | edwards25519.FeMul(&(p.T), &(p.X), &(p.Y)) 494 | } 495 | -------------------------------------------------------------------------------- /edwards25519/edwards25519.go: -------------------------------------------------------------------------------- 1 | // Copyright 2013 The Go Authors. All rights reserved. 2 | // Use of this source code is governed by a BSD-style 3 | // license that can be found in the LICENSE file. 4 | 5 | // Package edwards25519 implements operations in GF(2**255-19) and on an 6 | // Edwards curve that is isomorphic to curve25519. See 7 | // http://ed25519.cr.yp.to/. 8 | package edwards25519 9 | 10 | // This code is a port of the public domain, "ref10" implementation of ed25519 11 | // from SUPERCOP. 12 | 13 | // FieldElement represents an element of the field GF(2^255 - 19). An element 14 | // t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 15 | // t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on 16 | // context. 17 | type FieldElement [10]int32 18 | 19 | func FeZero(fe *FieldElement) { 20 | for i := range fe { 21 | fe[i] = 0 22 | } 23 | } 24 | 25 | func FeOne(fe *FieldElement) { 26 | FeZero(fe) 27 | fe[0] = 1 28 | } 29 | 30 | func FeAdd(dst, a, b *FieldElement) { 31 | for i := range dst { 32 | dst[i] = a[i] + b[i] 33 | } 34 | } 35 | 36 | func FeSub(dst, a, b *FieldElement) { 37 | for i := range dst { 38 | dst[i] = a[i] - b[i] 39 | } 40 | } 41 | 42 | func FeCopy(dst, src *FieldElement) { 43 | for i := range dst { 44 | dst[i] = src[i] 45 | } 46 | } 47 | 48 | // Replace (f,g) with (g,g) if b == 1; 49 | // replace (f,g) with (f,g) if b == 0. 50 | // 51 | // Preconditions: b in {0,1}. 52 | func FeCMove(f, g *FieldElement, b int32) { 53 | var x FieldElement 54 | b = -b 55 | for i := range x { 56 | x[i] = b & (f[i] ^ g[i]) 57 | } 58 | 59 | for i := range f { 60 | f[i] ^= x[i] 61 | } 62 | } 63 | 64 | func load3(in []byte) int64 { 65 | var r int64 66 | r = int64(in[0]) 67 | r |= int64(in[1]) << 8 68 | r |= int64(in[2]) << 16 69 | return r 70 | } 71 | 72 | func load4(in []byte) int64 { 73 | var r int64 74 | r = int64(in[0]) 75 | r |= int64(in[1]) << 8 76 | r |= int64(in[2]) << 16 77 | r |= int64(in[3]) << 24 78 | return r 79 | } 80 | 81 | func FeFromBytes(dst *FieldElement, src *[32]byte) { 82 | h0 := load4(src[:]) 83 | h1 := load3(src[4:]) << 6 84 | h2 := load3(src[7:]) << 5 85 | h3 := load3(src[10:]) << 3 86 | h4 := load3(src[13:]) << 2 87 | h5 := load4(src[16:]) 88 | h6 := load3(src[20:]) << 7 89 | h7 := load3(src[23:]) << 5 90 | h8 := load3(src[26:]) << 4 91 | h9 := (load3(src[29:]) & 8388607) << 2 92 | 93 | var carry [10]int64 94 | carry[9] = (h9 + 1<<24) >> 25 95 | h0 += carry[9] * 19 96 | h9 -= carry[9] << 25 97 | carry[1] = (h1 + 1<<24) >> 25 98 | h2 += carry[1] 99 | h1 -= carry[1] << 25 100 | carry[3] = (h3 + 1<<24) >> 25 101 | h4 += carry[3] 102 | h3 -= carry[3] << 25 103 | carry[5] = (h5 + 1<<24) >> 25 104 | h6 += carry[5] 105 | h5 -= carry[5] << 25 106 | carry[7] = (h7 + 1<<24) >> 25 107 | h8 += carry[7] 108 | h7 -= carry[7] << 25 109 | 110 | carry[0] = (h0 + 1<<25) >> 26 111 | h1 += carry[0] 112 | h0 -= carry[0] << 26 113 | carry[2] = (h2 + 1<<25) >> 26 114 | h3 += carry[2] 115 | h2 -= carry[2] << 26 116 | carry[4] = (h4 + 1<<25) >> 26 117 | h5 += carry[4] 118 | h4 -= carry[4] << 26 119 | carry[6] = (h6 + 1<<25) >> 26 120 | h7 += carry[6] 121 | h6 -= carry[6] << 26 122 | carry[8] = (h8 + 1<<25) >> 26 123 | h9 += carry[8] 124 | h8 -= carry[8] << 26 125 | 126 | dst[0] = int32(h0) 127 | dst[1] = int32(h1) 128 | dst[2] = int32(h2) 129 | dst[3] = int32(h3) 130 | dst[4] = int32(h4) 131 | dst[5] = int32(h5) 132 | dst[6] = int32(h6) 133 | dst[7] = int32(h7) 134 | dst[8] = int32(h8) 135 | dst[9] = int32(h9) 136 | } 137 | 138 | // FeToBytes marshals h to s. 139 | // Preconditions: 140 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 141 | // 142 | // Write p=2^255-19; q=floor(h/p). 143 | // Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). 144 | // 145 | // Proof: 146 | // Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. 147 | // Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. 148 | // 149 | // Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). 150 | // Then 0> 25 165 | q = (h[0] + q) >> 26 166 | q = (h[1] + q) >> 25 167 | q = (h[2] + q) >> 26 168 | q = (h[3] + q) >> 25 169 | q = (h[4] + q) >> 26 170 | q = (h[5] + q) >> 25 171 | q = (h[6] + q) >> 26 172 | q = (h[7] + q) >> 25 173 | q = (h[8] + q) >> 26 174 | q = (h[9] + q) >> 25 175 | 176 | // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. 177 | h[0] += 19 * q 178 | // Goal: Output h-2^255 q, which is between 0 and 2^255-20. 179 | 180 | carry[0] = h[0] >> 26 181 | h[1] += carry[0] 182 | h[0] -= carry[0] << 26 183 | carry[1] = h[1] >> 25 184 | h[2] += carry[1] 185 | h[1] -= carry[1] << 25 186 | carry[2] = h[2] >> 26 187 | h[3] += carry[2] 188 | h[2] -= carry[2] << 26 189 | carry[3] = h[3] >> 25 190 | h[4] += carry[3] 191 | h[3] -= carry[3] << 25 192 | carry[4] = h[4] >> 26 193 | h[5] += carry[4] 194 | h[4] -= carry[4] << 26 195 | carry[5] = h[5] >> 25 196 | h[6] += carry[5] 197 | h[5] -= carry[5] << 25 198 | carry[6] = h[6] >> 26 199 | h[7] += carry[6] 200 | h[6] -= carry[6] << 26 201 | carry[7] = h[7] >> 25 202 | h[8] += carry[7] 203 | h[7] -= carry[7] << 25 204 | carry[8] = h[8] >> 26 205 | h[9] += carry[8] 206 | h[8] -= carry[8] << 26 207 | carry[9] = h[9] >> 25 208 | h[9] -= carry[9] << 25 209 | // h10 = carry9 210 | 211 | // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. 212 | // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; 213 | // evidently 2^255 h10-2^255 q = 0. 214 | // Goal: Output h[0]+...+2^230 h[9]. 215 | 216 | s[0] = byte(h[0] >> 0) 217 | s[1] = byte(h[0] >> 8) 218 | s[2] = byte(h[0] >> 16) 219 | s[3] = byte((h[0] >> 24) | (h[1] << 2)) 220 | s[4] = byte(h[1] >> 6) 221 | s[5] = byte(h[1] >> 14) 222 | s[6] = byte((h[1] >> 22) | (h[2] << 3)) 223 | s[7] = byte(h[2] >> 5) 224 | s[8] = byte(h[2] >> 13) 225 | s[9] = byte((h[2] >> 21) | (h[3] << 5)) 226 | s[10] = byte(h[3] >> 3) 227 | s[11] = byte(h[3] >> 11) 228 | s[12] = byte((h[3] >> 19) | (h[4] << 6)) 229 | s[13] = byte(h[4] >> 2) 230 | s[14] = byte(h[4] >> 10) 231 | s[15] = byte(h[4] >> 18) 232 | s[16] = byte(h[5] >> 0) 233 | s[17] = byte(h[5] >> 8) 234 | s[18] = byte(h[5] >> 16) 235 | s[19] = byte((h[5] >> 24) | (h[6] << 1)) 236 | s[20] = byte(h[6] >> 7) 237 | s[21] = byte(h[6] >> 15) 238 | s[22] = byte((h[6] >> 23) | (h[7] << 3)) 239 | s[23] = byte(h[7] >> 5) 240 | s[24] = byte(h[7] >> 13) 241 | s[25] = byte((h[7] >> 21) | (h[8] << 4)) 242 | s[26] = byte(h[8] >> 4) 243 | s[27] = byte(h[8] >> 12) 244 | s[28] = byte((h[8] >> 20) | (h[9] << 6)) 245 | s[29] = byte(h[9] >> 2) 246 | s[30] = byte(h[9] >> 10) 247 | s[31] = byte(h[9] >> 18) 248 | } 249 | 250 | func FeIsNegative(f *FieldElement) byte { 251 | var s [32]byte 252 | FeToBytes(&s, f) 253 | return s[0] & 1 254 | } 255 | 256 | func FeIsNonZero(f *FieldElement) int32 { 257 | var s [32]byte 258 | FeToBytes(&s, f) 259 | var x uint8 260 | for _, b := range s { 261 | x |= b 262 | } 263 | x |= x >> 4 264 | x |= x >> 2 265 | x |= x >> 1 266 | return int32(x & 1) 267 | } 268 | 269 | // FeNeg sets h = -f 270 | // 271 | // Preconditions: 272 | // |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 273 | // 274 | // Postconditions: 275 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 276 | func FeNeg(h, f *FieldElement) { 277 | for i := range h { 278 | h[i] = -f[i] 279 | } 280 | } 281 | 282 | // FeMul calculates h = f * g 283 | // Can overlap h with f or g. 284 | // 285 | // Preconditions: 286 | // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 287 | // |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 288 | // 289 | // Postconditions: 290 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 291 | // 292 | // Notes on implementation strategy: 293 | // 294 | // Using schoolbook multiplication. 295 | // Karatsuba would save a little in some cost models. 296 | // 297 | // Most multiplications by 2 and 19 are 32-bit precomputations; 298 | // cheaper than 64-bit postcomputations. 299 | // 300 | // There is one remaining multiplication by 19 in the carry chain; 301 | // one *19 precomputation can be merged into this, 302 | // but the resulting data flow is considerably less clean. 303 | // 304 | // There are 12 carries below. 305 | // 10 of them are 2-way parallelizable and vectorizable. 306 | // Can get away with 11 carries, but then data flow is much deeper. 307 | // 308 | // With tighter constraints on inputs can squeeze carries into int32. 309 | func FeMul(h, f, g *FieldElement) { 310 | f0 := f[0] 311 | f1 := f[1] 312 | f2 := f[2] 313 | f3 := f[3] 314 | f4 := f[4] 315 | f5 := f[5] 316 | f6 := f[6] 317 | f7 := f[7] 318 | f8 := f[8] 319 | f9 := f[9] 320 | g0 := g[0] 321 | g1 := g[1] 322 | g2 := g[2] 323 | g3 := g[3] 324 | g4 := g[4] 325 | g5 := g[5] 326 | g6 := g[6] 327 | g7 := g[7] 328 | g8 := g[8] 329 | g9 := g[9] 330 | g1_19 := 19 * g1 /* 1.4*2^29 */ 331 | g2_19 := 19 * g2 /* 1.4*2^30; still ok */ 332 | g3_19 := 19 * g3 333 | g4_19 := 19 * g4 334 | g5_19 := 19 * g5 335 | g6_19 := 19 * g6 336 | g7_19 := 19 * g7 337 | g8_19 := 19 * g8 338 | g9_19 := 19 * g9 339 | f1_2 := 2 * f1 340 | f3_2 := 2 * f3 341 | f5_2 := 2 * f5 342 | f7_2 := 2 * f7 343 | f9_2 := 2 * f9 344 | f0g0 := int64(f0) * int64(g0) 345 | f0g1 := int64(f0) * int64(g1) 346 | f0g2 := int64(f0) * int64(g2) 347 | f0g3 := int64(f0) * int64(g3) 348 | f0g4 := int64(f0) * int64(g4) 349 | f0g5 := int64(f0) * int64(g5) 350 | f0g6 := int64(f0) * int64(g6) 351 | f0g7 := int64(f0) * int64(g7) 352 | f0g8 := int64(f0) * int64(g8) 353 | f0g9 := int64(f0) * int64(g9) 354 | f1g0 := int64(f1) * int64(g0) 355 | f1g1_2 := int64(f1_2) * int64(g1) 356 | f1g2 := int64(f1) * int64(g2) 357 | f1g3_2 := int64(f1_2) * int64(g3) 358 | f1g4 := int64(f1) * int64(g4) 359 | f1g5_2 := int64(f1_2) * int64(g5) 360 | f1g6 := int64(f1) * int64(g6) 361 | f1g7_2 := int64(f1_2) * int64(g7) 362 | f1g8 := int64(f1) * int64(g8) 363 | f1g9_38 := int64(f1_2) * int64(g9_19) 364 | f2g0 := int64(f2) * int64(g0) 365 | f2g1 := int64(f2) * int64(g1) 366 | f2g2 := int64(f2) * int64(g2) 367 | f2g3 := int64(f2) * int64(g3) 368 | f2g4 := int64(f2) * int64(g4) 369 | f2g5 := int64(f2) * int64(g5) 370 | f2g6 := int64(f2) * int64(g6) 371 | f2g7 := int64(f2) * int64(g7) 372 | f2g8_19 := int64(f2) * int64(g8_19) 373 | f2g9_19 := int64(f2) * int64(g9_19) 374 | f3g0 := int64(f3) * int64(g0) 375 | f3g1_2 := int64(f3_2) * int64(g1) 376 | f3g2 := int64(f3) * int64(g2) 377 | f3g3_2 := int64(f3_2) * int64(g3) 378 | f3g4 := int64(f3) * int64(g4) 379 | f3g5_2 := int64(f3_2) * int64(g5) 380 | f3g6 := int64(f3) * int64(g6) 381 | f3g7_38 := int64(f3_2) * int64(g7_19) 382 | f3g8_19 := int64(f3) * int64(g8_19) 383 | f3g9_38 := int64(f3_2) * int64(g9_19) 384 | f4g0 := int64(f4) * int64(g0) 385 | f4g1 := int64(f4) * int64(g1) 386 | f4g2 := int64(f4) * int64(g2) 387 | f4g3 := int64(f4) * int64(g3) 388 | f4g4 := int64(f4) * int64(g4) 389 | f4g5 := int64(f4) * int64(g5) 390 | f4g6_19 := int64(f4) * int64(g6_19) 391 | f4g7_19 := int64(f4) * int64(g7_19) 392 | f4g8_19 := int64(f4) * int64(g8_19) 393 | f4g9_19 := int64(f4) * int64(g9_19) 394 | f5g0 := int64(f5) * int64(g0) 395 | f5g1_2 := int64(f5_2) * int64(g1) 396 | f5g2 := int64(f5) * int64(g2) 397 | f5g3_2 := int64(f5_2) * int64(g3) 398 | f5g4 := int64(f5) * int64(g4) 399 | f5g5_38 := int64(f5_2) * int64(g5_19) 400 | f5g6_19 := int64(f5) * int64(g6_19) 401 | f5g7_38 := int64(f5_2) * int64(g7_19) 402 | f5g8_19 := int64(f5) * int64(g8_19) 403 | f5g9_38 := int64(f5_2) * int64(g9_19) 404 | f6g0 := int64(f6) * int64(g0) 405 | f6g1 := int64(f6) * int64(g1) 406 | f6g2 := int64(f6) * int64(g2) 407 | f6g3 := int64(f6) * int64(g3) 408 | f6g4_19 := int64(f6) * int64(g4_19) 409 | f6g5_19 := int64(f6) * int64(g5_19) 410 | f6g6_19 := int64(f6) * int64(g6_19) 411 | f6g7_19 := int64(f6) * int64(g7_19) 412 | f6g8_19 := int64(f6) * int64(g8_19) 413 | f6g9_19 := int64(f6) * int64(g9_19) 414 | f7g0 := int64(f7) * int64(g0) 415 | f7g1_2 := int64(f7_2) * int64(g1) 416 | f7g2 := int64(f7) * int64(g2) 417 | f7g3_38 := int64(f7_2) * int64(g3_19) 418 | f7g4_19 := int64(f7) * int64(g4_19) 419 | f7g5_38 := int64(f7_2) * int64(g5_19) 420 | f7g6_19 := int64(f7) * int64(g6_19) 421 | f7g7_38 := int64(f7_2) * int64(g7_19) 422 | f7g8_19 := int64(f7) * int64(g8_19) 423 | f7g9_38 := int64(f7_2) * int64(g9_19) 424 | f8g0 := int64(f8) * int64(g0) 425 | f8g1 := int64(f8) * int64(g1) 426 | f8g2_19 := int64(f8) * int64(g2_19) 427 | f8g3_19 := int64(f8) * int64(g3_19) 428 | f8g4_19 := int64(f8) * int64(g4_19) 429 | f8g5_19 := int64(f8) * int64(g5_19) 430 | f8g6_19 := int64(f8) * int64(g6_19) 431 | f8g7_19 := int64(f8) * int64(g7_19) 432 | f8g8_19 := int64(f8) * int64(g8_19) 433 | f8g9_19 := int64(f8) * int64(g9_19) 434 | f9g0 := int64(f9) * int64(g0) 435 | f9g1_38 := int64(f9_2) * int64(g1_19) 436 | f9g2_19 := int64(f9) * int64(g2_19) 437 | f9g3_38 := int64(f9_2) * int64(g3_19) 438 | f9g4_19 := int64(f9) * int64(g4_19) 439 | f9g5_38 := int64(f9_2) * int64(g5_19) 440 | f9g6_19 := int64(f9) * int64(g6_19) 441 | f9g7_38 := int64(f9_2) * int64(g7_19) 442 | f9g8_19 := int64(f9) * int64(g8_19) 443 | f9g9_38 := int64(f9_2) * int64(g9_19) 444 | h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38 445 | h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19 446 | h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38 447 | h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19 448 | h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38 449 | h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19 450 | h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38 451 | h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19 452 | h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38 453 | h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 454 | var carry [10]int64 455 | 456 | /* 457 | |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) 458 | i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8 459 | |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) 460 | i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9 461 | */ 462 | 463 | carry[0] = (h0 + (1 << 25)) >> 26 464 | h1 += carry[0] 465 | h0 -= carry[0] << 26 466 | carry[4] = (h4 + (1 << 25)) >> 26 467 | h5 += carry[4] 468 | h4 -= carry[4] << 26 469 | /* |h0| <= 2^25 */ 470 | /* |h4| <= 2^25 */ 471 | /* |h1| <= 1.51*2^58 */ 472 | /* |h5| <= 1.51*2^58 */ 473 | 474 | carry[1] = (h1 + (1 << 24)) >> 25 475 | h2 += carry[1] 476 | h1 -= carry[1] << 25 477 | carry[5] = (h5 + (1 << 24)) >> 25 478 | h6 += carry[5] 479 | h5 -= carry[5] << 25 480 | /* |h1| <= 2^24; from now on fits into int32 */ 481 | /* |h5| <= 2^24; from now on fits into int32 */ 482 | /* |h2| <= 1.21*2^59 */ 483 | /* |h6| <= 1.21*2^59 */ 484 | 485 | carry[2] = (h2 + (1 << 25)) >> 26 486 | h3 += carry[2] 487 | h2 -= carry[2] << 26 488 | carry[6] = (h6 + (1 << 25)) >> 26 489 | h7 += carry[6] 490 | h6 -= carry[6] << 26 491 | /* |h2| <= 2^25; from now on fits into int32 unchanged */ 492 | /* |h6| <= 2^25; from now on fits into int32 unchanged */ 493 | /* |h3| <= 1.51*2^58 */ 494 | /* |h7| <= 1.51*2^58 */ 495 | 496 | carry[3] = (h3 + (1 << 24)) >> 25 497 | h4 += carry[3] 498 | h3 -= carry[3] << 25 499 | carry[7] = (h7 + (1 << 24)) >> 25 500 | h8 += carry[7] 501 | h7 -= carry[7] << 25 502 | /* |h3| <= 2^24; from now on fits into int32 unchanged */ 503 | /* |h7| <= 2^24; from now on fits into int32 unchanged */ 504 | /* |h4| <= 1.52*2^33 */ 505 | /* |h8| <= 1.52*2^33 */ 506 | 507 | carry[4] = (h4 + (1 << 25)) >> 26 508 | h5 += carry[4] 509 | h4 -= carry[4] << 26 510 | carry[8] = (h8 + (1 << 25)) >> 26 511 | h9 += carry[8] 512 | h8 -= carry[8] << 26 513 | /* |h4| <= 2^25; from now on fits into int32 unchanged */ 514 | /* |h8| <= 2^25; from now on fits into int32 unchanged */ 515 | /* |h5| <= 1.01*2^24 */ 516 | /* |h9| <= 1.51*2^58 */ 517 | 518 | carry[9] = (h9 + (1 << 24)) >> 25 519 | h0 += carry[9] * 19 520 | h9 -= carry[9] << 25 521 | /* |h9| <= 2^24; from now on fits into int32 unchanged */ 522 | /* |h0| <= 1.8*2^37 */ 523 | 524 | carry[0] = (h0 + (1 << 25)) >> 26 525 | h1 += carry[0] 526 | h0 -= carry[0] << 26 527 | /* |h0| <= 2^25; from now on fits into int32 unchanged */ 528 | /* |h1| <= 1.01*2^24 */ 529 | 530 | h[0] = int32(h0) 531 | h[1] = int32(h1) 532 | h[2] = int32(h2) 533 | h[3] = int32(h3) 534 | h[4] = int32(h4) 535 | h[5] = int32(h5) 536 | h[6] = int32(h6) 537 | h[7] = int32(h7) 538 | h[8] = int32(h8) 539 | h[9] = int32(h9) 540 | } 541 | 542 | // FeSquare calculates h = f*f. Can overlap h with f. 543 | // 544 | // Preconditions: 545 | // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 546 | // 547 | // Postconditions: 548 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 549 | func FeSquare(h, f *FieldElement) { 550 | f0 := f[0] 551 | f1 := f[1] 552 | f2 := f[2] 553 | f3 := f[3] 554 | f4 := f[4] 555 | f5 := f[5] 556 | f6 := f[6] 557 | f7 := f[7] 558 | f8 := f[8] 559 | f9 := f[9] 560 | f0_2 := 2 * f0 561 | f1_2 := 2 * f1 562 | f2_2 := 2 * f2 563 | f3_2 := 2 * f3 564 | f4_2 := 2 * f4 565 | f5_2 := 2 * f5 566 | f6_2 := 2 * f6 567 | f7_2 := 2 * f7 568 | f5_38 := 38 * f5 // 1.31*2^30 569 | f6_19 := 19 * f6 // 1.31*2^30 570 | f7_38 := 38 * f7 // 1.31*2^30 571 | f8_19 := 19 * f8 // 1.31*2^30 572 | f9_38 := 38 * f9 // 1.31*2^30 573 | f0f0 := int64(f0) * int64(f0) 574 | f0f1_2 := int64(f0_2) * int64(f1) 575 | f0f2_2 := int64(f0_2) * int64(f2) 576 | f0f3_2 := int64(f0_2) * int64(f3) 577 | f0f4_2 := int64(f0_2) * int64(f4) 578 | f0f5_2 := int64(f0_2) * int64(f5) 579 | f0f6_2 := int64(f0_2) * int64(f6) 580 | f0f7_2 := int64(f0_2) * int64(f7) 581 | f0f8_2 := int64(f0_2) * int64(f8) 582 | f0f9_2 := int64(f0_2) * int64(f9) 583 | f1f1_2 := int64(f1_2) * int64(f1) 584 | f1f2_2 := int64(f1_2) * int64(f2) 585 | f1f3_4 := int64(f1_2) * int64(f3_2) 586 | f1f4_2 := int64(f1_2) * int64(f4) 587 | f1f5_4 := int64(f1_2) * int64(f5_2) 588 | f1f6_2 := int64(f1_2) * int64(f6) 589 | f1f7_4 := int64(f1_2) * int64(f7_2) 590 | f1f8_2 := int64(f1_2) * int64(f8) 591 | f1f9_76 := int64(f1_2) * int64(f9_38) 592 | f2f2 := int64(f2) * int64(f2) 593 | f2f3_2 := int64(f2_2) * int64(f3) 594 | f2f4_2 := int64(f2_2) * int64(f4) 595 | f2f5_2 := int64(f2_2) * int64(f5) 596 | f2f6_2 := int64(f2_2) * int64(f6) 597 | f2f7_2 := int64(f2_2) * int64(f7) 598 | f2f8_38 := int64(f2_2) * int64(f8_19) 599 | f2f9_38 := int64(f2) * int64(f9_38) 600 | f3f3_2 := int64(f3_2) * int64(f3) 601 | f3f4_2 := int64(f3_2) * int64(f4) 602 | f3f5_4 := int64(f3_2) * int64(f5_2) 603 | f3f6_2 := int64(f3_2) * int64(f6) 604 | f3f7_76 := int64(f3_2) * int64(f7_38) 605 | f3f8_38 := int64(f3_2) * int64(f8_19) 606 | f3f9_76 := int64(f3_2) * int64(f9_38) 607 | f4f4 := int64(f4) * int64(f4) 608 | f4f5_2 := int64(f4_2) * int64(f5) 609 | f4f6_38 := int64(f4_2) * int64(f6_19) 610 | f4f7_38 := int64(f4) * int64(f7_38) 611 | f4f8_38 := int64(f4_2) * int64(f8_19) 612 | f4f9_38 := int64(f4) * int64(f9_38) 613 | f5f5_38 := int64(f5) * int64(f5_38) 614 | f5f6_38 := int64(f5_2) * int64(f6_19) 615 | f5f7_76 := int64(f5_2) * int64(f7_38) 616 | f5f8_38 := int64(f5_2) * int64(f8_19) 617 | f5f9_76 := int64(f5_2) * int64(f9_38) 618 | f6f6_19 := int64(f6) * int64(f6_19) 619 | f6f7_38 := int64(f6) * int64(f7_38) 620 | f6f8_38 := int64(f6_2) * int64(f8_19) 621 | f6f9_38 := int64(f6) * int64(f9_38) 622 | f7f7_38 := int64(f7) * int64(f7_38) 623 | f7f8_38 := int64(f7_2) * int64(f8_19) 624 | f7f9_76 := int64(f7_2) * int64(f9_38) 625 | f8f8_19 := int64(f8) * int64(f8_19) 626 | f8f9_38 := int64(f8) * int64(f9_38) 627 | f9f9_38 := int64(f9) * int64(f9_38) 628 | h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38 629 | h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38 630 | h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19 631 | h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38 632 | h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38 633 | h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38 634 | h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19 635 | h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38 636 | h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38 637 | h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2 638 | var carry [10]int64 639 | 640 | carry[0] = (h0 + (1 << 25)) >> 26 641 | h1 += carry[0] 642 | h0 -= carry[0] << 26 643 | carry[4] = (h4 + (1 << 25)) >> 26 644 | h5 += carry[4] 645 | h4 -= carry[4] << 26 646 | 647 | carry[1] = (h1 + (1 << 24)) >> 25 648 | h2 += carry[1] 649 | h1 -= carry[1] << 25 650 | carry[5] = (h5 + (1 << 24)) >> 25 651 | h6 += carry[5] 652 | h5 -= carry[5] << 25 653 | 654 | carry[2] = (h2 + (1 << 25)) >> 26 655 | h3 += carry[2] 656 | h2 -= carry[2] << 26 657 | carry[6] = (h6 + (1 << 25)) >> 26 658 | h7 += carry[6] 659 | h6 -= carry[6] << 26 660 | 661 | carry[3] = (h3 + (1 << 24)) >> 25 662 | h4 += carry[3] 663 | h3 -= carry[3] << 25 664 | carry[7] = (h7 + (1 << 24)) >> 25 665 | h8 += carry[7] 666 | h7 -= carry[7] << 25 667 | 668 | carry[4] = (h4 + (1 << 25)) >> 26 669 | h5 += carry[4] 670 | h4 -= carry[4] << 26 671 | carry[8] = (h8 + (1 << 25)) >> 26 672 | h9 += carry[8] 673 | h8 -= carry[8] << 26 674 | 675 | carry[9] = (h9 + (1 << 24)) >> 25 676 | h0 += carry[9] * 19 677 | h9 -= carry[9] << 25 678 | 679 | carry[0] = (h0 + (1 << 25)) >> 26 680 | h1 += carry[0] 681 | h0 -= carry[0] << 26 682 | 683 | h[0] = int32(h0) 684 | h[1] = int32(h1) 685 | h[2] = int32(h2) 686 | h[3] = int32(h3) 687 | h[4] = int32(h4) 688 | h[5] = int32(h5) 689 | h[6] = int32(h6) 690 | h[7] = int32(h7) 691 | h[8] = int32(h8) 692 | h[9] = int32(h9) 693 | } 694 | 695 | // FeSquare2 sets h = 2 * f * f 696 | // 697 | // Can overlap h with f. 698 | // 699 | // Preconditions: 700 | // |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 701 | // 702 | // Postconditions: 703 | // |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 704 | // See fe_mul.c for discussion of implementation strategy. 705 | func FeSquare2(h, f *FieldElement) { 706 | f0 := f[0] 707 | f1 := f[1] 708 | f2 := f[2] 709 | f3 := f[3] 710 | f4 := f[4] 711 | f5 := f[5] 712 | f6 := f[6] 713 | f7 := f[7] 714 | f8 := f[8] 715 | f9 := f[9] 716 | f0_2 := 2 * f0 717 | f1_2 := 2 * f1 718 | f2_2 := 2 * f2 719 | f3_2 := 2 * f3 720 | f4_2 := 2 * f4 721 | f5_2 := 2 * f5 722 | f6_2 := 2 * f6 723 | f7_2 := 2 * f7 724 | f5_38 := 38 * f5 // 1.959375*2^30 725 | f6_19 := 19 * f6 // 1.959375*2^30 726 | f7_38 := 38 * f7 // 1.959375*2^30 727 | f8_19 := 19 * f8 // 1.959375*2^30 728 | f9_38 := 38 * f9 // 1.959375*2^30 729 | f0f0 := int64(f0) * int64(f0) 730 | f0f1_2 := int64(f0_2) * int64(f1) 731 | f0f2_2 := int64(f0_2) * int64(f2) 732 | f0f3_2 := int64(f0_2) * int64(f3) 733 | f0f4_2 := int64(f0_2) * int64(f4) 734 | f0f5_2 := int64(f0_2) * int64(f5) 735 | f0f6_2 := int64(f0_2) * int64(f6) 736 | f0f7_2 := int64(f0_2) * int64(f7) 737 | f0f8_2 := int64(f0_2) * int64(f8) 738 | f0f9_2 := int64(f0_2) * int64(f9) 739 | f1f1_2 := int64(f1_2) * int64(f1) 740 | f1f2_2 := int64(f1_2) * int64(f2) 741 | f1f3_4 := int64(f1_2) * int64(f3_2) 742 | f1f4_2 := int64(f1_2) * int64(f4) 743 | f1f5_4 := int64(f1_2) * int64(f5_2) 744 | f1f6_2 := int64(f1_2) * int64(f6) 745 | f1f7_4 := int64(f1_2) * int64(f7_2) 746 | f1f8_2 := int64(f1_2) * int64(f8) 747 | f1f9_76 := int64(f1_2) * int64(f9_38) 748 | f2f2 := int64(f2) * int64(f2) 749 | f2f3_2 := int64(f2_2) * int64(f3) 750 | f2f4_2 := int64(f2_2) * int64(f4) 751 | f2f5_2 := int64(f2_2) * int64(f5) 752 | f2f6_2 := int64(f2_2) * int64(f6) 753 | f2f7_2 := int64(f2_2) * int64(f7) 754 | f2f8_38 := int64(f2_2) * int64(f8_19) 755 | f2f9_38 := int64(f2) * int64(f9_38) 756 | f3f3_2 := int64(f3_2) * int64(f3) 757 | f3f4_2 := int64(f3_2) * int64(f4) 758 | f3f5_4 := int64(f3_2) * int64(f5_2) 759 | f3f6_2 := int64(f3_2) * int64(f6) 760 | f3f7_76 := int64(f3_2) * int64(f7_38) 761 | f3f8_38 := int64(f3_2) * int64(f8_19) 762 | f3f9_76 := int64(f3_2) * int64(f9_38) 763 | f4f4 := int64(f4) * int64(f4) 764 | f4f5_2 := int64(f4_2) * int64(f5) 765 | f4f6_38 := int64(f4_2) * int64(f6_19) 766 | f4f7_38 := int64(f4) * int64(f7_38) 767 | f4f8_38 := int64(f4_2) * int64(f8_19) 768 | f4f9_38 := int64(f4) * int64(f9_38) 769 | f5f5_38 := int64(f5) * int64(f5_38) 770 | f5f6_38 := int64(f5_2) * int64(f6_19) 771 | f5f7_76 := int64(f5_2) * int64(f7_38) 772 | f5f8_38 := int64(f5_2) * int64(f8_19) 773 | f5f9_76 := int64(f5_2) * int64(f9_38) 774 | f6f6_19 := int64(f6) * int64(f6_19) 775 | f6f7_38 := int64(f6) * int64(f7_38) 776 | f6f8_38 := int64(f6_2) * int64(f8_19) 777 | f6f9_38 := int64(f6) * int64(f9_38) 778 | f7f7_38 := int64(f7) * int64(f7_38) 779 | f7f8_38 := int64(f7_2) * int64(f8_19) 780 | f7f9_76 := int64(f7_2) * int64(f9_38) 781 | f8f8_19 := int64(f8) * int64(f8_19) 782 | f8f9_38 := int64(f8) * int64(f9_38) 783 | f9f9_38 := int64(f9) * int64(f9_38) 784 | h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38 785 | h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38 786 | h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19 787 | h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38 788 | h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38 789 | h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38 790 | h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19 791 | h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38 792 | h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38 793 | h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2 794 | var carry [10]int64 795 | 796 | h0 += h0 797 | h1 += h1 798 | h2 += h2 799 | h3 += h3 800 | h4 += h4 801 | h5 += h5 802 | h6 += h6 803 | h7 += h7 804 | h8 += h8 805 | h9 += h9 806 | 807 | carry[0] = (h0 + (1 << 25)) >> 26 808 | h1 += carry[0] 809 | h0 -= carry[0] << 26 810 | carry[4] = (h4 + (1 << 25)) >> 26 811 | h5 += carry[4] 812 | h4 -= carry[4] << 26 813 | 814 | carry[1] = (h1 + (1 << 24)) >> 25 815 | h2 += carry[1] 816 | h1 -= carry[1] << 25 817 | carry[5] = (h5 + (1 << 24)) >> 25 818 | h6 += carry[5] 819 | h5 -= carry[5] << 25 820 | 821 | carry[2] = (h2 + (1 << 25)) >> 26 822 | h3 += carry[2] 823 | h2 -= carry[2] << 26 824 | carry[6] = (h6 + (1 << 25)) >> 26 825 | h7 += carry[6] 826 | h6 -= carry[6] << 26 827 | 828 | carry[3] = (h3 + (1 << 24)) >> 25 829 | h4 += carry[3] 830 | h3 -= carry[3] << 25 831 | carry[7] = (h7 + (1 << 24)) >> 25 832 | h8 += carry[7] 833 | h7 -= carry[7] << 25 834 | 835 | carry[4] = (h4 + (1 << 25)) >> 26 836 | h5 += carry[4] 837 | h4 -= carry[4] << 26 838 | carry[8] = (h8 + (1 << 25)) >> 26 839 | h9 += carry[8] 840 | h8 -= carry[8] << 26 841 | 842 | carry[9] = (h9 + (1 << 24)) >> 25 843 | h0 += carry[9] * 19 844 | h9 -= carry[9] << 25 845 | 846 | carry[0] = (h0 + (1 << 25)) >> 26 847 | h1 += carry[0] 848 | h0 -= carry[0] << 26 849 | 850 | h[0] = int32(h0) 851 | h[1] = int32(h1) 852 | h[2] = int32(h2) 853 | h[3] = int32(h3) 854 | h[4] = int32(h4) 855 | h[5] = int32(h5) 856 | h[6] = int32(h6) 857 | h[7] = int32(h7) 858 | h[8] = int32(h8) 859 | h[9] = int32(h9) 860 | } 861 | 862 | func FeSqrt(out, a *FieldElement) { 863 | var exp, b, b2, bi, i FieldElement 864 | 865 | i = SqrtM1 866 | FePow22523(&exp, a) /* b = a^(q-5)/8 */ 867 | FeMul(&b, a, &exp) /* b = a * a^(q-5)/8 */ 868 | FeSquare(&b2, &b) /* b^2 = a * a^(q-1)/4 */ 869 | 870 | /* note b^4 == a^2, so b^2 == a or -a 871 | * if b^2 != a, multiply it by sqrt(-1) */ 872 | FeMul(&bi, &b, &i) 873 | FeCMove(&b, &bi, int32(1^FeIsequal(b2, *a))) 874 | 875 | FeCopy(out, &b) 876 | } 877 | 878 | func FeInvert(out, z *FieldElement) { 879 | var t0, t1, t2, t3 FieldElement 880 | var i int 881 | 882 | FeSquare(&t0, z) // 2^1 883 | FeSquare(&t1, &t0) // 2^2 884 | for i = 1; i < 2; i++ { // 2^3 885 | FeSquare(&t1, &t1) 886 | } 887 | FeMul(&t1, z, &t1) // 2^3 + 2^0 888 | FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0 889 | FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1 890 | FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0 891 | FeSquare(&t2, &t1) // 5,4,3,2,1 892 | for i = 1; i < 5; i++ { // 9,8,7,6,5 893 | FeSquare(&t2, &t2) 894 | } 895 | FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 896 | FeSquare(&t2, &t1) // 10..1 897 | for i = 1; i < 10; i++ { // 19..10 898 | FeSquare(&t2, &t2) 899 | } 900 | FeMul(&t2, &t2, &t1) // 19..0 901 | FeSquare(&t3, &t2) // 20..1 902 | for i = 1; i < 20; i++ { // 39..20 903 | FeSquare(&t3, &t3) 904 | } 905 | FeMul(&t2, &t3, &t2) // 39..0 906 | FeSquare(&t2, &t2) // 40..1 907 | for i = 1; i < 10; i++ { // 49..10 908 | FeSquare(&t2, &t2) 909 | } 910 | FeMul(&t1, &t2, &t1) // 49..0 911 | FeSquare(&t2, &t1) // 50..1 912 | for i = 1; i < 50; i++ { // 99..50 913 | FeSquare(&t2, &t2) 914 | } 915 | FeMul(&t2, &t2, &t1) // 99..0 916 | FeSquare(&t3, &t2) // 100..1 917 | for i = 1; i < 100; i++ { // 199..100 918 | FeSquare(&t3, &t3) 919 | } 920 | FeMul(&t2, &t3, &t2) // 199..0 921 | FeSquare(&t2, &t2) // 200..1 922 | for i = 1; i < 50; i++ { // 249..50 923 | FeSquare(&t2, &t2) 924 | } 925 | FeMul(&t1, &t2, &t1) // 249..0 926 | FeSquare(&t1, &t1) // 250..1 927 | for i = 1; i < 5; i++ { // 254..5 928 | FeSquare(&t1, &t1) 929 | } 930 | FeMul(out, &t1, &t0) // 254..5,3,1,0 931 | } 932 | 933 | func FePow22523(out, z *FieldElement) { 934 | var t0, t1, t2 FieldElement 935 | var i int 936 | 937 | FeSquare(&t0, z) 938 | for i = 1; i < 1; i++ { 939 | FeSquare(&t0, &t0) 940 | } 941 | FeSquare(&t1, &t0) 942 | for i = 1; i < 2; i++ { 943 | FeSquare(&t1, &t1) 944 | } 945 | FeMul(&t1, z, &t1) 946 | FeMul(&t0, &t0, &t1) 947 | FeSquare(&t0, &t0) 948 | for i = 1; i < 1; i++ { 949 | FeSquare(&t0, &t0) 950 | } 951 | FeMul(&t0, &t1, &t0) 952 | FeSquare(&t1, &t0) 953 | for i = 1; i < 5; i++ { 954 | FeSquare(&t1, &t1) 955 | } 956 | FeMul(&t0, &t1, &t0) 957 | FeSquare(&t1, &t0) 958 | for i = 1; i < 10; i++ { 959 | FeSquare(&t1, &t1) 960 | } 961 | FeMul(&t1, &t1, &t0) 962 | FeSquare(&t2, &t1) 963 | for i = 1; i < 20; i++ { 964 | FeSquare(&t2, &t2) 965 | } 966 | FeMul(&t1, &t2, &t1) 967 | FeSquare(&t1, &t1) 968 | for i = 1; i < 10; i++ { 969 | FeSquare(&t1, &t1) 970 | } 971 | FeMul(&t0, &t1, &t0) 972 | FeSquare(&t1, &t0) 973 | for i = 1; i < 50; i++ { 974 | FeSquare(&t1, &t1) 975 | } 976 | FeMul(&t1, &t1, &t0) 977 | FeSquare(&t2, &t1) 978 | for i = 1; i < 100; i++ { 979 | FeSquare(&t2, &t2) 980 | } 981 | FeMul(&t1, &t2, &t1) 982 | FeSquare(&t1, &t1) 983 | for i = 1; i < 50; i++ { 984 | FeSquare(&t1, &t1) 985 | } 986 | FeMul(&t0, &t1, &t0) 987 | FeSquare(&t0, &t0) 988 | for i = 1; i < 2; i++ { 989 | FeSquare(&t0, &t0) 990 | } 991 | FeMul(out, &t0, z) 992 | } 993 | 994 | func FeIsequal(f, g FieldElement) int { 995 | var h FieldElement 996 | FeSub(&h, &f, &g) 997 | return 1 ^ (1 & (feIsNonzero(h) >> 8)) 998 | } 999 | 1000 | func feIsNonzero(f FieldElement) int { 1001 | var s [32]byte 1002 | FeToBytes(&s, &f) 1003 | var zero [32]byte 1004 | 1005 | return FeCompare(s, zero) 1006 | } 1007 | 1008 | func FeCompare(x, y [32]byte) int { 1009 | d := 0 1010 | for i := 0; i < 32; i++ { 1011 | d |= int(x[i]) ^ int(y[i]) 1012 | } 1013 | return (1 & ((d - 1) >> 8)) - 1 1014 | } 1015 | 1016 | // Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 * 1017 | // y^2 where d = -121665/121666. 1018 | // 1019 | // Several representations are used: 1020 | // ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z 1021 | // ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT 1022 | // CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T 1023 | // PreComputedGroupElement: (y+x,y-x,2dxy) 1024 | 1025 | type ProjectiveGroupElement struct { 1026 | X, Y, Z FieldElement 1027 | } 1028 | 1029 | type ExtendedGroupElement struct { 1030 | X, Y, Z, T FieldElement 1031 | } 1032 | 1033 | type CompletedGroupElement struct { 1034 | X, Y, Z, T FieldElement 1035 | } 1036 | 1037 | type PreComputedGroupElement struct { 1038 | yPlusX, yMinusX, xy2d FieldElement 1039 | } 1040 | 1041 | type CachedGroupElement struct { 1042 | yPlusX, yMinusX, Z, T2d FieldElement 1043 | } 1044 | 1045 | func (p *ProjectiveGroupElement) Zero() { 1046 | FeZero(&p.X) 1047 | FeOne(&p.Y) 1048 | FeOne(&p.Z) 1049 | } 1050 | 1051 | func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) { 1052 | var t0 FieldElement 1053 | 1054 | FeSquare(&r.X, &p.X) 1055 | FeSquare(&r.Z, &p.Y) 1056 | FeSquare2(&r.T, &p.Z) 1057 | FeAdd(&r.Y, &p.X, &p.Y) 1058 | FeSquare(&t0, &r.Y) 1059 | FeAdd(&r.Y, &r.Z, &r.X) 1060 | FeSub(&r.Z, &r.Z, &r.X) 1061 | FeSub(&r.X, &t0, &r.Y) 1062 | FeSub(&r.T, &r.T, &r.Z) 1063 | } 1064 | 1065 | func (p *ProjectiveGroupElement) ToExtended(s *ExtendedGroupElement) { 1066 | var b [32]byte 1067 | p.ToBytes(&b) 1068 | s.FromBytes(&b) 1069 | } 1070 | 1071 | func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) { 1072 | var recip, x, y FieldElement 1073 | 1074 | FeInvert(&recip, &p.Z) 1075 | FeMul(&x, &p.X, &recip) 1076 | FeMul(&y, &p.Y, &recip) 1077 | FeToBytes(s, &y) 1078 | s[31] ^= FeIsNegative(&x) << 7 1079 | } 1080 | 1081 | func (p *ExtendedGroupElement) Zero() { 1082 | FeZero(&p.X) 1083 | FeOne(&p.Y) 1084 | FeOne(&p.Z) 1085 | FeZero(&p.T) 1086 | } 1087 | 1088 | func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) { 1089 | var q ProjectiveGroupElement 1090 | p.ToProjective(&q) 1091 | q.Double(r) 1092 | } 1093 | 1094 | func GeNeg(r *ExtendedGroupElement, p ExtendedGroupElement) { 1095 | FeNeg(&r.X, &p.X) 1096 | FeCopy(&r.Y, &p.Y) 1097 | FeCopy(&r.Z, &p.Z) 1098 | FeNeg(&r.T, &p.T) 1099 | } 1100 | 1101 | func GeDouble(r, p *ExtendedGroupElement) { 1102 | var q ProjectiveGroupElement 1103 | p.ToProjective(&q) 1104 | var rco CompletedGroupElement 1105 | q.Double(&rco) 1106 | rco.ToExtended(r) 1107 | } 1108 | 1109 | func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) { 1110 | FeAdd(&r.yPlusX, &p.Y, &p.X) 1111 | FeSub(&r.yMinusX, &p.Y, &p.X) 1112 | FeCopy(&r.Z, &p.Z) 1113 | FeMul(&r.T2d, &p.T, &d2) 1114 | } 1115 | 1116 | func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) { 1117 | FeCopy(&r.X, &p.X) 1118 | FeCopy(&r.Y, &p.Y) 1119 | FeCopy(&r.Z, &p.Z) 1120 | } 1121 | 1122 | func (p *ExtendedGroupElement) ToBytes(s *[32]byte) { 1123 | var recip, x, y FieldElement 1124 | 1125 | FeInvert(&recip, &p.Z) 1126 | FeMul(&x, &p.X, &recip) 1127 | FeMul(&y, &p.Y, &recip) 1128 | FeToBytes(s, &y) 1129 | s[31] ^= FeIsNegative(&x) << 7 1130 | } 1131 | 1132 | // FromBytesBaseGroup unmarshals an elliptic curve point returns true iff the 1133 | // point point is in the order l subgroup generated by the base point. This 1134 | // implementation is based on 1135 | // https://www.iacr.org/archive/pkc2003/25670211/25670211.pdf Definition 1. 1136 | // Validation of an elliptic curve public key P ensures that P is a point of 1137 | // order BasePointOrder in E. 1138 | func (p *ExtendedGroupElement) FromBytesBaseGroup(s *[32]byte) bool { 1139 | 1140 | // condition 1: P != infinity 1141 | if *s == [32]byte{1} { 1142 | return false 1143 | } 1144 | 1145 | // condition 3 is implied by successful point decompression 1146 | if !p.FromBytes(s) { 1147 | return false 1148 | } 1149 | // condition 2: ToBytes produces canonical encodings, check against that 1150 | var sCheck [32]byte 1151 | p.ToBytes(&sCheck) 1152 | if sCheck != *s { 1153 | return false 1154 | } 1155 | 1156 | // condition 4: order * P == infinity 1157 | var out ExtendedGroupElement 1158 | GeScalarMult(&out, &BasePointOrder, p) 1159 | out.ToBytes(&sCheck) 1160 | if sCheck != [32]byte{1} { 1161 | return false 1162 | } 1163 | 1164 | // an array of all zeros would not result from any randomly generated point 1165 | // and might easily be caused by bugs in ToBytes or FromBytes 1166 | if *s == [32]byte{} { 1167 | return false 1168 | } 1169 | return true 1170 | } 1171 | 1172 | func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool { 1173 | FeFromBytes(&p.Y, s) 1174 | return p.FromParityAndY((s[31] >> 7), &p.Y) 1175 | } 1176 | 1177 | func (p *ExtendedGroupElement) FromParityAndY(bit byte, y *FieldElement) bool { 1178 | // compare to ge_frombytes_negate_vartime 1179 | var u, v, v3, vxx, check FieldElement 1180 | 1181 | FeCopy(&p.Y, y) 1182 | FeOne(&p.Z) 1183 | FeSquare(&u, &p.Y) 1184 | FeMul(&v, &u, &d) 1185 | FeSub(&u, &u, &p.Z) // y = y^2-1 1186 | FeAdd(&v, &v, &p.Z) // v = dy^2+1 1187 | 1188 | FeSquare(&v3, &v) 1189 | FeMul(&v3, &v3, &v) // v3 = v^3 1190 | FeSquare(&p.X, &v3) 1191 | FeMul(&p.X, &p.X, &v) 1192 | FeMul(&p.X, &p.X, &u) // x = uv^7 1193 | 1194 | FePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8) 1195 | FeMul(&p.X, &p.X, &v3) 1196 | FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8) 1197 | 1198 | var tmpX, tmp2 [32]byte 1199 | 1200 | FeSquare(&vxx, &p.X) 1201 | FeMul(&vxx, &vxx, &v) 1202 | FeSub(&check, &vxx, &u) // vx^2-u 1203 | if FeIsNonZero(&check) == 1 { 1204 | FeAdd(&check, &vxx, &u) // vx^2+u 1205 | if FeIsNonZero(&check) == 1 { 1206 | return false 1207 | } 1208 | FeMul(&p.X, &p.X, &SqrtM1) 1209 | 1210 | FeToBytes(&tmpX, &p.X) 1211 | for i, v := range tmpX { 1212 | tmp2[31-i] = v 1213 | } 1214 | } 1215 | if FeIsNegative(&p.X) != bit { 1216 | FeNeg(&p.X, &p.X) 1217 | } 1218 | 1219 | FeMul(&p.T, &p.X, &p.Y) 1220 | return true 1221 | } 1222 | 1223 | func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) { 1224 | FeMul(&r.X, &p.X, &p.T) 1225 | FeMul(&r.Y, &p.Y, &p.Z) 1226 | FeMul(&r.Z, &p.Z, &p.T) 1227 | } 1228 | 1229 | func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) { 1230 | FeMul(&r.X, &p.X, &p.T) 1231 | FeMul(&r.Y, &p.Y, &p.Z) 1232 | FeMul(&r.Z, &p.Z, &p.T) 1233 | FeMul(&r.T, &p.X, &p.Y) 1234 | } 1235 | 1236 | func (p *PreComputedGroupElement) Zero() { 1237 | FeOne(&p.yPlusX) 1238 | FeOne(&p.yMinusX) 1239 | FeZero(&p.xy2d) 1240 | } 1241 | 1242 | func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { 1243 | var t0 FieldElement 1244 | 1245 | FeAdd(&r.X, &p.Y, &p.X) 1246 | FeSub(&r.Y, &p.Y, &p.X) 1247 | FeMul(&r.Z, &r.X, &q.yPlusX) 1248 | FeMul(&r.Y, &r.Y, &q.yMinusX) 1249 | FeMul(&r.T, &q.T2d, &p.T) 1250 | FeMul(&r.X, &p.Z, &q.Z) 1251 | FeAdd(&t0, &r.X, &r.X) 1252 | FeSub(&r.X, &r.Z, &r.Y) 1253 | FeAdd(&r.Y, &r.Z, &r.Y) 1254 | FeAdd(&r.Z, &t0, &r.T) 1255 | FeSub(&r.T, &t0, &r.T) 1256 | } 1257 | 1258 | func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { 1259 | var t0 FieldElement 1260 | 1261 | FeAdd(&r.X, &p.Y, &p.X) 1262 | FeSub(&r.Y, &p.Y, &p.X) 1263 | FeMul(&r.Z, &r.X, &q.yMinusX) 1264 | FeMul(&r.Y, &r.Y, &q.yPlusX) 1265 | FeMul(&r.T, &q.T2d, &p.T) 1266 | FeMul(&r.X, &p.Z, &q.Z) 1267 | FeAdd(&t0, &r.X, &r.X) 1268 | FeSub(&r.X, &r.Z, &r.Y) 1269 | FeAdd(&r.Y, &r.Z, &r.Y) 1270 | FeSub(&r.Z, &t0, &r.T) 1271 | FeAdd(&r.T, &t0, &r.T) 1272 | } 1273 | 1274 | func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { 1275 | var t0 FieldElement 1276 | 1277 | FeAdd(&r.X, &p.Y, &p.X) 1278 | FeSub(&r.Y, &p.Y, &p.X) 1279 | FeMul(&r.Z, &r.X, &q.yPlusX) 1280 | FeMul(&r.Y, &r.Y, &q.yMinusX) 1281 | FeMul(&r.T, &q.xy2d, &p.T) 1282 | FeAdd(&t0, &p.Z, &p.Z) 1283 | FeSub(&r.X, &r.Z, &r.Y) 1284 | FeAdd(&r.Y, &r.Z, &r.Y) 1285 | FeAdd(&r.Z, &t0, &r.T) 1286 | FeSub(&r.T, &t0, &r.T) 1287 | } 1288 | 1289 | func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { 1290 | var t0 FieldElement 1291 | 1292 | FeAdd(&r.X, &p.Y, &p.X) 1293 | FeSub(&r.Y, &p.Y, &p.X) 1294 | FeMul(&r.Z, &r.X, &q.yMinusX) 1295 | FeMul(&r.Y, &r.Y, &q.yPlusX) 1296 | FeMul(&r.T, &q.xy2d, &p.T) 1297 | FeAdd(&t0, &p.Z, &p.Z) 1298 | FeSub(&r.X, &r.Z, &r.Y) 1299 | FeAdd(&r.Y, &r.Z, &r.Y) 1300 | FeSub(&r.Z, &t0, &r.T) 1301 | FeAdd(&r.T, &t0, &r.T) 1302 | } 1303 | 1304 | func slide(r *[256]int8, a *[32]byte) { 1305 | for i := range r { 1306 | r[i] = int8(1 & (a[i>>3] >> uint(i&7))) 1307 | } 1308 | 1309 | for i := range r { 1310 | if r[i] != 0 { 1311 | for b := 1; b <= 6 && i+b < 256; b++ { 1312 | if r[i+b] != 0 { 1313 | if r[i]+(r[i+b]<= -15 { 1317 | r[i] -= r[i+b] << uint(b) 1318 | for k := i + b; k < 256; k++ { 1319 | if r[k] == 0 { 1320 | r[k] = 1 1321 | break 1322 | } 1323 | r[k] = 0 1324 | } 1325 | } else { 1326 | break 1327 | } 1328 | } 1329 | } 1330 | } 1331 | } 1332 | } 1333 | 1334 | // GeAdd sets r = a+b. r may overlaop with a and b. 1335 | func GeAdd(r, a, b *ExtendedGroupElement) { 1336 | var bca CachedGroupElement 1337 | b.ToCached(&bca) 1338 | var rc CompletedGroupElement 1339 | geAdd(&rc, a, &bca) 1340 | rc.ToExtended(r) 1341 | } 1342 | 1343 | func ExtendedGroupElementCopy(t, u *ExtendedGroupElement) { 1344 | FeCopy(&t.X, &u.X) 1345 | FeCopy(&t.Y, &u.Y) 1346 | FeCopy(&t.Z, &u.Z) 1347 | FeCopy(&t.T, &u.T) 1348 | } 1349 | 1350 | func ExtendedGroupElementCMove(t, u *ExtendedGroupElement, b int32) { 1351 | FeCMove(&t.X, &u.X, b) 1352 | FeCMove(&t.Y, &u.Y, b) 1353 | FeCMove(&t.Z, &u.Z, b) 1354 | FeCMove(&t.T, &u.T, b) 1355 | } 1356 | 1357 | // GeScalarMult sets r = a*A 1358 | // where a = a[0]+256*a[1]+...+256^31 a[31]. 1359 | func GeScalarMult(r *ExtendedGroupElement, a *[32]byte, A *ExtendedGroupElement) { 1360 | var p, q ExtendedGroupElement 1361 | q.Zero() 1362 | ExtendedGroupElementCopy(&p, A) 1363 | for i := uint(0); i < 256; i++ { 1364 | bit := int32(a[i>>3]>>(i&7)) & 1 1365 | var t ExtendedGroupElement 1366 | GeAdd(&t, &q, &p) 1367 | ExtendedGroupElementCMove(&q, &t, bit) 1368 | GeDouble(&p, &p) 1369 | } 1370 | ExtendedGroupElementCopy(r, &q) 1371 | } 1372 | 1373 | // GeIsNeutral 1374 | // returns 1 if p is the neutral point 1375 | // returns 0 otherwise 1376 | func GeIsNeutral(p *ExtendedGroupElement) bool { 1377 | var zero FieldElement 1378 | FeZero(&zero) 1379 | // Check if p == neutral element == (0, 1) 1380 | return FeIsequal(p.X, zero)&FeIsequal(p.Y, p.Z) == 1 1381 | } 1382 | 1383 | // GeDoubleScalarMultVartime sets r = a*A + b*B 1384 | // where a = a[0]+256*a[1]+...+256^31 a[31]. 1385 | // and b = b[0]+256*b[1]+...+256^31 b[31]. 1386 | // B is the Ed25519 base point (x,4/5) with x positive. 1387 | func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { 1388 | var aSlide, bSlide [256]int8 1389 | var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A 1390 | var t CompletedGroupElement 1391 | var u, A2 ExtendedGroupElement 1392 | var i int 1393 | 1394 | slide(&aSlide, a) 1395 | slide(&bSlide, b) 1396 | 1397 | A.ToCached(&Ai[0]) 1398 | A.Double(&t) 1399 | t.ToExtended(&A2) 1400 | 1401 | for i := 0; i < 7; i++ { 1402 | geAdd(&t, &A2, &Ai[i]) 1403 | t.ToExtended(&u) 1404 | u.ToCached(&Ai[i+1]) 1405 | } 1406 | 1407 | r.Zero() 1408 | 1409 | for i = 255; i >= 0; i-- { 1410 | if aSlide[i] != 0 || bSlide[i] != 0 { 1411 | break 1412 | } 1413 | } 1414 | 1415 | for ; i >= 0; i-- { 1416 | r.Double(&t) 1417 | 1418 | if aSlide[i] > 0 { 1419 | t.ToExtended(&u) 1420 | geAdd(&t, &u, &Ai[aSlide[i]/2]) 1421 | } else if aSlide[i] < 0 { 1422 | t.ToExtended(&u) 1423 | geSub(&t, &u, &Ai[(-aSlide[i])/2]) 1424 | } 1425 | 1426 | if bSlide[i] > 0 { 1427 | t.ToExtended(&u) 1428 | geMixedAdd(&t, &u, &bi[bSlide[i]/2]) 1429 | } else if bSlide[i] < 0 { 1430 | t.ToExtended(&u) 1431 | geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) 1432 | } 1433 | 1434 | t.ToProjective(r) 1435 | } 1436 | } 1437 | 1438 | func GeToMontX(u *FieldElement, ed *ExtendedGroupElement) { 1439 | /* 1440 | u = (y + 1) / (1 - y) 1441 | or 1442 | u = (y + z) / (z - y) 1443 | 1444 | NOTE: y=1 is converted to u=0 since fe_invert is mod-exp 1445 | */ 1446 | var yPlusOne, oneMinusY, invOneMinusY FieldElement 1447 | 1448 | FeAdd(&yPlusOne, &ed.Y, &ed.Z) 1449 | FeSub(&oneMinusY, &ed.Z, &ed.Y) 1450 | FeInvert(&invOneMinusY, &oneMinusY) 1451 | FeMul(u, &yPlusOne, &invOneMinusY) 1452 | 1453 | } 1454 | 1455 | // equal returns 1 if b == c and 0 otherwise. 1456 | func equal(b, c int32) int32 { 1457 | x := uint32(b ^ c) 1458 | x-- 1459 | return int32(x >> 31) 1460 | } 1461 | 1462 | // negative returns 1 if b < 0 and 0 otherwise. 1463 | func negative(b int32) int32 { 1464 | return (b >> 31) & 1 1465 | } 1466 | 1467 | func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { 1468 | FeCMove(&t.yPlusX, &u.yPlusX, b) 1469 | FeCMove(&t.yMinusX, &u.yMinusX, b) 1470 | FeCMove(&t.xy2d, &u.xy2d, b) 1471 | } 1472 | 1473 | func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { 1474 | var minusT PreComputedGroupElement 1475 | bNegative := negative(b) 1476 | bAbs := b - (((-bNegative) & b) << 1) 1477 | 1478 | t.Zero() 1479 | for i := int32(0); i < 8; i++ { 1480 | PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) 1481 | } 1482 | FeCopy(&minusT.yPlusX, &t.yMinusX) 1483 | FeCopy(&minusT.yMinusX, &t.yPlusX) 1484 | FeNeg(&minusT.xy2d, &t.xy2d) 1485 | PreComputedGroupElementCMove(t, &minusT, bNegative) 1486 | } 1487 | 1488 | // GeScalarMultBase computes h = a*B, where 1489 | // a = a[0]+256*a[1]+...+256^31 a[31] 1490 | // B is the Ed25519 base point (x,4/5) with x positive. 1491 | // 1492 | // Preconditions: 1493 | // a[31] <= 127 1494 | func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { 1495 | var e [64]int8 1496 | 1497 | for i, v := range a { 1498 | e[2*i] = int8(v & 15) 1499 | e[2*i+1] = int8((v >> 4) & 15) 1500 | } 1501 | 1502 | // each e[i] is between 0 and 15 and e[63] is between 0 and 7. 1503 | 1504 | carry := int8(0) 1505 | for i := 0; i < 63; i++ { 1506 | e[i] += carry 1507 | carry = (e[i] + 8) >> 4 1508 | e[i] -= carry << 4 1509 | } 1510 | e[63] += carry 1511 | // each e[i] is between -8 and 8. 1512 | 1513 | h.Zero() 1514 | var t PreComputedGroupElement 1515 | var r CompletedGroupElement 1516 | for i := int32(1); i < 64; i += 2 { 1517 | selectPoint(&t, i/2, int32(e[i])) 1518 | geMixedAdd(&r, h, &t) 1519 | r.ToExtended(h) 1520 | } 1521 | 1522 | var s ProjectiveGroupElement 1523 | 1524 | h.Double(&r) 1525 | r.ToProjective(&s) 1526 | s.Double(&r) 1527 | r.ToProjective(&s) 1528 | s.Double(&r) 1529 | r.ToProjective(&s) 1530 | s.Double(&r) 1531 | r.ToExtended(h) 1532 | 1533 | for i := int32(0); i < 64; i += 2 { 1534 | selectPoint(&t, i/2, int32(e[i])) 1535 | geMixedAdd(&r, h, &t) 1536 | r.ToExtended(h) 1537 | } 1538 | } 1539 | 1540 | // The scalars are GF(2^252 + 27742317777372353535851937790883648493). 1541 | 1542 | // Input: 1543 | // a[0]+256*a[1]+...+256^31*a[31] = a 1544 | // b[0]+256*b[1]+...+256^31*b[31] = b 1545 | // c[0]+256*c[1]+...+256^31*c[31] = c 1546 | // 1547 | // Output: 1548 | // s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l 1549 | // where l = 2^252 + 27742317777372353535851937790883648493. 1550 | func ScMulAdd(s, a, b, c *[32]byte) { 1551 | a0 := 2097151 & load3(a[:]) 1552 | a1 := 2097151 & (load4(a[2:]) >> 5) 1553 | a2 := 2097151 & (load3(a[5:]) >> 2) 1554 | a3 := 2097151 & (load4(a[7:]) >> 7) 1555 | a4 := 2097151 & (load4(a[10:]) >> 4) 1556 | a5 := 2097151 & (load3(a[13:]) >> 1) 1557 | a6 := 2097151 & (load4(a[15:]) >> 6) 1558 | a7 := 2097151 & (load3(a[18:]) >> 3) 1559 | a8 := 2097151 & load3(a[21:]) 1560 | a9 := 2097151 & (load4(a[23:]) >> 5) 1561 | a10 := 2097151 & (load3(a[26:]) >> 2) 1562 | a11 := (load4(a[28:]) >> 7) 1563 | b0 := 2097151 & load3(b[:]) 1564 | b1 := 2097151 & (load4(b[2:]) >> 5) 1565 | b2 := 2097151 & (load3(b[5:]) >> 2) 1566 | b3 := 2097151 & (load4(b[7:]) >> 7) 1567 | b4 := 2097151 & (load4(b[10:]) >> 4) 1568 | b5 := 2097151 & (load3(b[13:]) >> 1) 1569 | b6 := 2097151 & (load4(b[15:]) >> 6) 1570 | b7 := 2097151 & (load3(b[18:]) >> 3) 1571 | b8 := 2097151 & load3(b[21:]) 1572 | b9 := 2097151 & (load4(b[23:]) >> 5) 1573 | b10 := 2097151 & (load3(b[26:]) >> 2) 1574 | b11 := (load4(b[28:]) >> 7) 1575 | c0 := 2097151 & load3(c[:]) 1576 | c1 := 2097151 & (load4(c[2:]) >> 5) 1577 | c2 := 2097151 & (load3(c[5:]) >> 2) 1578 | c3 := 2097151 & (load4(c[7:]) >> 7) 1579 | c4 := 2097151 & (load4(c[10:]) >> 4) 1580 | c5 := 2097151 & (load3(c[13:]) >> 1) 1581 | c6 := 2097151 & (load4(c[15:]) >> 6) 1582 | c7 := 2097151 & (load3(c[18:]) >> 3) 1583 | c8 := 2097151 & load3(c[21:]) 1584 | c9 := 2097151 & (load4(c[23:]) >> 5) 1585 | c10 := 2097151 & (load3(c[26:]) >> 2) 1586 | c11 := (load4(c[28:]) >> 7) 1587 | var carry [23]int64 1588 | 1589 | s0 := c0 + a0*b0 1590 | s1 := c1 + a0*b1 + a1*b0 1591 | s2 := c2 + a0*b2 + a1*b1 + a2*b0 1592 | s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 1593 | s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 1594 | s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 1595 | s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 1596 | s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 1597 | s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 1598 | s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 1599 | s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 1600 | 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 1601 | s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 1602 | s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 1603 | s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 1604 | s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 1605 | s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 1606 | s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 1607 | s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 1608 | s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 1609 | s20 := a9*b11 + a10*b10 + a11*b9 1610 | s21 := a10*b11 + a11*b10 1611 | s22 := a11 * b11 1612 | s23 := int64(0) 1613 | 1614 | carry[0] = (s0 + (1 << 20)) >> 21 1615 | s1 += carry[0] 1616 | s0 -= carry[0] << 21 1617 | carry[2] = (s2 + (1 << 20)) >> 21 1618 | s3 += carry[2] 1619 | s2 -= carry[2] << 21 1620 | carry[4] = (s4 + (1 << 20)) >> 21 1621 | s5 += carry[4] 1622 | s4 -= carry[4] << 21 1623 | carry[6] = (s6 + (1 << 20)) >> 21 1624 | s7 += carry[6] 1625 | s6 -= carry[6] << 21 1626 | carry[8] = (s8 + (1 << 20)) >> 21 1627 | s9 += carry[8] 1628 | s8 -= carry[8] << 21 1629 | carry[10] = (s10 + (1 << 20)) >> 21 1630 | s11 += carry[10] 1631 | s10 -= carry[10] << 21 1632 | carry[12] = (s12 + (1 << 20)) >> 21 1633 | s13 += carry[12] 1634 | s12 -= carry[12] << 21 1635 | carry[14] = (s14 + (1 << 20)) >> 21 1636 | s15 += carry[14] 1637 | s14 -= carry[14] << 21 1638 | carry[16] = (s16 + (1 << 20)) >> 21 1639 | s17 += carry[16] 1640 | s16 -= carry[16] << 21 1641 | carry[18] = (s18 + (1 << 20)) >> 21 1642 | s19 += carry[18] 1643 | s18 -= carry[18] << 21 1644 | carry[20] = (s20 + (1 << 20)) >> 21 1645 | s21 += carry[20] 1646 | s20 -= carry[20] << 21 1647 | carry[22] = (s22 + (1 << 20)) >> 21 1648 | s23 += carry[22] 1649 | s22 -= carry[22] << 21 1650 | 1651 | carry[1] = (s1 + (1 << 20)) >> 21 1652 | s2 += carry[1] 1653 | s1 -= carry[1] << 21 1654 | carry[3] = (s3 + (1 << 20)) >> 21 1655 | s4 += carry[3] 1656 | s3 -= carry[3] << 21 1657 | carry[5] = (s5 + (1 << 20)) >> 21 1658 | s6 += carry[5] 1659 | s5 -= carry[5] << 21 1660 | carry[7] = (s7 + (1 << 20)) >> 21 1661 | s8 += carry[7] 1662 | s7 -= carry[7] << 21 1663 | carry[9] = (s9 + (1 << 20)) >> 21 1664 | s10 += carry[9] 1665 | s9 -= carry[9] << 21 1666 | carry[11] = (s11 + (1 << 20)) >> 21 1667 | s12 += carry[11] 1668 | s11 -= carry[11] << 21 1669 | carry[13] = (s13 + (1 << 20)) >> 21 1670 | s14 += carry[13] 1671 | s13 -= carry[13] << 21 1672 | carry[15] = (s15 + (1 << 20)) >> 21 1673 | s16 += carry[15] 1674 | s15 -= carry[15] << 21 1675 | carry[17] = (s17 + (1 << 20)) >> 21 1676 | s18 += carry[17] 1677 | s17 -= carry[17] << 21 1678 | carry[19] = (s19 + (1 << 20)) >> 21 1679 | s20 += carry[19] 1680 | s19 -= carry[19] << 21 1681 | carry[21] = (s21 + (1 << 20)) >> 21 1682 | s22 += carry[21] 1683 | s21 -= carry[21] << 21 1684 | 1685 | s11 += s23 * 666643 1686 | s12 += s23 * 470296 1687 | s13 += s23 * 654183 1688 | s14 -= s23 * 997805 1689 | s15 += s23 * 136657 1690 | s16 -= s23 * 683901 1691 | s23 = 0 1692 | 1693 | s10 += s22 * 666643 1694 | s11 += s22 * 470296 1695 | s12 += s22 * 654183 1696 | s13 -= s22 * 997805 1697 | s14 += s22 * 136657 1698 | s15 -= s22 * 683901 1699 | s22 = 0 1700 | 1701 | s9 += s21 * 666643 1702 | s10 += s21 * 470296 1703 | s11 += s21 * 654183 1704 | s12 -= s21 * 997805 1705 | s13 += s21 * 136657 1706 | s14 -= s21 * 683901 1707 | s21 = 0 1708 | 1709 | s8 += s20 * 666643 1710 | s9 += s20 * 470296 1711 | s10 += s20 * 654183 1712 | s11 -= s20 * 997805 1713 | s12 += s20 * 136657 1714 | s13 -= s20 * 683901 1715 | s20 = 0 1716 | 1717 | s7 += s19 * 666643 1718 | s8 += s19 * 470296 1719 | s9 += s19 * 654183 1720 | s10 -= s19 * 997805 1721 | s11 += s19 * 136657 1722 | s12 -= s19 * 683901 1723 | s19 = 0 1724 | 1725 | s6 += s18 * 666643 1726 | s7 += s18 * 470296 1727 | s8 += s18 * 654183 1728 | s9 -= s18 * 997805 1729 | s10 += s18 * 136657 1730 | s11 -= s18 * 683901 1731 | s18 = 0 1732 | 1733 | carry[6] = (s6 + (1 << 20)) >> 21 1734 | s7 += carry[6] 1735 | s6 -= carry[6] << 21 1736 | carry[8] = (s8 + (1 << 20)) >> 21 1737 | s9 += carry[8] 1738 | s8 -= carry[8] << 21 1739 | carry[10] = (s10 + (1 << 20)) >> 21 1740 | s11 += carry[10] 1741 | s10 -= carry[10] << 21 1742 | carry[12] = (s12 + (1 << 20)) >> 21 1743 | s13 += carry[12] 1744 | s12 -= carry[12] << 21 1745 | carry[14] = (s14 + (1 << 20)) >> 21 1746 | s15 += carry[14] 1747 | s14 -= carry[14] << 21 1748 | carry[16] = (s16 + (1 << 20)) >> 21 1749 | s17 += carry[16] 1750 | s16 -= carry[16] << 21 1751 | 1752 | carry[7] = (s7 + (1 << 20)) >> 21 1753 | s8 += carry[7] 1754 | s7 -= carry[7] << 21 1755 | carry[9] = (s9 + (1 << 20)) >> 21 1756 | s10 += carry[9] 1757 | s9 -= carry[9] << 21 1758 | carry[11] = (s11 + (1 << 20)) >> 21 1759 | s12 += carry[11] 1760 | s11 -= carry[11] << 21 1761 | carry[13] = (s13 + (1 << 20)) >> 21 1762 | s14 += carry[13] 1763 | s13 -= carry[13] << 21 1764 | carry[15] = (s15 + (1 << 20)) >> 21 1765 | s16 += carry[15] 1766 | s15 -= carry[15] << 21 1767 | 1768 | s5 += s17 * 666643 1769 | s6 += s17 * 470296 1770 | s7 += s17 * 654183 1771 | s8 -= s17 * 997805 1772 | s9 += s17 * 136657 1773 | s10 -= s17 * 683901 1774 | s17 = 0 1775 | 1776 | s4 += s16 * 666643 1777 | s5 += s16 * 470296 1778 | s6 += s16 * 654183 1779 | s7 -= s16 * 997805 1780 | s8 += s16 * 136657 1781 | s9 -= s16 * 683901 1782 | s16 = 0 1783 | 1784 | s3 += s15 * 666643 1785 | s4 += s15 * 470296 1786 | s5 += s15 * 654183 1787 | s6 -= s15 * 997805 1788 | s7 += s15 * 136657 1789 | s8 -= s15 * 683901 1790 | s15 = 0 1791 | 1792 | s2 += s14 * 666643 1793 | s3 += s14 * 470296 1794 | s4 += s14 * 654183 1795 | s5 -= s14 * 997805 1796 | s6 += s14 * 136657 1797 | s7 -= s14 * 683901 1798 | s14 = 0 1799 | 1800 | s1 += s13 * 666643 1801 | s2 += s13 * 470296 1802 | s3 += s13 * 654183 1803 | s4 -= s13 * 997805 1804 | s5 += s13 * 136657 1805 | s6 -= s13 * 683901 1806 | s13 = 0 1807 | 1808 | s0 += s12 * 666643 1809 | s1 += s12 * 470296 1810 | s2 += s12 * 654183 1811 | s3 -= s12 * 997805 1812 | s4 += s12 * 136657 1813 | s5 -= s12 * 683901 1814 | s12 = 0 1815 | 1816 | carry[0] = (s0 + (1 << 20)) >> 21 1817 | s1 += carry[0] 1818 | s0 -= carry[0] << 21 1819 | carry[2] = (s2 + (1 << 20)) >> 21 1820 | s3 += carry[2] 1821 | s2 -= carry[2] << 21 1822 | carry[4] = (s4 + (1 << 20)) >> 21 1823 | s5 += carry[4] 1824 | s4 -= carry[4] << 21 1825 | carry[6] = (s6 + (1 << 20)) >> 21 1826 | s7 += carry[6] 1827 | s6 -= carry[6] << 21 1828 | carry[8] = (s8 + (1 << 20)) >> 21 1829 | s9 += carry[8] 1830 | s8 -= carry[8] << 21 1831 | carry[10] = (s10 + (1 << 20)) >> 21 1832 | s11 += carry[10] 1833 | s10 -= carry[10] << 21 1834 | 1835 | carry[1] = (s1 + (1 << 20)) >> 21 1836 | s2 += carry[1] 1837 | s1 -= carry[1] << 21 1838 | carry[3] = (s3 + (1 << 20)) >> 21 1839 | s4 += carry[3] 1840 | s3 -= carry[3] << 21 1841 | carry[5] = (s5 + (1 << 20)) >> 21 1842 | s6 += carry[5] 1843 | s5 -= carry[5] << 21 1844 | carry[7] = (s7 + (1 << 20)) >> 21 1845 | s8 += carry[7] 1846 | s7 -= carry[7] << 21 1847 | carry[9] = (s9 + (1 << 20)) >> 21 1848 | s10 += carry[9] 1849 | s9 -= carry[9] << 21 1850 | carry[11] = (s11 + (1 << 20)) >> 21 1851 | s12 += carry[11] 1852 | s11 -= carry[11] << 21 1853 | 1854 | s0 += s12 * 666643 1855 | s1 += s12 * 470296 1856 | s2 += s12 * 654183 1857 | s3 -= s12 * 997805 1858 | s4 += s12 * 136657 1859 | s5 -= s12 * 683901 1860 | s12 = 0 1861 | 1862 | carry[0] = s0 >> 21 1863 | s1 += carry[0] 1864 | s0 -= carry[0] << 21 1865 | carry[1] = s1 >> 21 1866 | s2 += carry[1] 1867 | s1 -= carry[1] << 21 1868 | carry[2] = s2 >> 21 1869 | s3 += carry[2] 1870 | s2 -= carry[2] << 21 1871 | carry[3] = s3 >> 21 1872 | s4 += carry[3] 1873 | s3 -= carry[3] << 21 1874 | carry[4] = s4 >> 21 1875 | s5 += carry[4] 1876 | s4 -= carry[4] << 21 1877 | carry[5] = s5 >> 21 1878 | s6 += carry[5] 1879 | s5 -= carry[5] << 21 1880 | carry[6] = s6 >> 21 1881 | s7 += carry[6] 1882 | s6 -= carry[6] << 21 1883 | carry[7] = s7 >> 21 1884 | s8 += carry[7] 1885 | s7 -= carry[7] << 21 1886 | carry[8] = s8 >> 21 1887 | s9 += carry[8] 1888 | s8 -= carry[8] << 21 1889 | carry[9] = s9 >> 21 1890 | s10 += carry[9] 1891 | s9 -= carry[9] << 21 1892 | carry[10] = s10 >> 21 1893 | s11 += carry[10] 1894 | s10 -= carry[10] << 21 1895 | carry[11] = s11 >> 21 1896 | s12 += carry[11] 1897 | s11 -= carry[11] << 21 1898 | 1899 | s0 += s12 * 666643 1900 | s1 += s12 * 470296 1901 | s2 += s12 * 654183 1902 | s3 -= s12 * 997805 1903 | s4 += s12 * 136657 1904 | s5 -= s12 * 683901 1905 | s12 = 0 1906 | 1907 | carry[0] = s0 >> 21 1908 | s1 += carry[0] 1909 | s0 -= carry[0] << 21 1910 | carry[1] = s1 >> 21 1911 | s2 += carry[1] 1912 | s1 -= carry[1] << 21 1913 | carry[2] = s2 >> 21 1914 | s3 += carry[2] 1915 | s2 -= carry[2] << 21 1916 | carry[3] = s3 >> 21 1917 | s4 += carry[3] 1918 | s3 -= carry[3] << 21 1919 | carry[4] = s4 >> 21 1920 | s5 += carry[4] 1921 | s4 -= carry[4] << 21 1922 | carry[5] = s5 >> 21 1923 | s6 += carry[5] 1924 | s5 -= carry[5] << 21 1925 | carry[6] = s6 >> 21 1926 | s7 += carry[6] 1927 | s6 -= carry[6] << 21 1928 | carry[7] = s7 >> 21 1929 | s8 += carry[7] 1930 | s7 -= carry[7] << 21 1931 | carry[8] = s8 >> 21 1932 | s9 += carry[8] 1933 | s8 -= carry[8] << 21 1934 | carry[9] = s9 >> 21 1935 | s10 += carry[9] 1936 | s9 -= carry[9] << 21 1937 | carry[10] = s10 >> 21 1938 | s11 += carry[10] 1939 | s10 -= carry[10] << 21 1940 | 1941 | s[0] = byte(s0 >> 0) 1942 | s[1] = byte(s0 >> 8) 1943 | s[2] = byte((s0 >> 16) | (s1 << 5)) 1944 | s[3] = byte(s1 >> 3) 1945 | s[4] = byte(s1 >> 11) 1946 | s[5] = byte((s1 >> 19) | (s2 << 2)) 1947 | s[6] = byte(s2 >> 6) 1948 | s[7] = byte((s2 >> 14) | (s3 << 7)) 1949 | s[8] = byte(s3 >> 1) 1950 | s[9] = byte(s3 >> 9) 1951 | s[10] = byte((s3 >> 17) | (s4 << 4)) 1952 | s[11] = byte(s4 >> 4) 1953 | s[12] = byte(s4 >> 12) 1954 | s[13] = byte((s4 >> 20) | (s5 << 1)) 1955 | s[14] = byte(s5 >> 7) 1956 | s[15] = byte((s5 >> 15) | (s6 << 6)) 1957 | s[16] = byte(s6 >> 2) 1958 | s[17] = byte(s6 >> 10) 1959 | s[18] = byte((s6 >> 18) | (s7 << 3)) 1960 | s[19] = byte(s7 >> 5) 1961 | s[20] = byte(s7 >> 13) 1962 | s[21] = byte(s8 >> 0) 1963 | s[22] = byte(s8 >> 8) 1964 | s[23] = byte((s8 >> 16) | (s9 << 5)) 1965 | s[24] = byte(s9 >> 3) 1966 | s[25] = byte(s9 >> 11) 1967 | s[26] = byte((s9 >> 19) | (s10 << 2)) 1968 | s[27] = byte(s10 >> 6) 1969 | s[28] = byte((s10 >> 14) | (s11 << 7)) 1970 | s[29] = byte(s11 >> 1) 1971 | s[30] = byte(s11 >> 9) 1972 | s[31] = byte(s11 >> 17) 1973 | } 1974 | 1975 | var lMinus1 = [32]byte{0xec, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 1976 | 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14, 1977 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 1978 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10} 1979 | 1980 | // ScNeg computes: 1981 | // b = -a (mod l) 1982 | // where l = 2^252 + 27742317777372353535851937790883648493. 1983 | func ScNeg(b, a *[32]byte) { 1984 | var zero [32]byte 1985 | ScMulAdd(b, &lMinus1, a, &zero) 1986 | } 1987 | 1988 | // Input: 1989 | // s[0]+256*s[1]+...+256^63*s[63] = s 1990 | // 1991 | // Output: 1992 | // s[0]+256*s[1]+...+256^31*s[31] = s mod l 1993 | // where l = 2^252 + 27742317777372353535851937790883648493. 1994 | func ScReduce(out *[32]byte, s *[64]byte) { 1995 | s0 := 2097151 & load3(s[:]) 1996 | s1 := 2097151 & (load4(s[2:]) >> 5) 1997 | s2 := 2097151 & (load3(s[5:]) >> 2) 1998 | s3 := 2097151 & (load4(s[7:]) >> 7) 1999 | s4 := 2097151 & (load4(s[10:]) >> 4) 2000 | s5 := 2097151 & (load3(s[13:]) >> 1) 2001 | s6 := 2097151 & (load4(s[15:]) >> 6) 2002 | s7 := 2097151 & (load3(s[18:]) >> 3) 2003 | s8 := 2097151 & load3(s[21:]) 2004 | s9 := 2097151 & (load4(s[23:]) >> 5) 2005 | s10 := 2097151 & (load3(s[26:]) >> 2) 2006 | s11 := 2097151 & (load4(s[28:]) >> 7) 2007 | s12 := 2097151 & (load4(s[31:]) >> 4) 2008 | s13 := 2097151 & (load3(s[34:]) >> 1) 2009 | s14 := 2097151 & (load4(s[36:]) >> 6) 2010 | s15 := 2097151 & (load3(s[39:]) >> 3) 2011 | s16 := 2097151 & load3(s[42:]) 2012 | s17 := 2097151 & (load4(s[44:]) >> 5) 2013 | s18 := 2097151 & (load3(s[47:]) >> 2) 2014 | s19 := 2097151 & (load4(s[49:]) >> 7) 2015 | s20 := 2097151 & (load4(s[52:]) >> 4) 2016 | s21 := 2097151 & (load3(s[55:]) >> 1) 2017 | s22 := 2097151 & (load4(s[57:]) >> 6) 2018 | s23 := (load4(s[60:]) >> 3) 2019 | 2020 | s11 += s23 * 666643 2021 | s12 += s23 * 470296 2022 | s13 += s23 * 654183 2023 | s14 -= s23 * 997805 2024 | s15 += s23 * 136657 2025 | s16 -= s23 * 683901 2026 | s23 = 0 2027 | 2028 | s10 += s22 * 666643 2029 | s11 += s22 * 470296 2030 | s12 += s22 * 654183 2031 | s13 -= s22 * 997805 2032 | s14 += s22 * 136657 2033 | s15 -= s22 * 683901 2034 | s22 = 0 2035 | 2036 | s9 += s21 * 666643 2037 | s10 += s21 * 470296 2038 | s11 += s21 * 654183 2039 | s12 -= s21 * 997805 2040 | s13 += s21 * 136657 2041 | s14 -= s21 * 683901 2042 | s21 = 0 2043 | 2044 | s8 += s20 * 666643 2045 | s9 += s20 * 470296 2046 | s10 += s20 * 654183 2047 | s11 -= s20 * 997805 2048 | s12 += s20 * 136657 2049 | s13 -= s20 * 683901 2050 | s20 = 0 2051 | 2052 | s7 += s19 * 666643 2053 | s8 += s19 * 470296 2054 | s9 += s19 * 654183 2055 | s10 -= s19 * 997805 2056 | s11 += s19 * 136657 2057 | s12 -= s19 * 683901 2058 | s19 = 0 2059 | 2060 | s6 += s18 * 666643 2061 | s7 += s18 * 470296 2062 | s8 += s18 * 654183 2063 | s9 -= s18 * 997805 2064 | s10 += s18 * 136657 2065 | s11 -= s18 * 683901 2066 | s18 = 0 2067 | 2068 | var carry [17]int64 2069 | 2070 | carry[6] = (s6 + (1 << 20)) >> 21 2071 | s7 += carry[6] 2072 | s6 -= carry[6] << 21 2073 | carry[8] = (s8 + (1 << 20)) >> 21 2074 | s9 += carry[8] 2075 | s8 -= carry[8] << 21 2076 | carry[10] = (s10 + (1 << 20)) >> 21 2077 | s11 += carry[10] 2078 | s10 -= carry[10] << 21 2079 | carry[12] = (s12 + (1 << 20)) >> 21 2080 | s13 += carry[12] 2081 | s12 -= carry[12] << 21 2082 | carry[14] = (s14 + (1 << 20)) >> 21 2083 | s15 += carry[14] 2084 | s14 -= carry[14] << 21 2085 | carry[16] = (s16 + (1 << 20)) >> 21 2086 | s17 += carry[16] 2087 | s16 -= carry[16] << 21 2088 | 2089 | carry[7] = (s7 + (1 << 20)) >> 21 2090 | s8 += carry[7] 2091 | s7 -= carry[7] << 21 2092 | carry[9] = (s9 + (1 << 20)) >> 21 2093 | s10 += carry[9] 2094 | s9 -= carry[9] << 21 2095 | carry[11] = (s11 + (1 << 20)) >> 21 2096 | s12 += carry[11] 2097 | s11 -= carry[11] << 21 2098 | carry[13] = (s13 + (1 << 20)) >> 21 2099 | s14 += carry[13] 2100 | s13 -= carry[13] << 21 2101 | carry[15] = (s15 + (1 << 20)) >> 21 2102 | s16 += carry[15] 2103 | s15 -= carry[15] << 21 2104 | 2105 | s5 += s17 * 666643 2106 | s6 += s17 * 470296 2107 | s7 += s17 * 654183 2108 | s8 -= s17 * 997805 2109 | s9 += s17 * 136657 2110 | s10 -= s17 * 683901 2111 | s17 = 0 2112 | 2113 | s4 += s16 * 666643 2114 | s5 += s16 * 470296 2115 | s6 += s16 * 654183 2116 | s7 -= s16 * 997805 2117 | s8 += s16 * 136657 2118 | s9 -= s16 * 683901 2119 | s16 = 0 2120 | 2121 | s3 += s15 * 666643 2122 | s4 += s15 * 470296 2123 | s5 += s15 * 654183 2124 | s6 -= s15 * 997805 2125 | s7 += s15 * 136657 2126 | s8 -= s15 * 683901 2127 | s15 = 0 2128 | 2129 | s2 += s14 * 666643 2130 | s3 += s14 * 470296 2131 | s4 += s14 * 654183 2132 | s5 -= s14 * 997805 2133 | s6 += s14 * 136657 2134 | s7 -= s14 * 683901 2135 | s14 = 0 2136 | 2137 | s1 += s13 * 666643 2138 | s2 += s13 * 470296 2139 | s3 += s13 * 654183 2140 | s4 -= s13 * 997805 2141 | s5 += s13 * 136657 2142 | s6 -= s13 * 683901 2143 | s13 = 0 2144 | 2145 | s0 += s12 * 666643 2146 | s1 += s12 * 470296 2147 | s2 += s12 * 654183 2148 | s3 -= s12 * 997805 2149 | s4 += s12 * 136657 2150 | s5 -= s12 * 683901 2151 | s12 = 0 2152 | 2153 | carry[0] = (s0 + (1 << 20)) >> 21 2154 | s1 += carry[0] 2155 | s0 -= carry[0] << 21 2156 | carry[2] = (s2 + (1 << 20)) >> 21 2157 | s3 += carry[2] 2158 | s2 -= carry[2] << 21 2159 | carry[4] = (s4 + (1 << 20)) >> 21 2160 | s5 += carry[4] 2161 | s4 -= carry[4] << 21 2162 | carry[6] = (s6 + (1 << 20)) >> 21 2163 | s7 += carry[6] 2164 | s6 -= carry[6] << 21 2165 | carry[8] = (s8 + (1 << 20)) >> 21 2166 | s9 += carry[8] 2167 | s8 -= carry[8] << 21 2168 | carry[10] = (s10 + (1 << 20)) >> 21 2169 | s11 += carry[10] 2170 | s10 -= carry[10] << 21 2171 | 2172 | carry[1] = (s1 + (1 << 20)) >> 21 2173 | s2 += carry[1] 2174 | s1 -= carry[1] << 21 2175 | carry[3] = (s3 + (1 << 20)) >> 21 2176 | s4 += carry[3] 2177 | s3 -= carry[3] << 21 2178 | carry[5] = (s5 + (1 << 20)) >> 21 2179 | s6 += carry[5] 2180 | s5 -= carry[5] << 21 2181 | carry[7] = (s7 + (1 << 20)) >> 21 2182 | s8 += carry[7] 2183 | s7 -= carry[7] << 21 2184 | carry[9] = (s9 + (1 << 20)) >> 21 2185 | s10 += carry[9] 2186 | s9 -= carry[9] << 21 2187 | carry[11] = (s11 + (1 << 20)) >> 21 2188 | s12 += carry[11] 2189 | s11 -= carry[11] << 21 2190 | 2191 | s0 += s12 * 666643 2192 | s1 += s12 * 470296 2193 | s2 += s12 * 654183 2194 | s3 -= s12 * 997805 2195 | s4 += s12 * 136657 2196 | s5 -= s12 * 683901 2197 | s12 = 0 2198 | 2199 | carry[0] = s0 >> 21 2200 | s1 += carry[0] 2201 | s0 -= carry[0] << 21 2202 | carry[1] = s1 >> 21 2203 | s2 += carry[1] 2204 | s1 -= carry[1] << 21 2205 | carry[2] = s2 >> 21 2206 | s3 += carry[2] 2207 | s2 -= carry[2] << 21 2208 | carry[3] = s3 >> 21 2209 | s4 += carry[3] 2210 | s3 -= carry[3] << 21 2211 | carry[4] = s4 >> 21 2212 | s5 += carry[4] 2213 | s4 -= carry[4] << 21 2214 | carry[5] = s5 >> 21 2215 | s6 += carry[5] 2216 | s5 -= carry[5] << 21 2217 | carry[6] = s6 >> 21 2218 | s7 += carry[6] 2219 | s6 -= carry[6] << 21 2220 | carry[7] = s7 >> 21 2221 | s8 += carry[7] 2222 | s7 -= carry[7] << 21 2223 | carry[8] = s8 >> 21 2224 | s9 += carry[8] 2225 | s8 -= carry[8] << 21 2226 | carry[9] = s9 >> 21 2227 | s10 += carry[9] 2228 | s9 -= carry[9] << 21 2229 | carry[10] = s10 >> 21 2230 | s11 += carry[10] 2231 | s10 -= carry[10] << 21 2232 | carry[11] = s11 >> 21 2233 | s12 += carry[11] 2234 | s11 -= carry[11] << 21 2235 | 2236 | s0 += s12 * 666643 2237 | s1 += s12 * 470296 2238 | s2 += s12 * 654183 2239 | s3 -= s12 * 997805 2240 | s4 += s12 * 136657 2241 | s5 -= s12 * 683901 2242 | s12 = 0 2243 | 2244 | carry[0] = s0 >> 21 2245 | s1 += carry[0] 2246 | s0 -= carry[0] << 21 2247 | carry[1] = s1 >> 21 2248 | s2 += carry[1] 2249 | s1 -= carry[1] << 21 2250 | carry[2] = s2 >> 21 2251 | s3 += carry[2] 2252 | s2 -= carry[2] << 21 2253 | carry[3] = s3 >> 21 2254 | s4 += carry[3] 2255 | s3 -= carry[3] << 21 2256 | carry[4] = s4 >> 21 2257 | s5 += carry[4] 2258 | s4 -= carry[4] << 21 2259 | carry[5] = s5 >> 21 2260 | s6 += carry[5] 2261 | s5 -= carry[5] << 21 2262 | carry[6] = s6 >> 21 2263 | s7 += carry[6] 2264 | s6 -= carry[6] << 21 2265 | carry[7] = s7 >> 21 2266 | s8 += carry[7] 2267 | s7 -= carry[7] << 21 2268 | carry[8] = s8 >> 21 2269 | s9 += carry[8] 2270 | s8 -= carry[8] << 21 2271 | carry[9] = s9 >> 21 2272 | s10 += carry[9] 2273 | s9 -= carry[9] << 21 2274 | carry[10] = s10 >> 21 2275 | s11 += carry[10] 2276 | s10 -= carry[10] << 21 2277 | 2278 | out[0] = byte(s0 >> 0) 2279 | out[1] = byte(s0 >> 8) 2280 | out[2] = byte((s0 >> 16) | (s1 << 5)) 2281 | out[3] = byte(s1 >> 3) 2282 | out[4] = byte(s1 >> 11) 2283 | out[5] = byte((s1 >> 19) | (s2 << 2)) 2284 | out[6] = byte(s2 >> 6) 2285 | out[7] = byte((s2 >> 14) | (s3 << 7)) 2286 | out[8] = byte(s3 >> 1) 2287 | out[9] = byte(s3 >> 9) 2288 | out[10] = byte((s3 >> 17) | (s4 << 4)) 2289 | out[11] = byte(s4 >> 4) 2290 | out[12] = byte(s4 >> 12) 2291 | out[13] = byte((s4 >> 20) | (s5 << 1)) 2292 | out[14] = byte(s5 >> 7) 2293 | out[15] = byte((s5 >> 15) | (s6 << 6)) 2294 | out[16] = byte(s6 >> 2) 2295 | out[17] = byte(s6 >> 10) 2296 | out[18] = byte((s6 >> 18) | (s7 << 3)) 2297 | out[19] = byte(s7 >> 5) 2298 | out[20] = byte(s7 >> 13) 2299 | out[21] = byte(s8 >> 0) 2300 | out[22] = byte(s8 >> 8) 2301 | out[23] = byte((s8 >> 16) | (s9 << 5)) 2302 | out[24] = byte(s9 >> 3) 2303 | out[25] = byte(s9 >> 11) 2304 | out[26] = byte((s9 >> 19) | (s10 << 2)) 2305 | out[27] = byte(s10 >> 6) 2306 | out[28] = byte((s10 >> 14) | (s11 << 7)) 2307 | out[29] = byte(s11 >> 1) 2308 | out[30] = byte(s11 >> 9) 2309 | out[31] = byte(s11 >> 17) 2310 | } 2311 | 2312 | // ScCMove is equivalent to FeCMove but operates directly on the [32]byte 2313 | // representation instead on the FieldElement. Can be used to spare a 2314 | // FieldElement.FromBytes operation. 2315 | func ScCMove(f, g *[32]byte, b int32) { 2316 | var x [32]byte 2317 | for i := range x { 2318 | x[i] = (f[i] ^ g[i]) 2319 | } 2320 | b = -b 2321 | for i := range x { 2322 | x[i] &= byte(b) 2323 | } 2324 | for i := range f { 2325 | f[i] ^= x[i] 2326 | } 2327 | } 2328 | 2329 | // ScClamp Sets and clears bits to make a random 32 bytes into a private key 2330 | func ScClamp(a *[32]byte) { 2331 | a[0] &= 248 2332 | a[31] &= 127 2333 | a[31] |= 64 2334 | } 2335 | -------------------------------------------------------------------------------- /edwards25519/const.go: -------------------------------------------------------------------------------- 1 | // Copyright 2013 The Go Authors. All rights reserved. 2 | // Use of this source code is governed by a BSD-style 3 | // license that can be found in the LICENSE file. 4 | 5 | package edwards25519 6 | 7 | // BasePointOrder is the number of points in the subgroup generated by the base point. 8 | var BasePointOrder = [32]byte{237, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16} 9 | 10 | var d = FieldElement{ 11 | -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, 12 | } 13 | 14 | var d2 = FieldElement{ 15 | -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, 16 | } 17 | 18 | var SqrtM1 = FieldElement{ 19 | -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482, 20 | } 21 | 22 | var A = FieldElement{ 23 | 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24 | } 25 | 26 | var bi = [8]PreComputedGroupElement{ 27 | { 28 | FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, 29 | FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, 30 | FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, 31 | }, 32 | { 33 | FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, 34 | FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, 35 | FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, 36 | }, 37 | { 38 | FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, 39 | FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, 40 | FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, 41 | }, 42 | { 43 | FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, 44 | FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, 45 | FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, 46 | }, 47 | { 48 | FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877}, 49 | FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951}, 50 | FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784}, 51 | }, 52 | { 53 | FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436}, 54 | FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918}, 55 | FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877}, 56 | }, 57 | { 58 | FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800}, 59 | FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305}, 60 | FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300}, 61 | }, 62 | { 63 | FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876}, 64 | FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619}, 65 | FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683}, 66 | }, 67 | } 68 | 69 | var base = [32][8]PreComputedGroupElement{ 70 | { 71 | { 72 | FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, 73 | FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, 74 | FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, 75 | }, 76 | { 77 | FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303}, 78 | FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081}, 79 | FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697}, 80 | }, 81 | { 82 | FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, 83 | FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, 84 | FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, 85 | }, 86 | { 87 | FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540}, 88 | FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397}, 89 | FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325}, 90 | }, 91 | { 92 | FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, 93 | FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, 94 | FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, 95 | }, 96 | { 97 | FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777}, 98 | FieldElement{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737}, 99 | FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652}, 100 | }, 101 | { 102 | FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, 103 | FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, 104 | FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, 105 | }, 106 | { 107 | FieldElement{14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726}, 108 | FieldElement{-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955}, 109 | FieldElement{27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425}, 110 | }, 111 | }, 112 | { 113 | { 114 | FieldElement{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171}, 115 | FieldElement{27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510}, 116 | FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660}, 117 | }, 118 | { 119 | FieldElement{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639}, 120 | FieldElement{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963}, 121 | FieldElement{5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950}, 122 | }, 123 | { 124 | FieldElement{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568}, 125 | FieldElement{12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335}, 126 | FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628}, 127 | }, 128 | { 129 | FieldElement{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007}, 130 | FieldElement{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772}, 131 | FieldElement{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653}, 132 | }, 133 | { 134 | FieldElement{2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, -7236665, 24316168, -5253567}, 135 | FieldElement{13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 33040651, -13424532, -20729456, 8321686}, 136 | FieldElement{21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 23845965, -11874838, -9984458, 608372}, 137 | }, 138 | { 139 | FieldElement{-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 1123968, -6780577, 27229399, 23887}, 140 | FieldElement{-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 12797024, -6440308, -1633405, 16678954}, 141 | FieldElement{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, -1508144, -4795045, -17169265, 4904953}, 142 | }, 143 | { 144 | FieldElement{24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 5169211, 16191880, 2128236, -4326833}, 145 | FieldElement{-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, -29806336, 916033, -6882542, -2986532}, 146 | FieldElement{-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 285431, 2763829, 15736322, 4143876}, 147 | }, 148 | { 149 | FieldElement{2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, -14594663, 23527084, -16458268}, 150 | FieldElement{33431127, -11130478, -17838966, -15626900, 8909499, 8376530, -32625340, 4087881, -15188911, -14416214}, 151 | FieldElement{1767683, 7197987, 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