├── README.md ├── crypto ├── doc.go ├── testutil.go ├── sign │ ├── sign_test.go │ └── sign.go ├── util_test.go ├── util.go ├── internal │ └── ed25519 │ │ ├── extra25519 │ │ ├── extra25519_test.go │ │ └── extra25519.go │ │ └── edwards25519 │ │ ├── edwards25519_test.go │ │ ├── edwards25519.go │ │ └── const.go └── vrf │ ├── vrf_test.go │ └── vrf.go ├── .gitignore └── LICENSE /README.md: -------------------------------------------------------------------------------- 1 | # CONIKS Go 2 | 3 | ## Golang Library 4 | 5 | The packages in this library implement the various components of the CONIKS 6 | system and may be imported individually. 7 | 8 | - `crypto`: Cryptographic algorithms and operations 9 | -------------------------------------------------------------------------------- /crypto/doc.go: -------------------------------------------------------------------------------- 1 | // Package crypto contains the cryptographic routines for CONIKS: 2 | // hashing, cryptographic commitments, digital signatures, VRF, 3 | // and random number generation. 4 | // 5 | // These cryptographic routines are used to: 6 | // 7 | // - hash arbitrary data (Digest) using SHA3 (SHAKE128), 8 | // 9 | // - create a cryptographic commit to arbitrary data, 10 | // 11 | // - generate a random slice of bytes, 12 | // 13 | // - sign data and verify signatures using Ed25519, 14 | // 15 | // - apply a VRF to data and verify the VRF proof. 16 | package crypto 17 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Compiled Object files, Static and Dynamic libs (Shared Objects) 2 | *.o 3 | *.a 4 | *.so 5 | *.dylib 6 | 7 | # Cgo compiled headers 8 | libconiks.h 9 | 10 | # Folders 11 | _obj 12 | _test 13 | 14 | # Architecture specific extensions/prefixes 15 | *.[568vq] 16 | [568vq].out 17 | 18 | *.cgo1.go 19 | *.cgo2.c 20 | _cgo_defun.c 21 | _cgo_gotypes.go 22 | _cgo_export.* 23 | 24 | _testmain.go 25 | 26 | *.exe 27 | *.test 28 | *.prof 29 | 30 | /vendor 31 | 32 | # vscode 33 | .vscode 34 | 35 | # CONIKS server 36 | config.toml 37 | botconfig.toml 38 | sign.priv 39 | sign.pub 40 | vrf.priv 41 | vrf.pub 42 | server.key 43 | server.pem 44 | coniks.db/ 45 | coniks.pid 46 | coniksserver.log 47 | 48 | .DS_Store 49 | *~ -------------------------------------------------------------------------------- /crypto/testutil.go: -------------------------------------------------------------------------------- 1 | package crypto 2 | 3 | import ( 4 | "bytes" 5 | 6 | "github.com/NKNetwork/coniks-go/crypto/sign" 7 | "github.com/NKNetwork/coniks-go/crypto/vrf" 8 | ) 9 | 10 | // NewStaticTestVRFKey returns a static VRF private key for _tests_. 11 | func NewStaticTestVRFKey() vrf.PrivateKey { 12 | sk, err := vrf.GenerateKey(bytes.NewReader( 13 | []byte("deterministic tests need 256 bit"))) 14 | if err != nil { 15 | panic(err) 16 | } 17 | return sk 18 | } 19 | 20 | // NewStaticTestSigningKey returns a static private signing key for _tests_. 21 | func NewStaticTestSigningKey() sign.PrivateKey { 22 | sk, err := sign.GenerateKey(bytes.NewReader( 23 | []byte("deterministic tests need 256 bit"))) 24 | if err != nil { 25 | panic(err) 26 | } 27 | return sk 28 | } 29 | -------------------------------------------------------------------------------- /crypto/sign/sign_test.go: -------------------------------------------------------------------------------- 1 | package sign 2 | 3 | import ( 4 | "bytes" 5 | "testing" 6 | ) 7 | 8 | // copied from official crypto.ed25519 tests 9 | func TestVerifySignature(t *testing.T) { 10 | key, err := GenerateKey(nil) 11 | if err != nil { 12 | t.Fatal(err) 13 | } 14 | 15 | message := []byte("test message") 16 | sig := key.Sign(message) 17 | 18 | pk, ok := key.Public() 19 | if !ok { 20 | t.Errorf("bad PK?") 21 | } 22 | 23 | if !pk.Verify(message, sig) { 24 | t.Errorf("valid signature rejected") 25 | } 26 | 27 | wrongMessage := []byte("wrong message") 28 | if pk.Verify(wrongMessage, sig) { 29 | t.Errorf("signature of different message accepted") 30 | } 31 | } 32 | 33 | func TestConvertPrivateKeyToPublicKey(t *testing.T) { 34 | sk, err := GenerateKey(nil) 35 | if err != nil { 36 | t.Fatal(err) 37 | } 38 | 39 | pk, ok := sk.Public() 40 | if !ok { 41 | t.Fatal("Couldn't obtain public key.") 42 | } 43 | if !bytes.Equal(pk, sk[32:]) { 44 | t.Fatal("Raw byte respresentation doesn't match public key.") 45 | } 46 | } 47 | -------------------------------------------------------------------------------- /crypto/util_test.go: -------------------------------------------------------------------------------- 1 | package crypto 2 | 3 | import ( 4 | "bytes" 5 | "crypto/rand" 6 | "errors" 7 | "testing" 8 | ) 9 | 10 | func TestDigest(t *testing.T) { 11 | msg := []byte("test message") 12 | d := Digest(msg) 13 | if len(d) != HashSizeByte { 14 | t.Fatal("Computation of Hash failed.") 15 | } 16 | if bytes.Equal(d, make([]byte, HashSizeByte)) { 17 | t.Fatal("Hash is all zeros.") 18 | } 19 | } 20 | 21 | type testErrorRandReader struct{} 22 | 23 | func (er testErrorRandReader) Read([]byte) (int, error) { 24 | return 0, errors.New("Not enough entropy!") 25 | } 26 | 27 | func TestMakeRand(t *testing.T) { 28 | r, err := MakeRand() 29 | if err != nil { 30 | t.Fatal(err) 31 | } 32 | // check if hashed the random output: 33 | if len(r) != HashSizeByte { 34 | t.Fatal("Looks like Digest wasn't called correctly.") 35 | } 36 | orig := rand.Reader 37 | rand.Reader = testErrorRandReader{} 38 | r, err = MakeRand() 39 | if err == nil { 40 | t.Fatal("No error returned") 41 | } 42 | rand.Reader = orig 43 | } 44 | 45 | func TestCommit(t *testing.T) { 46 | stuff := []byte("123") 47 | commit, err := NewCommit(stuff) 48 | if err != nil { 49 | t.Fatal(err) 50 | } 51 | if !commit.Verify(stuff) { 52 | t.Fatal("Commit doesn't verify!") 53 | } 54 | } 55 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | This license applies to all parts of the CONIKS Golang library and reference implementation except the following: 2 | - The ed25519 subpackage, located in crypto/ed25519 3 | This subpackage is copyrighted by the Go Authors. 4 | - Coname utility functions, located in utils/coname.go 5 | - The vrf subpackage, located in crypto/vrf 6 | - The kv subpackage, located in storage/kv 7 | - The leveldbkv subpackage, located in storage/kv/leveldbkv 8 | These subpackages are copyrighted by the Coname Authors. 9 | 10 | Copyright (c) 2016, Princeton University. 11 | Copyright (c) 2016, Ecole Polytechnique Federale de Lausanne (EPFL). 12 | Copyright (c) 2016, The Tor Project, Inc. 13 | Copyright (c) 2016, Vu Quoc Huy. 14 | All rights reserved. 15 | 16 | Redistribution and use in source and binary forms, with or without 17 | modification, are permitted provided that the following conditions are met: 18 | 19 | * Redistributions of source code must retain the above copyright notice, this 20 | list of conditions and the following disclaimer. 21 | 22 | * Redistributions in binary form must reproduce the above copyright notice, 23 | this list of conditions and the following disclaimer in the documentation 24 | and/or other materials provided with the distribution. 25 | 26 | * Neither the names of Princeton University, EPFL and The Tor Project nor the names of its 27 | contributors may be used to endorse or promote products derived from 28 | this software without specific prior written permission. 29 | 30 | THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 31 | AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 32 | IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE 33 | DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE 34 | FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 35 | DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR 36 | SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER 37 | CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, 38 | OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 39 | OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 40 | -------------------------------------------------------------------------------- /crypto/util.go: -------------------------------------------------------------------------------- 1 | package crypto 2 | 3 | import ( 4 | "bytes" 5 | "crypto/rand" 6 | 7 | "golang.org/x/crypto/sha3" 8 | ) 9 | 10 | const ( 11 | // HashSizeByte is the size of the hash output in bytes. 12 | HashSizeByte = 32 13 | // HashID identifies the used hash as a string. 14 | HashID = "SHAKE128" 15 | ) 16 | 17 | // Digest hashes all passed byte slices. 18 | // The passed slices won't be mutated. 19 | func Digest(ms ...[]byte) []byte { 20 | h := sha3.NewShake128() 21 | for _, m := range ms { 22 | h.Write(m) 23 | } 24 | ret := make([]byte, HashSizeByte) 25 | h.Read(ret) 26 | return ret 27 | } 28 | 29 | // MakeRand returns a random slice of bytes. 30 | // It returns an error if there was a problem while generating 31 | // the random slice. 32 | // It is different from the 'standard' random byte generation as it 33 | // hashes its output before returning it; by hashing the system's 34 | // PRNG output before it is send over the wire, we aim to make the 35 | // random output less predictable (even if the system's PRNG isn't 36 | // as unpredictable as desired). 37 | // See https://trac.torproject.org/projects/tor/ticket/17694 38 | func MakeRand() ([]byte, error) { 39 | r := make([]byte, HashSizeByte) 40 | if _, err := rand.Read(r); err != nil { 41 | return nil, err 42 | } 43 | // Do not directly reveal bytes from rand.Read on the wire 44 | return Digest(r), nil 45 | } 46 | 47 | // Commit can be used to create a cryptographic commit to some value (use 48 | // NewCommit() for this purpose. 49 | type Commit struct { 50 | // Salt is a cryptographic salt which will be hashed in addition 51 | // to the value. 52 | Salt []byte 53 | // Value is the actual value to commit to. 54 | Value []byte 55 | } 56 | 57 | // NewCommit creates a new cryptographic commit to the passed byte slices 58 | // stuff (which won't be mutated). It creates a random salt before 59 | // committing to the values. 60 | func NewCommit(stuff ...[]byte) (*Commit, error) { 61 | salt, err := MakeRand() 62 | if err != nil { 63 | return nil, err 64 | } 65 | return &Commit{ 66 | Salt: salt, 67 | Value: Digest(append([][]byte{salt}, stuff...)...), 68 | }, nil 69 | } 70 | 71 | // Verify verifies that the underlying commit c was a commit to the passed 72 | // byte slices stuff (which won't be mutated). 73 | func (c *Commit) Verify(stuff ...[]byte) bool { 74 | return bytes.Equal(c.Value, Digest(append([][]byte{c.Salt}, stuff...)...)) 75 | } 76 | -------------------------------------------------------------------------------- /crypto/sign/sign.go: -------------------------------------------------------------------------------- 1 | // Package sign implements a digital signature scheme using the Edwards 2 | // form of Curve25519. 3 | package sign 4 | 5 | import ( 6 | "crypto/rand" 7 | "io" 8 | 9 | "golang.org/x/crypto/ed25519" 10 | ) 11 | 12 | const ( 13 | // PrivateKeySize is the size of the private-key in bytes. 14 | PrivateKeySize = 64 15 | // PublicKeySize is the size of the public-key in bytes. 16 | PublicKeySize = 32 17 | // SignatureSize is the size of the created signature in bytes. 18 | SignatureSize = 64 19 | ) 20 | 21 | // PrivateKey wraps the underlying private-key (ed25519.PrivateKey). 22 | // It provides some wrapper methods: Sign(), Public() 23 | type PrivateKey ed25519.PrivateKey 24 | 25 | // PublicKey wraps the underlying public-key type. It can be used to verify a 26 | // signature which was created by using a corresponding PrivateKey 27 | type PublicKey ed25519.PublicKey 28 | 29 | // GenerateKey generates and returns a fresh random private-key, from 30 | // which the corresponding public-key can be derived (by calling Public() 31 | // on it). 32 | // It will use the passed io.Reader rnd as a source of randomness, or, if 33 | // rnd is nil it will use a sane default (rand.Reader). 34 | // 35 | // It returns an error if the key couldn't be properly generated. This, for 36 | // example, can happen if there isn't enough entropy for the randomness. 37 | func GenerateKey(rnd io.Reader) (PrivateKey, error) { 38 | if rnd == nil { 39 | rnd = rand.Reader 40 | } 41 | _, sk, err := ed25519.GenerateKey(rnd) 42 | return PrivateKey(sk), err 43 | } 44 | 45 | // Sign returns a signature on the passed byte slice message using the 46 | // underlying private-key. 47 | // The passed slice won't be modified. 48 | func (key PrivateKey) Sign(message []byte) []byte { 49 | return ed25519.Sign(ed25519.PrivateKey(key), message) 50 | } 51 | 52 | // Public derives the corresponding public-key from the underlying 53 | // private-key. 54 | // It returns the derived key, if possible, and a boolean flag which 55 | // indicates if the operations to derive the public-key were successful. 56 | func (key PrivateKey) Public() (PublicKey, bool) { 57 | pk, ok := ed25519.PrivateKey(key).Public().(ed25519.PublicKey) 58 | return PublicKey(pk), ok 59 | } 60 | 61 | // Verify verifies a signature sig on message using the underlying 62 | // public-key. It returns true if and only if the signature is valid. 63 | // The passed slices aren't modified. 64 | func (pk PublicKey) Verify(message, sig []byte) bool { 65 | return ed25519.Verify(ed25519.PublicKey(pk), message, sig) 66 | } 67 | -------------------------------------------------------------------------------- /crypto/internal/ed25519/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 | "testing" 12 | 13 | "github.com/NKNetwork/coniks-go/crypto/internal/ed25519/edwards25519" 14 | "golang.org/x/crypto/curve25519" 15 | "golang.org/x/crypto/ed25519" 16 | ) 17 | 18 | func TestCurve25519Conversion(t *testing.T) { 19 | public, private, _ := ed25519.GenerateKey(rand.Reader) 20 | var pubBytes [32]byte 21 | copy(pubBytes[:], public) 22 | var privBytes [64]byte 23 | copy(privBytes[:], private) 24 | 25 | var curve25519Public, curve25519Public2, curve25519Private [32]byte 26 | PrivateKeyToCurve25519(&curve25519Private, &privBytes) 27 | curve25519.ScalarBaseMult(&curve25519Public, &curve25519Private) 28 | 29 | if !PublicKeyToCurve25519(&curve25519Public2, &pubBytes) { 30 | t.Fatalf("PublicKeyToCurve25519 failed") 31 | } 32 | 33 | if !bytes.Equal(curve25519Public[:], curve25519Public2[:]) { 34 | t.Errorf("Values didn't match: curve25519 produced %x, conversion produced %x", curve25519Public[:], curve25519Public2[:]) 35 | } 36 | } 37 | 38 | func TestHashNoCollisions(t *testing.T) { 39 | type intpair struct { 40 | i int 41 | j uint 42 | } 43 | rainbow := make(map[[32]byte]intpair) 44 | N := 25 45 | if testing.Short() { 46 | N = 3 47 | } 48 | var h [32]byte 49 | // NOTE: hash values 0b100000000000... and 0b00000000000... both map to 50 | // the identity. this is a core part of the elligator function and not a 51 | // collision we need to worry about because an attacker would need to find 52 | // the preimages of these hashes to exploit it. 53 | h[0] = 1 54 | for i := 0; i < N; i++ { 55 | for j := uint(0); j < 257; j++ { 56 | if j < 256 { 57 | h[j>>3] ^= byte(1) << (j & 7) 58 | } 59 | 60 | var P edwards25519.ExtendedGroupElement 61 | HashToEdwards(&P, &h) 62 | var p [32]byte 63 | P.ToBytes(&p) 64 | if c, ok := rainbow[p]; ok { 65 | t.Fatalf("found collision: (%d, %d) and (%d, %d)", i, j, c.i, c.j) 66 | } 67 | rainbow[p] = intpair{i, j} 68 | 69 | if j < 256 { 70 | h[j>>3] ^= byte(1) << (j & 7) 71 | } 72 | } 73 | hh := sha512.Sum512(h[:]) // this package already imports sha512 74 | copy(h[:], hh[:]) 75 | } 76 | } 77 | 78 | func TestElligator(t *testing.T) { 79 | var publicKey, publicKey2, publicKey3, representative, privateKey [32]byte 80 | 81 | for i := 0; i < 1000; i++ { 82 | rand.Reader.Read(privateKey[:]) 83 | 84 | if !ScalarBaseMult(&publicKey, &representative, &privateKey) { 85 | continue 86 | } 87 | RepresentativeToPublicKey(&publicKey2, &representative) 88 | if !bytes.Equal(publicKey[:], publicKey2[:]) { 89 | t.Fatal("The resulting public key doesn't match the initial one.") 90 | } 91 | 92 | curve25519.ScalarBaseMult(&publicKey3, &privateKey) 93 | if !bytes.Equal(publicKey[:], publicKey3[:]) { 94 | t.Fatal("The public key doesn't match the value that curve25519 produced.") 95 | } 96 | } 97 | } 98 | 99 | func BenchmarkKeyGeneration(b *testing.B) { 100 | var publicKey, representative, privateKey [32]byte 101 | 102 | // Find the private key that results in a point that's in the image of the map. 103 | for { 104 | rand.Reader.Read(privateKey[:]) 105 | if ScalarBaseMult(&publicKey, &representative, &privateKey) { 106 | break 107 | } 108 | } 109 | 110 | b.ResetTimer() 111 | for i := 0; i < b.N; i++ { 112 | ScalarBaseMult(&publicKey, &representative, &privateKey) 113 | } 114 | } 115 | 116 | func BenchmarkMap(b *testing.B) { 117 | var publicKey, representative [32]byte 118 | rand.Reader.Read(representative[:]) 119 | 120 | b.ResetTimer() 121 | for i := 0; i < b.N; i++ { 122 | RepresentativeToPublicKey(&publicKey, &representative) 123 | } 124 | } 125 | -------------------------------------------------------------------------------- /crypto/internal/ed25519/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 | -------------------------------------------------------------------------------- /crypto/vrf/vrf_test.go: -------------------------------------------------------------------------------- 1 | package vrf 2 | 3 | import ( 4 | "bytes" 5 | "testing" 6 | // "fmt" 7 | ) 8 | 9 | func TestHonestComplete(t *testing.T) { 10 | sk, err := GenerateKey(nil) 11 | if err != nil { 12 | t.Fatal(err) 13 | } 14 | pk, _ := sk.Public() 15 | alice := []byte("alice") 16 | aliceVRF := sk.Compute(alice) 17 | aliceVRFFromProof, aliceProof := sk.Prove(alice) 18 | 19 | // fmt.Printf("pk: %X\n", pk) 20 | // fmt.Printf("sk: %X\n", *sk) 21 | // fmt.Printf("alice(bytes): %X\n", alice) 22 | // fmt.Printf("aliceVRF: %X\n", aliceVRF) 23 | // fmt.Printf("aliceProof: %X\n", aliceProof) 24 | 25 | if !pk.Verify(alice, aliceVRF, aliceProof) { 26 | t.Error("Gen -> Compute -> Prove -> Verify -> FALSE") 27 | } 28 | if !bytes.Equal(aliceVRF, aliceVRFFromProof) { 29 | t.Error("Compute != Prove") 30 | } 31 | } 32 | 33 | func TestConvertPrivateKeyToPublicKey(t *testing.T) { 34 | sk, err := GenerateKey(nil) 35 | if err != nil { 36 | t.Fatal(err) 37 | } 38 | 39 | pk, ok := sk.Public() 40 | if !ok { 41 | t.Fatal("Couldn't obtain public key.") 42 | } 43 | if !bytes.Equal(sk[32:], pk) { 44 | t.Fatal("Raw byte respresentation doesn't match public key.") 45 | } 46 | } 47 | 48 | func TestFlipBitForgery(t *testing.T) { 49 | sk, err := GenerateKey(nil) 50 | if err != nil { 51 | t.Fatal(err) 52 | } 53 | pk, _ := sk.Public() 54 | alice := []byte("alice") 55 | for i := 0; i < 32; i++ { 56 | for j := uint(0); j < 8; j++ { 57 | aliceVRF := sk.Compute(alice) 58 | aliceVRF[i] ^= 1 << j 59 | _, aliceProof := sk.Prove(alice) 60 | if pk.Verify(alice, aliceVRF, aliceProof) { 61 | t.Fatalf("forged by using aliceVRF[%d]^=%d:\n (sk=%x)", i, j, sk) 62 | } 63 | } 64 | } 65 | } 66 | 67 | func sampleVectorTest(pk PublicKey, aliceVRF, aliceProof []byte, t *testing.T) { 68 | alice := []byte{97, 108, 105, 99, 101} 69 | 70 | // Positive test case 71 | if !pk.Verify(alice, aliceVRF, aliceProof) { 72 | t.Error("TestSampleVectors HonestVector Failed") 73 | } 74 | 75 | // Negative test cases - try increment the first byte of every vector 76 | pk[0]++ 77 | if pk.Verify(alice, aliceVRF, aliceProof) { 78 | t.Error("TestSampleVectors ForgedVector (pk modified) Passed") 79 | } 80 | pk[0]-- 81 | 82 | alice[0]++ 83 | if pk.Verify(alice, aliceVRF, aliceProof) { 84 | t.Error("TestSampleVectors ForgedVector (alice modified) Passed") 85 | } 86 | alice[0]-- 87 | 88 | aliceVRF[0]++ 89 | if pk.Verify(alice, aliceVRF, aliceProof) { 90 | t.Error("TestSampleVectors ForgedVector (aliceVRF modified) Passed") 91 | } 92 | aliceVRF[0]-- 93 | 94 | aliceProof[0]++ 95 | if pk.Verify(alice, aliceVRF, aliceProof) { 96 | t.Error("TestSampleVectors ForgedVector (aliceProof modified) Passed") 97 | } 98 | aliceProof[0]-- 99 | } 100 | 101 | func TestSampleVectorSets(t *testing.T) { 102 | t.Skip("TODO: generate new test vectors or remove test") 103 | var aliceVRF, aliceProof []byte 104 | var pk []byte 105 | 106 | // Following sets of test vectors are collected from TestHonestComplete(), 107 | // and are used for testing the JS implementation of vrf.verify() 108 | // Reference: https://github.com/yahoo/end-to-end/pull/58 109 | 110 | pk = []byte{194, 191, 96, 139, 106, 249, 24, 253, 198, 131, 88, 169, 100, 231, 7, 211, 70, 171, 171, 207, 24, 30, 150, 114, 77, 124, 240, 123, 191, 14, 29, 111} 111 | aliceVRF = []byte{68, 98, 55, 78, 153, 189, 11, 15, 8, 238, 132, 5, 53, 28, 232, 22, 222, 98, 21, 139, 89, 67, 111, 197, 213, 75, 86, 226, 178, 71, 245, 159} 112 | aliceProof = []byte{49, 128, 4, 253, 103, 241, 164, 51, 21, 45, 168, 55, 18, 103, 22, 233, 245, 136, 242, 238, 113, 218, 160, 122, 129, 89, 72, 103, 250, 222, 3, 3, 239, 235, 93, 98, 173, 115, 168, 24, 222, 165, 186, 224, 138, 76, 201, 237, 130, 201, 47, 18, 191, 24, 61, 80, 113, 139, 246, 233, 23, 94, 177, 12, 193, 106, 38, 172, 66, 192, 22, 188, 177, 14, 144, 100, 38, 179, 96, 70, 55, 157, 80, 139, 145, 62, 94, 195, 181, 224, 183, 42, 64, 66, 145, 162} 113 | sampleVectorTest(pk, aliceVRF, aliceProof, t) 114 | 115 | pk = []byte{133, 36, 180, 21, 60, 103, 35, 92, 204, 245, 236, 174, 242, 50, 212, 69, 124, 230, 1, 106, 94, 95, 201, 55, 208, 252, 195, 13, 12, 96, 87, 170} 116 | aliceVRF = []byte{35, 127, 188, 177, 246, 242, 213, 46, 16, 72, 1, 196, 69, 181, 160, 204, 69, 230, 17, 147, 251, 207, 203, 184, 154, 122, 118, 10, 144, 76, 229, 234} 117 | aliceProof = []byte{253, 33, 80, 241, 250, 172, 198, 28, 16, 171, 161, 194, 110, 175, 158, 233, 250, 89, 35, 174, 221, 101, 98, 136, 32, 191, 82, 127, 92, 208, 199, 10, 123, 46, 70, 95, 56, 102, 63, 137, 53, 160, 128, 216, 134, 152, 87, 58, 19, 244, 167, 108, 144, 13, 97, 232, 207, 75, 107, 57, 193, 124, 231, 5, 242, 122, 182, 247, 155, 187, 86, 165, 114, 46, 188, 52, 21, 121, 238, 100, 85, 32, 119, 116, 250, 208, 32, 60, 145, 53, 145, 76, 84, 153, 185, 28} 118 | sampleVectorTest(pk, aliceVRF, aliceProof, t) 119 | 120 | pk = []byte{85, 126, 176, 228, 114, 43, 110, 223, 111, 129, 204, 38, 215, 110, 165, 148, 223, 232, 79, 254, 150, 107, 61, 29, 216, 14, 238, 104, 55, 163, 121, 185} 121 | aliceVRF = []byte{171, 240, 42, 215, 128, 5, 247, 64, 164, 154, 198, 231, 6, 174, 207, 10, 95, 231, 117, 189, 88, 103, 72, 229, 43, 218, 184, 162, 44, 183, 196, 159} 122 | aliceProof = []byte{99, 103, 243, 119, 251, 30, 21, 57, 69, 162, 192, 80, 7, 49, 244, 136, 13, 252, 150, 165, 215, 181, 55, 203, 141, 124, 197, 36, 20, 183, 239, 14, 238, 213, 240, 96, 181, 187, 24, 137, 152, 152, 38, 186, 80, 141, 72, 15, 209, 178, 60, 205, 22, 31, 101, 185, 225, 159, 22, 118, 84, 179, 95, 0, 124, 140, 237, 187, 8, 77, 233, 213, 207, 211, 251, 153, 71, 112, 61, 89, 53, 26, 195, 167, 254, 73, 218, 135, 145, 89, 12, 4, 16, 255, 63, 89} 123 | sampleVectorTest(pk, aliceVRF, aliceProof, t) 124 | 125 | } 126 | 127 | func BenchmarkHashToGE(b *testing.B) { 128 | alice := []byte("alice") 129 | b.ResetTimer() 130 | for n := 0; n < b.N; n++ { 131 | hashToCurve(alice) 132 | } 133 | } 134 | 135 | func BenchmarkCompute(b *testing.B) { 136 | sk, err := GenerateKey(nil) 137 | if err != nil { 138 | b.Fatal(err) 139 | } 140 | alice := []byte("alice") 141 | b.ResetTimer() 142 | for n := 0; n < b.N; n++ { 143 | sk.Compute(alice) 144 | } 145 | } 146 | 147 | func BenchmarkProve(b *testing.B) { 148 | sk, err := GenerateKey(nil) 149 | if err != nil { 150 | b.Fatal(err) 151 | } 152 | alice := []byte("alice") 153 | b.ResetTimer() 154 | for n := 0; n < b.N; n++ { 155 | sk.Prove(alice) 156 | } 157 | } 158 | 159 | func BenchmarkVerify(b *testing.B) { 160 | sk, err := GenerateKey(nil) 161 | if err != nil { 162 | b.Fatal(err) 163 | } 164 | alice := []byte("alice") 165 | aliceVRF := sk.Compute(alice) 166 | _, aliceProof := sk.Prove(alice) 167 | pk, _ := sk.Public() 168 | b.ResetTimer() 169 | for n := 0; n < b.N; n++ { 170 | pk.Verify(alice, aliceVRF, aliceProof) 171 | } 172 | } 173 | -------------------------------------------------------------------------------- /crypto/vrf/vrf.go: -------------------------------------------------------------------------------- 1 | // Package ed25519 implements a verifiable random function using the Edwards form 2 | // of Curve25519, SHA512 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 ed25519 19 | 20 | import ( 21 | "bytes" 22 | "crypto/rand" 23 | "crypto/sha512" 24 | "errors" 25 | "io" 26 | 27 | "github.com/coniks-sys/coniks-go/crypto/internal/ed25519/edwards25519" 28 | "github.com/coniks-sys/coniks-go/crypto/internal/ed25519/extra25519" 29 | 30 | "golang.org/x/crypto/ed25519" 31 | ) 32 | 33 | const ( 34 | PublicKeySize = ed25519.PublicKeySize 35 | PrivateKeySize = ed25519.PrivateKeySize 36 | Size = 32 37 | intermediateSize = ed25519.PublicKeySize 38 | ProofSize = 32 + 32 + intermediateSize 39 | ) 40 | 41 | var ( 42 | ErrGetPubKey = errors.New("[vrf] Couldn't get corresponding public-key from private-key") 43 | ) 44 | 45 | type PrivateKey []byte 46 | type PublicKey []byte 47 | 48 | // GenerateKey creates a public/private key pair using rnd for randomness. 49 | // If rnd is nil, crypto/rand is used. 50 | func GenerateKey(rnd io.Reader) (sk PrivateKey, err error) { 51 | if rnd == nil { 52 | rnd = rand.Reader 53 | } 54 | _, skr, err := ed25519.GenerateKey(rnd) 55 | return PrivateKey(skr), err 56 | } 57 | 58 | // Public extracts the public VRF key from the underlying private-key 59 | // and returns a boolean indicating if the operation was successful. 60 | func (sk PrivateKey) Public() (PublicKey, bool) { 61 | pk, ok := ed25519.PrivateKey(sk).Public().(ed25519.PublicKey) 62 | return PublicKey(pk), ok 63 | } 64 | 65 | func (sk PrivateKey) expandSecret() (x, skhr *[32]byte) { 66 | x, skhr = new([32]byte), new([32]byte) 67 | skh := sha512.Sum512(sk[:32]) 68 | copy(x[:], skh[:]) 69 | copy(skhr[:], skh[32:]) 70 | x[0] &= 248 71 | x[31] &= 127 72 | x[31] |= 64 73 | return 74 | } 75 | 76 | // Compute generates the vrf value for the byte slice m using the 77 | // underlying private key sk. 78 | func (sk PrivateKey) Compute(m []byte) []byte { 79 | x, _ := sk.expandSecret() 80 | var ii edwards25519.ExtendedGroupElement 81 | var iiB [32]byte 82 | edwards25519.GeScalarMult(&ii, x, hashToCurve(m)) 83 | ii.ToBytes(&iiB) 84 | 85 | vrf := sha512.New() 86 | vrf.Write(iiB[:]) // const length: Size 87 | vrf.Write(m) 88 | return vrf.Sum(nil)[:32] 89 | } 90 | 91 | func hashToCurve(m []byte) *edwards25519.ExtendedGroupElement { 92 | // H(n) = (f(h(n))^8) 93 | hmbH := sha512.Sum512(m) 94 | var hmb [32]byte 95 | copy(hmb[:], hmbH[:]) 96 | var hm edwards25519.ExtendedGroupElement 97 | extra25519.HashToEdwards(&hm, &hmb) 98 | edwards25519.GeDouble(&hm, &hm) 99 | edwards25519.GeDouble(&hm, &hm) 100 | edwards25519.GeDouble(&hm, &hm) 101 | return &hm 102 | } 103 | 104 | // Prove returns the vrf value and a proof such that 105 | // Verify(m, vrf, proof) == true. The vrf value is the 106 | // same as returned by Compute(m). 107 | func (sk PrivateKey) Prove(m []byte) (vrf, proof []byte) { 108 | x, skhr := sk.expandSecret() 109 | var sH, rH [64]byte 110 | var r, s, minusS, t, gB, grB, hrB, hxB, hB [32]byte 111 | var ii, gr, hr edwards25519.ExtendedGroupElement 112 | 113 | h := hashToCurve(m) 114 | h.ToBytes(&hB) 115 | edwards25519.GeScalarMult(&ii, x, h) 116 | ii.ToBytes(&hxB) 117 | 118 | // use hash of private-, public-key and msg as randomness source: 119 | hash := sha512.New() 120 | hash.Write(skhr[:]) 121 | hash.Write(sk[32:]) // public key, as in ed25519 122 | hash.Write(m) 123 | hash.Sum(rH[:0]) 124 | hash.Reset() 125 | edwards25519.ScReduce(&r, &rH) 126 | 127 | edwards25519.GeScalarMultBase(&gr, &r) 128 | edwards25519.GeScalarMult(&hr, &r, h) 129 | gr.ToBytes(&grB) 130 | hr.ToBytes(&hrB) 131 | gB = edwards25519.BaseBytes 132 | 133 | // H2(g, h, g^x, h^x, g^r, h^r, m) 134 | hash.Write(gB[:]) 135 | hash.Write(hB[:]) 136 | hash.Write(sk[32:]) // ed25519 public-key 137 | hash.Write(hxB[:]) 138 | hash.Write(grB[:]) 139 | hash.Write(hrB[:]) 140 | hash.Write(m) 141 | hash.Sum(sH[:0]) 142 | hash.Reset() 143 | edwards25519.ScReduce(&s, &sH) 144 | 145 | edwards25519.ScNeg(&minusS, &s) 146 | edwards25519.ScMulAdd(&t, x, &minusS, &r) 147 | 148 | proof = make([]byte, ProofSize) 149 | copy(proof[:32], s[:]) 150 | copy(proof[32:64], t[:]) 151 | copy(proof[64:96], hxB[:]) 152 | 153 | hash.Reset() 154 | hash.Write(hxB[:]) 155 | hash.Write(m) 156 | vrf = hash.Sum(nil)[:Size] 157 | return 158 | } 159 | 160 | // Verify returns true iff vrf=Compute(m) for the sk that 161 | // corresponds to pk. 162 | func (pkBytes PublicKey) Verify(m, vrfBytes, proof []byte) bool { 163 | if len(proof) != ProofSize || len(vrfBytes) != Size || len(pkBytes) != PublicKeySize { 164 | return false 165 | } 166 | var pk, s, sRef, t, vrf, hxB, hB, gB, ABytes, BBytes [32]byte 167 | copy(vrf[:], vrfBytes) 168 | copy(pk[:], pkBytes[:]) 169 | copy(s[:32], proof[:32]) 170 | copy(t[:32], proof[32:64]) 171 | copy(hxB[:], proof[64:96]) 172 | 173 | hash := sha512.New() 174 | hash.Write(hxB[:]) // const length 175 | hash.Write(m) 176 | if !bytes.Equal(hash.Sum(nil)[:Size], vrf[:Size]) { 177 | return false 178 | } 179 | hash.Reset() 180 | 181 | var P, B, ii, iic edwards25519.ExtendedGroupElement 182 | var A, hmtP, iicP edwards25519.ProjectiveGroupElement 183 | if !P.FromBytesBaseGroup(&pk) { 184 | return false 185 | } 186 | if !ii.FromBytesBaseGroup(&hxB) { 187 | return false 188 | } 189 | edwards25519.GeDoubleScalarMultVartime(&A, &s, &P, &t) 190 | A.ToBytes(&ABytes) 191 | gB = edwards25519.BaseBytes 192 | 193 | h := hashToCurve(m) // h = H1(m) 194 | h.ToBytes(&hB) 195 | edwards25519.GeDoubleScalarMultVartime(&hmtP, &t, h, &[32]byte{}) 196 | edwards25519.GeDoubleScalarMultVartime(&iicP, &s, &ii, &[32]byte{}) 197 | iicP.ToExtended(&iic) 198 | hmtP.ToExtended(&B) 199 | edwards25519.GeAdd(&B, &B, &iic) 200 | B.ToBytes(&BBytes) 201 | 202 | var sH [64]byte 203 | // sRef = H2(g, h, g^x, v, g^t·G^s,H1(m)^t·v^s, m), with v=H1(m)^x=h^x 204 | hash.Write(gB[:]) 205 | hash.Write(hB[:]) 206 | hash.Write(pkBytes) 207 | hash.Write(hxB[:]) 208 | hash.Write(ABytes[:]) // const length (g^t*G^s) 209 | hash.Write(BBytes[:]) // const length (H1(m)^t*v^s) 210 | hash.Write(m) 211 | hash.Sum(sH[:0]) 212 | 213 | edwards25519.ScReduce(&sRef, &sH) 214 | return sRef == s 215 | } 216 | -------------------------------------------------------------------------------- /crypto/internal/ed25519/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/NKNetwork/coniks-go/crypto/internal/ed25519/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 montgomeryXToEdwardsY(out, x *edwards25519.FieldElement) { 351 | var t, tt edwards25519.FieldElement 352 | edwards25519.FeOne(&t) 353 | edwards25519.FeAdd(&tt, x, &t) // u+1 354 | edwards25519.FeInvert(&tt, &tt) // 1/(u+1) 355 | edwards25519.FeSub(&t, x, &t) // u-1 356 | edwards25519.FeMul(out, &tt, &t) // (u-1)/(u+1) 357 | } 358 | 359 | // HashToEdwards converts a 256-bit hash output into a point on the Edwards 360 | // curve isomorphic to Curve25519 in a manner that preserves 361 | // collision-resistance. The returned curve points are NOT indistinguishable 362 | // from random even if the hash value is. 363 | // Specifically, first one bit of the hash output is set aside for parity and 364 | // the rest is truncated and fed into the elligator bijection (which covers half 365 | // of the points on the elliptic curve). 366 | func HashToEdwards(out *edwards25519.ExtendedGroupElement, h *[32]byte) { 367 | hh := *h 368 | bit := hh[31] >> 7 369 | hh[31] &= 127 370 | edwards25519.FeFromBytes(&out.Y, &hh) 371 | representativeToMontgomeryX(&out.X, &out.Y) 372 | montgomeryXToEdwardsY(&out.Y, &out.X) 373 | if ok := out.FromParityAndY(bit, &out.Y); !ok { 374 | panic("HashToEdwards: point not on curve") 375 | } 376 | } 377 | -------------------------------------------------------------------------------- /crypto/internal/ed25519/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 | import "math" 11 | 12 | // This code is a port of the public domain, "ref10" implementation of ed25519 13 | // from SUPERCOP. 14 | 15 | // FieldElement represents an element of the field GF(2^255 - 19). An element 16 | // t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 17 | // t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on 18 | // context. 19 | type FieldElement [10]int32 20 | 21 | func FeZero(fe *FieldElement) { 22 | for i := range fe { 23 | fe[i] = 0 24 | } 25 | } 26 | 27 | func FeOne(fe *FieldElement) { 28 | FeZero(fe) 29 | fe[0] = 1 30 | } 31 | 32 | func FeAdd(dst, a, b *FieldElement) { 33 | for i := range dst { 34 | dst[i] = a[i] + b[i] 35 | } 36 | } 37 | 38 | func FeSub(dst, a, b *FieldElement) { 39 | for i := range dst { 40 | dst[i] = a[i] - b[i] 41 | } 42 | } 43 | 44 | func FeCopy(dst, src *FieldElement) { 45 | for i := range dst { 46 | dst[i] = src[i] 47 | } 48 | } 49 | 50 | // Replace (f,g) with (g,g) if b == 1; 51 | // replace (f,g) with (f,g) if b == 0. 52 | // 53 | // Preconditions: b in {0,1}. 54 | func FeCMove(f, g *FieldElement, b int32) { 55 | var x FieldElement 56 | b = -b 57 | for i := range x { 58 | x[i] = b & (f[i] ^ g[i]) 59 | } 60 | 61 | for i := range f { 62 | f[i] ^= x[i] 63 | } 64 | } 65 | 66 | func load3(in []byte) int64 { 67 | var r int64 68 | r = int64(in[0]) 69 | r |= int64(in[1]) << 8 70 | r |= int64(in[2]) << 16 71 | return r 72 | } 73 | 74 | func load4(in []byte) int64 { 75 | var r int64 76 | r = int64(in[0]) 77 | r |= int64(in[1]) << 8 78 | r |= int64(in[2]) << 16 79 | r |= int64(in[3]) << 24 80 | return r 81 | } 82 | 83 | func FeFromBytes(dst *FieldElement, src *[32]byte) { 84 | h0 := load4(src[:]) 85 | h1 := load3(src[4:]) << 6 86 | h2 := load3(src[7:]) << 5 87 | h3 := load3(src[10:]) << 3 88 | h4 := load3(src[13:]) << 2 89 | h5 := load4(src[16:]) 90 | h6 := load3(src[20:]) << 7 91 | h7 := load3(src[23:]) << 5 92 | h8 := load3(src[26:]) << 4 93 | h9 := (load3(src[29:]) & 8388607) << 2 94 | 95 | var carry [10]int64 96 | carry[9] = (h9 + 1<<24) >> 25 97 | h0 += carry[9] * 19 98 | h9 -= carry[9] << 25 99 | carry[1] = (h1 + 1<<24) >> 25 100 | h2 += carry[1] 101 | h1 -= carry[1] << 25 102 | carry[3] = (h3 + 1<<24) >> 25 103 | h4 += carry[3] 104 | h3 -= carry[3] << 25 105 | carry[5] = (h5 + 1<<24) >> 25 106 | h6 += carry[5] 107 | h5 -= carry[5] << 25 108 | carry[7] = (h7 + 1<<24) >> 25 109 | h8 += carry[7] 110 | h7 -= carry[7] << 25 111 | 112 | carry[0] = (h0 + 1<<25) >> 26 113 | h1 += carry[0] 114 | h0 -= carry[0] << 26 115 | carry[2] = (h2 + 1<<25) >> 26 116 | h3 += carry[2] 117 | h2 -= carry[2] << 26 118 | carry[4] = (h4 + 1<<25) >> 26 119 | h5 += carry[4] 120 | h4 -= carry[4] << 26 121 | carry[6] = (h6 + 1<<25) >> 26 122 | h7 += carry[6] 123 | h6 -= carry[6] << 26 124 | carry[8] = (h8 + 1<<25) >> 26 125 | h9 += carry[8] 126 | h8 -= carry[8] << 26 127 | 128 | dst[0] = int32(h0) 129 | dst[1] = int32(h1) 130 | dst[2] = int32(h2) 131 | dst[3] = int32(h3) 132 | dst[4] = int32(h4) 133 | dst[5] = int32(h5) 134 | dst[6] = int32(h6) 135 | dst[7] = int32(h7) 136 | dst[8] = int32(h8) 137 | dst[9] = int32(h9) 138 | } 139 | 140 | // FeToBytes marshals h to s. 141 | // Preconditions: 142 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 143 | // 144 | // Write p=2^255-19; q=floor(h/p). 145 | // Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). 146 | // 147 | // Proof: 148 | // Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. 149 | // Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. 150 | // 151 | // Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). 152 | // Then 0> 25 167 | q = (h[0] + q) >> 26 168 | q = (h[1] + q) >> 25 169 | q = (h[2] + q) >> 26 170 | q = (h[3] + q) >> 25 171 | q = (h[4] + q) >> 26 172 | q = (h[5] + q) >> 25 173 | q = (h[6] + q) >> 26 174 | q = (h[7] + q) >> 25 175 | q = (h[8] + q) >> 26 176 | q = (h[9] + q) >> 25 177 | 178 | // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. 179 | h[0] += 19 * q 180 | // Goal: Output h-2^255 q, which is between 0 and 2^255-20. 181 | 182 | carry[0] = h[0] >> 26 183 | h[1] += carry[0] 184 | h[0] -= carry[0] << 26 185 | carry[1] = h[1] >> 25 186 | h[2] += carry[1] 187 | h[1] -= carry[1] << 25 188 | carry[2] = h[2] >> 26 189 | h[3] += carry[2] 190 | h[2] -= carry[2] << 26 191 | carry[3] = h[3] >> 25 192 | h[4] += carry[3] 193 | h[3] -= carry[3] << 25 194 | carry[4] = h[4] >> 26 195 | h[5] += carry[4] 196 | h[4] -= carry[4] << 26 197 | carry[5] = h[5] >> 25 198 | h[6] += carry[5] 199 | h[5] -= carry[5] << 25 200 | carry[6] = h[6] >> 26 201 | h[7] += carry[6] 202 | h[6] -= carry[6] << 26 203 | carry[7] = h[7] >> 25 204 | h[8] += carry[7] 205 | h[7] -= carry[7] << 25 206 | carry[8] = h[8] >> 26 207 | h[9] += carry[8] 208 | h[8] -= carry[8] << 26 209 | carry[9] = h[9] >> 25 210 | h[9] -= carry[9] << 25 211 | // h10 = carry9 212 | 213 | // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. 214 | // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; 215 | // evidently 2^255 h10-2^255 q = 0. 216 | // Goal: Output h[0]+...+2^230 h[9]. 217 | 218 | s[0] = byte(h[0] >> 0) 219 | s[1] = byte(h[0] >> 8) 220 | s[2] = byte(h[0] >> 16) 221 | s[3] = byte((h[0] >> 24) | (h[1] << 2)) 222 | s[4] = byte(h[1] >> 6) 223 | s[5] = byte(h[1] >> 14) 224 | s[6] = byte((h[1] >> 22) | (h[2] << 3)) 225 | s[7] = byte(h[2] >> 5) 226 | s[8] = byte(h[2] >> 13) 227 | s[9] = byte((h[2] >> 21) | (h[3] << 5)) 228 | s[10] = byte(h[3] >> 3) 229 | s[11] = byte(h[3] >> 11) 230 | s[12] = byte((h[3] >> 19) | (h[4] << 6)) 231 | s[13] = byte(h[4] >> 2) 232 | s[14] = byte(h[4] >> 10) 233 | s[15] = byte(h[4] >> 18) 234 | s[16] = byte(h[5] >> 0) 235 | s[17] = byte(h[5] >> 8) 236 | s[18] = byte(h[5] >> 16) 237 | s[19] = byte((h[5] >> 24) | (h[6] << 1)) 238 | s[20] = byte(h[6] >> 7) 239 | s[21] = byte(h[6] >> 15) 240 | s[22] = byte((h[6] >> 23) | (h[7] << 3)) 241 | s[23] = byte(h[7] >> 5) 242 | s[24] = byte(h[7] >> 13) 243 | s[25] = byte((h[7] >> 21) | (h[8] << 4)) 244 | s[26] = byte(h[8] >> 4) 245 | s[27] = byte(h[8] >> 12) 246 | s[28] = byte((h[8] >> 20) | (h[9] << 6)) 247 | s[29] = byte(h[9] >> 2) 248 | s[30] = byte(h[9] >> 10) 249 | s[31] = byte(h[9] >> 18) 250 | } 251 | 252 | func FeIsNegative(f *FieldElement) byte { 253 | var s [32]byte 254 | FeToBytes(&s, f) 255 | return s[0] & 1 256 | } 257 | 258 | func FeIsNonZero(f *FieldElement) int32 { 259 | var s [32]byte 260 | FeToBytes(&s, f) 261 | var x uint8 262 | for _, b := range s { 263 | x |= b 264 | } 265 | x |= x >> 4 266 | x |= x >> 2 267 | x |= x >> 1 268 | return int32(x & 1) 269 | } 270 | 271 | // FeNeg sets h = -f 272 | // 273 | // Preconditions: 274 | // |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 275 | // 276 | // Postconditions: 277 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 278 | func FeNeg(h, f *FieldElement) { 279 | for i := range h { 280 | h[i] = -f[i] 281 | } 282 | } 283 | 284 | // FeMul calculates h = f * g 285 | // Can overlap h with f or g. 286 | // 287 | // Preconditions: 288 | // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 289 | // |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 290 | // 291 | // Postconditions: 292 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 293 | // 294 | // Notes on implementation strategy: 295 | // 296 | // Using schoolbook multiplication. 297 | // Karatsuba would save a little in some cost models. 298 | // 299 | // Most multiplications by 2 and 19 are 32-bit precomputations; 300 | // cheaper than 64-bit postcomputations. 301 | // 302 | // There is one remaining multiplication by 19 in the carry chain; 303 | // one *19 precomputation can be merged into this, 304 | // but the resulting data flow is considerably less clean. 305 | // 306 | // There are 12 carries below. 307 | // 10 of them are 2-way parallelizable and vectorizable. 308 | // Can get away with 11 carries, but then data flow is much deeper. 309 | // 310 | // With tighter constraints on inputs can squeeze carries into int32. 311 | func FeMul(h, f, g *FieldElement) { 312 | f0 := f[0] 313 | f1 := f[1] 314 | f2 := f[2] 315 | f3 := f[3] 316 | f4 := f[4] 317 | f5 := f[5] 318 | f6 := f[6] 319 | f7 := f[7] 320 | f8 := f[8] 321 | f9 := f[9] 322 | g0 := g[0] 323 | g1 := g[1] 324 | g2 := g[2] 325 | g3 := g[3] 326 | g4 := g[4] 327 | g5 := g[5] 328 | g6 := g[6] 329 | g7 := g[7] 330 | g8 := g[8] 331 | g9 := g[9] 332 | g1_19 := 19 * g1 /* 1.4*2^29 */ 333 | g2_19 := 19 * g2 /* 1.4*2^30; still ok */ 334 | g3_19 := 19 * g3 335 | g4_19 := 19 * g4 336 | g5_19 := 19 * g5 337 | g6_19 := 19 * g6 338 | g7_19 := 19 * g7 339 | g8_19 := 19 * g8 340 | g9_19 := 19 * g9 341 | f1_2 := 2 * f1 342 | f3_2 := 2 * f3 343 | f5_2 := 2 * f5 344 | f7_2 := 2 * f7 345 | f9_2 := 2 * f9 346 | f0g0 := int64(f0) * int64(g0) 347 | f0g1 := int64(f0) * int64(g1) 348 | f0g2 := int64(f0) * int64(g2) 349 | f0g3 := int64(f0) * int64(g3) 350 | f0g4 := int64(f0) * int64(g4) 351 | f0g5 := int64(f0) * int64(g5) 352 | f0g6 := int64(f0) * int64(g6) 353 | f0g7 := int64(f0) * int64(g7) 354 | f0g8 := int64(f0) * int64(g8) 355 | f0g9 := int64(f0) * int64(g9) 356 | f1g0 := int64(f1) * int64(g0) 357 | f1g1_2 := int64(f1_2) * int64(g1) 358 | f1g2 := int64(f1) * int64(g2) 359 | f1g3_2 := int64(f1_2) * int64(g3) 360 | f1g4 := int64(f1) * int64(g4) 361 | f1g5_2 := int64(f1_2) * int64(g5) 362 | f1g6 := int64(f1) * int64(g6) 363 | f1g7_2 := int64(f1_2) * int64(g7) 364 | f1g8 := int64(f1) * int64(g8) 365 | f1g9_38 := int64(f1_2) * int64(g9_19) 366 | f2g0 := int64(f2) * int64(g0) 367 | f2g1 := int64(f2) * int64(g1) 368 | f2g2 := int64(f2) * int64(g2) 369 | f2g3 := int64(f2) * int64(g3) 370 | f2g4 := int64(f2) * int64(g4) 371 | f2g5 := int64(f2) * int64(g5) 372 | f2g6 := int64(f2) * int64(g6) 373 | f2g7 := int64(f2) * int64(g7) 374 | f2g8_19 := int64(f2) * int64(g8_19) 375 | f2g9_19 := int64(f2) * int64(g9_19) 376 | f3g0 := int64(f3) * int64(g0) 377 | f3g1_2 := int64(f3_2) * int64(g1) 378 | f3g2 := int64(f3) * int64(g2) 379 | f3g3_2 := int64(f3_2) * int64(g3) 380 | f3g4 := int64(f3) * int64(g4) 381 | f3g5_2 := int64(f3_2) * int64(g5) 382 | f3g6 := int64(f3) * int64(g6) 383 | f3g7_38 := int64(f3_2) * int64(g7_19) 384 | f3g8_19 := int64(f3) * int64(g8_19) 385 | f3g9_38 := int64(f3_2) * int64(g9_19) 386 | f4g0 := int64(f4) * int64(g0) 387 | f4g1 := int64(f4) * int64(g1) 388 | f4g2 := int64(f4) * int64(g2) 389 | f4g3 := int64(f4) * int64(g3) 390 | f4g4 := int64(f4) * int64(g4) 391 | f4g5 := int64(f4) * int64(g5) 392 | f4g6_19 := int64(f4) * int64(g6_19) 393 | f4g7_19 := int64(f4) * int64(g7_19) 394 | f4g8_19 := int64(f4) * int64(g8_19) 395 | f4g9_19 := int64(f4) * int64(g9_19) 396 | f5g0 := int64(f5) * int64(g0) 397 | f5g1_2 := int64(f5_2) * int64(g1) 398 | f5g2 := int64(f5) * int64(g2) 399 | f5g3_2 := int64(f5_2) * int64(g3) 400 | f5g4 := int64(f5) * int64(g4) 401 | f5g5_38 := int64(f5_2) * int64(g5_19) 402 | f5g6_19 := int64(f5) * int64(g6_19) 403 | f5g7_38 := int64(f5_2) * int64(g7_19) 404 | f5g8_19 := int64(f5) * int64(g8_19) 405 | f5g9_38 := int64(f5_2) * int64(g9_19) 406 | f6g0 := int64(f6) * int64(g0) 407 | f6g1 := int64(f6) * int64(g1) 408 | f6g2 := int64(f6) * int64(g2) 409 | f6g3 := int64(f6) * int64(g3) 410 | f6g4_19 := int64(f6) * int64(g4_19) 411 | f6g5_19 := int64(f6) * int64(g5_19) 412 | f6g6_19 := int64(f6) * int64(g6_19) 413 | f6g7_19 := int64(f6) * int64(g7_19) 414 | f6g8_19 := int64(f6) * int64(g8_19) 415 | f6g9_19 := int64(f6) * int64(g9_19) 416 | f7g0 := int64(f7) * int64(g0) 417 | f7g1_2 := int64(f7_2) * int64(g1) 418 | f7g2 := int64(f7) * int64(g2) 419 | f7g3_38 := int64(f7_2) * int64(g3_19) 420 | f7g4_19 := int64(f7) * int64(g4_19) 421 | f7g5_38 := int64(f7_2) * int64(g5_19) 422 | f7g6_19 := int64(f7) * int64(g6_19) 423 | f7g7_38 := int64(f7_2) * int64(g7_19) 424 | f7g8_19 := int64(f7) * int64(g8_19) 425 | f7g9_38 := int64(f7_2) * int64(g9_19) 426 | f8g0 := int64(f8) * int64(g0) 427 | f8g1 := int64(f8) * int64(g1) 428 | f8g2_19 := int64(f8) * int64(g2_19) 429 | f8g3_19 := int64(f8) * int64(g3_19) 430 | f8g4_19 := int64(f8) * int64(g4_19) 431 | f8g5_19 := int64(f8) * int64(g5_19) 432 | f8g6_19 := int64(f8) * int64(g6_19) 433 | f8g7_19 := int64(f8) * int64(g7_19) 434 | f8g8_19 := int64(f8) * int64(g8_19) 435 | f8g9_19 := int64(f8) * int64(g9_19) 436 | f9g0 := int64(f9) * int64(g0) 437 | f9g1_38 := int64(f9_2) * int64(g1_19) 438 | f9g2_19 := int64(f9) * int64(g2_19) 439 | f9g3_38 := int64(f9_2) * int64(g3_19) 440 | f9g4_19 := int64(f9) * int64(g4_19) 441 | f9g5_38 := int64(f9_2) * int64(g5_19) 442 | f9g6_19 := int64(f9) * int64(g6_19) 443 | f9g7_38 := int64(f9_2) * int64(g7_19) 444 | f9g8_19 := int64(f9) * int64(g8_19) 445 | f9g9_38 := int64(f9_2) * int64(g9_19) 446 | h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38 447 | h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19 448 | h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38 449 | h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19 450 | h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38 451 | h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19 452 | h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38 453 | h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19 454 | h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38 455 | h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 456 | var carry [10]int64 457 | 458 | /* 459 | |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) 460 | i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8 461 | |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) 462 | i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9 463 | */ 464 | 465 | carry[0] = (h0 + (1 << 25)) >> 26 466 | h1 += carry[0] 467 | h0 -= carry[0] << 26 468 | carry[4] = (h4 + (1 << 25)) >> 26 469 | h5 += carry[4] 470 | h4 -= carry[4] << 26 471 | /* |h0| <= 2^25 */ 472 | /* |h4| <= 2^25 */ 473 | /* |h1| <= 1.51*2^58 */ 474 | /* |h5| <= 1.51*2^58 */ 475 | 476 | carry[1] = (h1 + (1 << 24)) >> 25 477 | h2 += carry[1] 478 | h1 -= carry[1] << 25 479 | carry[5] = (h5 + (1 << 24)) >> 25 480 | h6 += carry[5] 481 | h5 -= carry[5] << 25 482 | /* |h1| <= 2^24; from now on fits into int32 */ 483 | /* |h5| <= 2^24; from now on fits into int32 */ 484 | /* |h2| <= 1.21*2^59 */ 485 | /* |h6| <= 1.21*2^59 */ 486 | 487 | carry[2] = (h2 + (1 << 25)) >> 26 488 | h3 += carry[2] 489 | h2 -= carry[2] << 26 490 | carry[6] = (h6 + (1 << 25)) >> 26 491 | h7 += carry[6] 492 | h6 -= carry[6] << 26 493 | /* |h2| <= 2^25; from now on fits into int32 unchanged */ 494 | /* |h6| <= 2^25; from now on fits into int32 unchanged */ 495 | /* |h3| <= 1.51*2^58 */ 496 | /* |h7| <= 1.51*2^58 */ 497 | 498 | carry[3] = (h3 + (1 << 24)) >> 25 499 | h4 += carry[3] 500 | h3 -= carry[3] << 25 501 | carry[7] = (h7 + (1 << 24)) >> 25 502 | h8 += carry[7] 503 | h7 -= carry[7] << 25 504 | /* |h3| <= 2^24; from now on fits into int32 unchanged */ 505 | /* |h7| <= 2^24; from now on fits into int32 unchanged */ 506 | /* |h4| <= 1.52*2^33 */ 507 | /* |h8| <= 1.52*2^33 */ 508 | 509 | carry[4] = (h4 + (1 << 25)) >> 26 510 | h5 += carry[4] 511 | h4 -= carry[4] << 26 512 | carry[8] = (h8 + (1 << 25)) >> 26 513 | h9 += carry[8] 514 | h8 -= carry[8] << 26 515 | /* |h4| <= 2^25; from now on fits into int32 unchanged */ 516 | /* |h8| <= 2^25; from now on fits into int32 unchanged */ 517 | /* |h5| <= 1.01*2^24 */ 518 | /* |h9| <= 1.51*2^58 */ 519 | 520 | carry[9] = (h9 + (1 << 24)) >> 25 521 | h0 += carry[9] * 19 522 | h9 -= carry[9] << 25 523 | /* |h9| <= 2^24; from now on fits into int32 unchanged */ 524 | /* |h0| <= 1.8*2^37 */ 525 | 526 | carry[0] = (h0 + (1 << 25)) >> 26 527 | h1 += carry[0] 528 | h0 -= carry[0] << 26 529 | /* |h0| <= 2^25; from now on fits into int32 unchanged */ 530 | /* |h1| <= 1.01*2^24 */ 531 | 532 | h[0] = int32(h0) 533 | h[1] = int32(h1) 534 | h[2] = int32(h2) 535 | h[3] = int32(h3) 536 | h[4] = int32(h4) 537 | h[5] = int32(h5) 538 | h[6] = int32(h6) 539 | h[7] = int32(h7) 540 | h[8] = int32(h8) 541 | h[9] = int32(h9) 542 | } 543 | 544 | // FeSquare calculates h = f*f. Can overlap h with f. 545 | // 546 | // Preconditions: 547 | // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 548 | // 549 | // Postconditions: 550 | // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 551 | func FeSquare(h, f *FieldElement) { 552 | f0 := f[0] 553 | f1 := f[1] 554 | f2 := f[2] 555 | f3 := f[3] 556 | f4 := f[4] 557 | f5 := f[5] 558 | f6 := f[6] 559 | f7 := f[7] 560 | f8 := f[8] 561 | f9 := f[9] 562 | f0_2 := 2 * f0 563 | f1_2 := 2 * f1 564 | f2_2 := 2 * f2 565 | f3_2 := 2 * f3 566 | f4_2 := 2 * f4 567 | f5_2 := 2 * f5 568 | f6_2 := 2 * f6 569 | f7_2 := 2 * f7 570 | f5_38 := 38 * f5 // 1.31*2^30 571 | f6_19 := 19 * f6 // 1.31*2^30 572 | f7_38 := 38 * f7 // 1.31*2^30 573 | f8_19 := 19 * f8 // 1.31*2^30 574 | f9_38 := 38 * f9 // 1.31*2^30 575 | f0f0 := int64(f0) * int64(f0) 576 | f0f1_2 := int64(f0_2) * int64(f1) 577 | f0f2_2 := int64(f0_2) * int64(f2) 578 | f0f3_2 := int64(f0_2) * int64(f3) 579 | f0f4_2 := int64(f0_2) * int64(f4) 580 | f0f5_2 := int64(f0_2) * int64(f5) 581 | f0f6_2 := int64(f0_2) * int64(f6) 582 | f0f7_2 := int64(f0_2) * int64(f7) 583 | f0f8_2 := int64(f0_2) * int64(f8) 584 | f0f9_2 := int64(f0_2) * int64(f9) 585 | f1f1_2 := int64(f1_2) * int64(f1) 586 | f1f2_2 := int64(f1_2) * int64(f2) 587 | f1f3_4 := int64(f1_2) * int64(f3_2) 588 | f1f4_2 := int64(f1_2) * int64(f4) 589 | f1f5_4 := int64(f1_2) * int64(f5_2) 590 | f1f6_2 := int64(f1_2) * int64(f6) 591 | f1f7_4 := int64(f1_2) * int64(f7_2) 592 | f1f8_2 := int64(f1_2) * int64(f8) 593 | f1f9_76 := int64(f1_2) * int64(f9_38) 594 | f2f2 := int64(f2) * int64(f2) 595 | f2f3_2 := int64(f2_2) * int64(f3) 596 | f2f4_2 := int64(f2_2) * int64(f4) 597 | f2f5_2 := int64(f2_2) * int64(f5) 598 | f2f6_2 := int64(f2_2) * int64(f6) 599 | f2f7_2 := int64(f2_2) * int64(f7) 600 | f2f8_38 := int64(f2_2) * int64(f8_19) 601 | f2f9_38 := int64(f2) * int64(f9_38) 602 | f3f3_2 := int64(f3_2) * int64(f3) 603 | f3f4_2 := int64(f3_2) * int64(f4) 604 | f3f5_4 := int64(f3_2) * int64(f5_2) 605 | f3f6_2 := int64(f3_2) * int64(f6) 606 | f3f7_76 := int64(f3_2) * int64(f7_38) 607 | f3f8_38 := int64(f3_2) * int64(f8_19) 608 | f3f9_76 := int64(f3_2) * int64(f9_38) 609 | f4f4 := int64(f4) * int64(f4) 610 | f4f5_2 := int64(f4_2) * int64(f5) 611 | f4f6_38 := int64(f4_2) * int64(f6_19) 612 | f4f7_38 := int64(f4) * int64(f7_38) 613 | f4f8_38 := int64(f4_2) * int64(f8_19) 614 | f4f9_38 := int64(f4) * int64(f9_38) 615 | f5f5_38 := int64(f5) * int64(f5_38) 616 | f5f6_38 := int64(f5_2) * int64(f6_19) 617 | f5f7_76 := int64(f5_2) * int64(f7_38) 618 | f5f8_38 := int64(f5_2) * int64(f8_19) 619 | f5f9_76 := int64(f5_2) * int64(f9_38) 620 | f6f6_19 := int64(f6) * int64(f6_19) 621 | f6f7_38 := int64(f6) * int64(f7_38) 622 | f6f8_38 := int64(f6_2) * int64(f8_19) 623 | f6f9_38 := int64(f6) * int64(f9_38) 624 | f7f7_38 := int64(f7) * int64(f7_38) 625 | f7f8_38 := int64(f7_2) * int64(f8_19) 626 | f7f9_76 := int64(f7_2) * int64(f9_38) 627 | f8f8_19 := int64(f8) * int64(f8_19) 628 | f8f9_38 := int64(f8) * int64(f9_38) 629 | f9f9_38 := int64(f9) * int64(f9_38) 630 | h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38 631 | h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38 632 | h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19 633 | h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38 634 | h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38 635 | h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38 636 | h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19 637 | h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38 638 | h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38 639 | h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2 640 | var carry [10]int64 641 | 642 | carry[0] = (h0 + (1 << 25)) >> 26 643 | h1 += carry[0] 644 | h0 -= carry[0] << 26 645 | carry[4] = (h4 + (1 << 25)) >> 26 646 | h5 += carry[4] 647 | h4 -= carry[4] << 26 648 | 649 | carry[1] = (h1 + (1 << 24)) >> 25 650 | h2 += carry[1] 651 | h1 -= carry[1] << 25 652 | carry[5] = (h5 + (1 << 24)) >> 25 653 | h6 += carry[5] 654 | h5 -= carry[5] << 25 655 | 656 | carry[2] = (h2 + (1 << 25)) >> 26 657 | h3 += carry[2] 658 | h2 -= carry[2] << 26 659 | carry[6] = (h6 + (1 << 25)) >> 26 660 | h7 += carry[6] 661 | h6 -= carry[6] << 26 662 | 663 | carry[3] = (h3 + (1 << 24)) >> 25 664 | h4 += carry[3] 665 | h3 -= carry[3] << 25 666 | carry[7] = (h7 + (1 << 24)) >> 25 667 | h8 += carry[7] 668 | h7 -= carry[7] << 25 669 | 670 | carry[4] = (h4 + (1 << 25)) >> 26 671 | h5 += carry[4] 672 | h4 -= carry[4] << 26 673 | carry[8] = (h8 + (1 << 25)) >> 26 674 | h9 += carry[8] 675 | h8 -= carry[8] << 26 676 | 677 | carry[9] = (h9 + (1 << 24)) >> 25 678 | h0 += carry[9] * 19 679 | h9 -= carry[9] << 25 680 | 681 | carry[0] = (h0 + (1 << 25)) >> 26 682 | h1 += carry[0] 683 | h0 -= carry[0] << 26 684 | 685 | h[0] = int32(h0) 686 | h[1] = int32(h1) 687 | h[2] = int32(h2) 688 | h[3] = int32(h3) 689 | h[4] = int32(h4) 690 | h[5] = int32(h5) 691 | h[6] = int32(h6) 692 | h[7] = int32(h7) 693 | h[8] = int32(h8) 694 | h[9] = int32(h9) 695 | } 696 | 697 | // FeSquare2 sets h = 2 * f * f 698 | // 699 | // Can overlap h with f. 700 | // 701 | // Preconditions: 702 | // |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 703 | // 704 | // Postconditions: 705 | // |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 706 | // See fe_mul.c for discussion of implementation strategy. 707 | func FeSquare2(h, f *FieldElement) { 708 | f0 := f[0] 709 | f1 := f[1] 710 | f2 := f[2] 711 | f3 := f[3] 712 | f4 := f[4] 713 | f5 := f[5] 714 | f6 := f[6] 715 | f7 := f[7] 716 | f8 := f[8] 717 | f9 := f[9] 718 | f0_2 := 2 * f0 719 | f1_2 := 2 * f1 720 | f2_2 := 2 * f2 721 | f3_2 := 2 * f3 722 | f4_2 := 2 * f4 723 | f5_2 := 2 * f5 724 | f6_2 := 2 * f6 725 | f7_2 := 2 * f7 726 | f5_38 := 38 * f5 // 1.959375*2^30 727 | f6_19 := 19 * f6 // 1.959375*2^30 728 | f7_38 := 38 * f7 // 1.959375*2^30 729 | f8_19 := 19 * f8 // 1.959375*2^30 730 | f9_38 := 38 * f9 // 1.959375*2^30 731 | f0f0 := int64(f0) * int64(f0) 732 | f0f1_2 := int64(f0_2) * int64(f1) 733 | f0f2_2 := int64(f0_2) * int64(f2) 734 | f0f3_2 := int64(f0_2) * int64(f3) 735 | f0f4_2 := int64(f0_2) * int64(f4) 736 | f0f5_2 := int64(f0_2) * int64(f5) 737 | f0f6_2 := int64(f0_2) * int64(f6) 738 | f0f7_2 := int64(f0_2) * int64(f7) 739 | f0f8_2 := int64(f0_2) * int64(f8) 740 | f0f9_2 := int64(f0_2) * int64(f9) 741 | f1f1_2 := int64(f1_2) * int64(f1) 742 | f1f2_2 := int64(f1_2) * int64(f2) 743 | f1f3_4 := int64(f1_2) * int64(f3_2) 744 | f1f4_2 := int64(f1_2) * int64(f4) 745 | f1f5_4 := int64(f1_2) * int64(f5_2) 746 | f1f6_2 := int64(f1_2) * int64(f6) 747 | f1f7_4 := int64(f1_2) * int64(f7_2) 748 | f1f8_2 := int64(f1_2) * int64(f8) 749 | f1f9_76 := int64(f1_2) * int64(f9_38) 750 | f2f2 := int64(f2) * int64(f2) 751 | f2f3_2 := int64(f2_2) * int64(f3) 752 | f2f4_2 := int64(f2_2) * int64(f4) 753 | f2f5_2 := int64(f2_2) * int64(f5) 754 | f2f6_2 := int64(f2_2) * int64(f6) 755 | f2f7_2 := int64(f2_2) * int64(f7) 756 | f2f8_38 := int64(f2_2) * int64(f8_19) 757 | f2f9_38 := int64(f2) * int64(f9_38) 758 | f3f3_2 := int64(f3_2) * int64(f3) 759 | f3f4_2 := int64(f3_2) * int64(f4) 760 | f3f5_4 := int64(f3_2) * int64(f5_2) 761 | f3f6_2 := int64(f3_2) * int64(f6) 762 | f3f7_76 := int64(f3_2) * int64(f7_38) 763 | f3f8_38 := int64(f3_2) * int64(f8_19) 764 | f3f9_76 := int64(f3_2) * int64(f9_38) 765 | f4f4 := int64(f4) * int64(f4) 766 | f4f5_2 := int64(f4_2) * int64(f5) 767 | f4f6_38 := int64(f4_2) * int64(f6_19) 768 | f4f7_38 := int64(f4) * int64(f7_38) 769 | f4f8_38 := int64(f4_2) * int64(f8_19) 770 | f4f9_38 := int64(f4) * int64(f9_38) 771 | f5f5_38 := int64(f5) * int64(f5_38) 772 | f5f6_38 := int64(f5_2) * int64(f6_19) 773 | f5f7_76 := int64(f5_2) * int64(f7_38) 774 | f5f8_38 := int64(f5_2) * int64(f8_19) 775 | f5f9_76 := int64(f5_2) * int64(f9_38) 776 | f6f6_19 := int64(f6) * int64(f6_19) 777 | f6f7_38 := int64(f6) * int64(f7_38) 778 | f6f8_38 := int64(f6_2) * int64(f8_19) 779 | f6f9_38 := int64(f6) * int64(f9_38) 780 | f7f7_38 := int64(f7) * int64(f7_38) 781 | f7f8_38 := int64(f7_2) * int64(f8_19) 782 | f7f9_76 := int64(f7_2) * int64(f9_38) 783 | f8f8_19 := int64(f8) * int64(f8_19) 784 | f8f9_38 := int64(f8) * int64(f9_38) 785 | f9f9_38 := int64(f9) * int64(f9_38) 786 | h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38 787 | h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38 788 | h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19 789 | h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38 790 | h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38 791 | h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38 792 | h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19 793 | h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38 794 | h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38 795 | h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2 796 | var carry [10]int64 797 | 798 | h0 += h0 799 | h1 += h1 800 | h2 += h2 801 | h3 += h3 802 | h4 += h4 803 | h5 += h5 804 | h6 += h6 805 | h7 += h7 806 | h8 += h8 807 | h9 += h9 808 | 809 | carry[0] = (h0 + (1 << 25)) >> 26 810 | h1 += carry[0] 811 | h0 -= carry[0] << 26 812 | carry[4] = (h4 + (1 << 25)) >> 26 813 | h5 += carry[4] 814 | h4 -= carry[4] << 26 815 | 816 | carry[1] = (h1 + (1 << 24)) >> 25 817 | h2 += carry[1] 818 | h1 -= carry[1] << 25 819 | carry[5] = (h5 + (1 << 24)) >> 25 820 | h6 += carry[5] 821 | h5 -= carry[5] << 25 822 | 823 | carry[2] = (h2 + (1 << 25)) >> 26 824 | h3 += carry[2] 825 | h2 -= carry[2] << 26 826 | carry[6] = (h6 + (1 << 25)) >> 26 827 | h7 += carry[6] 828 | h6 -= carry[6] << 26 829 | 830 | carry[3] = (h3 + (1 << 24)) >> 25 831 | h4 += carry[3] 832 | h3 -= carry[3] << 25 833 | carry[7] = (h7 + (1 << 24)) >> 25 834 | h8 += carry[7] 835 | h7 -= carry[7] << 25 836 | 837 | carry[4] = (h4 + (1 << 25)) >> 26 838 | h5 += carry[4] 839 | h4 -= carry[4] << 26 840 | carry[8] = (h8 + (1 << 25)) >> 26 841 | h9 += carry[8] 842 | h8 -= carry[8] << 26 843 | 844 | carry[9] = (h9 + (1 << 24)) >> 25 845 | h0 += carry[9] * 19 846 | h9 -= carry[9] << 25 847 | 848 | carry[0] = (h0 + (1 << 25)) >> 26 849 | h1 += carry[0] 850 | h0 -= carry[0] << 26 851 | 852 | h[0] = int32(h0) 853 | h[1] = int32(h1) 854 | h[2] = int32(h2) 855 | h[3] = int32(h3) 856 | h[4] = int32(h4) 857 | h[5] = int32(h5) 858 | h[6] = int32(h6) 859 | h[7] = int32(h7) 860 | h[8] = int32(h8) 861 | h[9] = int32(h9) 862 | } 863 | 864 | func FeInvert(out, z *FieldElement) { 865 | var t0, t1, t2, t3 FieldElement 866 | var i int 867 | 868 | FeSquare(&t0, z) // 2^1 869 | FeSquare(&t1, &t0) // 2^2 870 | for i = 1; i < 2; i++ { // 2^3 871 | FeSquare(&t1, &t1) 872 | } 873 | FeMul(&t1, z, &t1) // 2^3 + 2^0 874 | FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0 875 | FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1 876 | FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0 877 | FeSquare(&t2, &t1) // 5,4,3,2,1 878 | for i = 1; i < 5; i++ { // 9,8,7,6,5 879 | FeSquare(&t2, &t2) 880 | } 881 | FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 882 | FeSquare(&t2, &t1) // 10..1 883 | for i = 1; i < 10; i++ { // 19..10 884 | FeSquare(&t2, &t2) 885 | } 886 | FeMul(&t2, &t2, &t1) // 19..0 887 | FeSquare(&t3, &t2) // 20..1 888 | for i = 1; i < 20; i++ { // 39..20 889 | FeSquare(&t3, &t3) 890 | } 891 | FeMul(&t2, &t3, &t2) // 39..0 892 | FeSquare(&t2, &t2) // 40..1 893 | for i = 1; i < 10; i++ { // 49..10 894 | FeSquare(&t2, &t2) 895 | } 896 | FeMul(&t1, &t2, &t1) // 49..0 897 | FeSquare(&t2, &t1) // 50..1 898 | for i = 1; i < 50; i++ { // 99..50 899 | FeSquare(&t2, &t2) 900 | } 901 | FeMul(&t2, &t2, &t1) // 99..0 902 | FeSquare(&t3, &t2) // 100..1 903 | for i = 1; i < 100; i++ { // 199..100 904 | FeSquare(&t3, &t3) 905 | } 906 | FeMul(&t2, &t3, &t2) // 199..0 907 | FeSquare(&t2, &t2) // 200..1 908 | for i = 1; i < 50; i++ { // 249..50 909 | FeSquare(&t2, &t2) 910 | } 911 | FeMul(&t1, &t2, &t1) // 249..0 912 | FeSquare(&t1, &t1) // 250..1 913 | for i = 1; i < 5; i++ { // 254..5 914 | FeSquare(&t1, &t1) 915 | } 916 | FeMul(out, &t1, &t0) // 254..5,3,1,0 917 | } 918 | 919 | func fePow22523(out, z *FieldElement) { 920 | var t0, t1, t2 FieldElement 921 | var i int 922 | 923 | FeSquare(&t0, z) 924 | for i = 1; i < 1; i++ { 925 | FeSquare(&t0, &t0) 926 | } 927 | FeSquare(&t1, &t0) 928 | for i = 1; i < 2; i++ { 929 | FeSquare(&t1, &t1) 930 | } 931 | FeMul(&t1, z, &t1) 932 | FeMul(&t0, &t0, &t1) 933 | FeSquare(&t0, &t0) 934 | for i = 1; i < 1; i++ { 935 | FeSquare(&t0, &t0) 936 | } 937 | FeMul(&t0, &t1, &t0) 938 | FeSquare(&t1, &t0) 939 | for i = 1; i < 5; i++ { 940 | FeSquare(&t1, &t1) 941 | } 942 | FeMul(&t0, &t1, &t0) 943 | FeSquare(&t1, &t0) 944 | for i = 1; i < 10; i++ { 945 | FeSquare(&t1, &t1) 946 | } 947 | FeMul(&t1, &t1, &t0) 948 | FeSquare(&t2, &t1) 949 | for i = 1; i < 20; i++ { 950 | FeSquare(&t2, &t2) 951 | } 952 | FeMul(&t1, &t2, &t1) 953 | FeSquare(&t1, &t1) 954 | for i = 1; i < 10; i++ { 955 | FeSquare(&t1, &t1) 956 | } 957 | FeMul(&t0, &t1, &t0) 958 | FeSquare(&t1, &t0) 959 | for i = 1; i < 50; i++ { 960 | FeSquare(&t1, &t1) 961 | } 962 | FeMul(&t1, &t1, &t0) 963 | FeSquare(&t2, &t1) 964 | for i = 1; i < 100; i++ { 965 | FeSquare(&t2, &t2) 966 | } 967 | FeMul(&t1, &t2, &t1) 968 | FeSquare(&t1, &t1) 969 | for i = 1; i < 50; i++ { 970 | FeSquare(&t1, &t1) 971 | } 972 | FeMul(&t0, &t1, &t0) 973 | FeSquare(&t0, &t0) 974 | for i = 1; i < 2; i++ { 975 | FeSquare(&t0, &t0) 976 | } 977 | FeMul(out, &t0, z) 978 | } 979 | 980 | // Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 * 981 | // y^2 where d = -121665/121666. 982 | // 983 | // Several representations are used: 984 | // ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z 985 | // ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT 986 | // CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T 987 | // PreComputedGroupElement: (y+x,y-x,2dxy) 988 | 989 | type ProjectiveGroupElement struct { 990 | X, Y, Z FieldElement 991 | } 992 | 993 | type ExtendedGroupElement struct { 994 | X, Y, Z, T FieldElement 995 | } 996 | 997 | type CompletedGroupElement struct { 998 | X, Y, Z, T FieldElement 999 | } 1000 | 1001 | type PreComputedGroupElement struct { 1002 | yPlusX, yMinusX, xy2d FieldElement 1003 | } 1004 | 1005 | type CachedGroupElement struct { 1006 | yPlusX, yMinusX, Z, T2d FieldElement 1007 | } 1008 | 1009 | func (p *ProjectiveGroupElement) Zero() { 1010 | FeZero(&p.X) 1011 | FeOne(&p.Y) 1012 | FeOne(&p.Z) 1013 | } 1014 | 1015 | func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) { 1016 | var t0 FieldElement 1017 | 1018 | FeSquare(&r.X, &p.X) 1019 | FeSquare(&r.Z, &p.Y) 1020 | FeSquare2(&r.T, &p.Z) 1021 | FeAdd(&r.Y, &p.X, &p.Y) 1022 | FeSquare(&t0, &r.Y) 1023 | FeAdd(&r.Y, &r.Z, &r.X) 1024 | FeSub(&r.Z, &r.Z, &r.X) 1025 | FeSub(&r.X, &t0, &r.Y) 1026 | FeSub(&r.T, &r.T, &r.Z) 1027 | } 1028 | 1029 | func (p *ProjectiveGroupElement) ToExtended(s *ExtendedGroupElement) { 1030 | var b [32]byte 1031 | p.ToBytes(&b) 1032 | s.FromBytes(&b) 1033 | } 1034 | 1035 | func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) { 1036 | var recip, x, y FieldElement 1037 | 1038 | FeInvert(&recip, &p.Z) 1039 | FeMul(&x, &p.X, &recip) 1040 | FeMul(&y, &p.Y, &recip) 1041 | FeToBytes(s, &y) 1042 | s[31] ^= FeIsNegative(&x) << 7 1043 | } 1044 | 1045 | func (p *ExtendedGroupElement) Zero() { 1046 | FeZero(&p.X) 1047 | FeOne(&p.Y) 1048 | FeOne(&p.Z) 1049 | FeZero(&p.T) 1050 | } 1051 | 1052 | func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) { 1053 | var q ProjectiveGroupElement 1054 | p.ToProjective(&q) 1055 | q.Double(r) 1056 | } 1057 | 1058 | func GeDouble(r, p *ExtendedGroupElement) { 1059 | var q ProjectiveGroupElement 1060 | p.ToProjective(&q) 1061 | var rco CompletedGroupElement 1062 | q.Double(&rco) 1063 | rco.ToExtended(r) 1064 | } 1065 | 1066 | func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) { 1067 | FeAdd(&r.yPlusX, &p.Y, &p.X) 1068 | FeSub(&r.yMinusX, &p.Y, &p.X) 1069 | FeCopy(&r.Z, &p.Z) 1070 | FeMul(&r.T2d, &p.T, &d2) 1071 | } 1072 | 1073 | func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) { 1074 | FeCopy(&r.X, &p.X) 1075 | FeCopy(&r.Y, &p.Y) 1076 | FeCopy(&r.Z, &p.Z) 1077 | } 1078 | 1079 | func (p *ExtendedGroupElement) ToBytes(s *[32]byte) { 1080 | var recip, x, y FieldElement 1081 | 1082 | FeInvert(&recip, &p.Z) 1083 | FeMul(&x, &p.X, &recip) 1084 | FeMul(&y, &p.Y, &recip) 1085 | FeToBytes(s, &y) 1086 | s[31] ^= FeIsNegative(&x) << 7 1087 | } 1088 | 1089 | // FromBytesBaseGroup unmarshals an elliptic curve point returns true iff the 1090 | // point point is in the order l subgroup generated by the base point. This 1091 | // implementation is based on 1092 | // https://www.iacr.org/archive/pkc2003/25670211/25670211.pdf Definition 1. 1093 | // Validation of an elliptic curve public key P ensures that P is a point of 1094 | // order BasePointOrder in E. 1095 | func (p *ExtendedGroupElement) FromBytesBaseGroup(s *[32]byte) bool { 1096 | 1097 | // condition 1: P != infinity 1098 | if *s == [32]byte{1} { 1099 | return false 1100 | } 1101 | 1102 | // condition 3 is implied by successful point decompression 1103 | if !p.FromBytes(s) { 1104 | return false 1105 | } 1106 | // condition 2: ToBytes produces canonical encodings, check against that 1107 | var sCheck [32]byte 1108 | p.ToBytes(&sCheck) 1109 | if sCheck != *s { 1110 | return false 1111 | } 1112 | 1113 | // condition 4: order * P == infinity 1114 | var out ExtendedGroupElement 1115 | GeScalarMult(&out, &BasePointOrder, p) 1116 | out.ToBytes(&sCheck) 1117 | if sCheck != [32]byte{1} { 1118 | return false 1119 | } 1120 | 1121 | // an array of all zeros would not result from any randomly generated point 1122 | // and might easily be caused by bugs in ToBytes or FromBytes 1123 | if *s == [32]byte{} { 1124 | return false 1125 | } 1126 | return true 1127 | } 1128 | 1129 | func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool { 1130 | FeFromBytes(&p.Y, s) 1131 | return p.FromParityAndY(s[31]>>7, &p.Y) 1132 | } 1133 | 1134 | func (p *ExtendedGroupElement) FromParityAndY(bit byte, y *FieldElement) bool { 1135 | var u, v, v3, vxx, check FieldElement 1136 | 1137 | FeCopy(&p.Y, y) 1138 | FeOne(&p.Z) 1139 | FeSquare(&u, &p.Y) 1140 | FeMul(&v, &u, &d) 1141 | FeSub(&u, &u, &p.Z) // y = y^2-1 1142 | FeAdd(&v, &v, &p.Z) // v = dy^2+1 1143 | 1144 | FeSquare(&v3, &v) 1145 | FeMul(&v3, &v3, &v) // v3 = v^3 1146 | FeSquare(&p.X, &v3) 1147 | FeMul(&p.X, &p.X, &v) 1148 | FeMul(&p.X, &p.X, &u) // x = uv^7 1149 | 1150 | fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8) 1151 | FeMul(&p.X, &p.X, &v3) 1152 | FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8) 1153 | 1154 | var tmpX, tmp2 [32]byte 1155 | 1156 | FeSquare(&vxx, &p.X) 1157 | FeMul(&vxx, &vxx, &v) 1158 | FeSub(&check, &vxx, &u) // vx^2-u 1159 | if FeIsNonZero(&check) == 1 { 1160 | FeAdd(&check, &vxx, &u) // vx^2+u 1161 | if FeIsNonZero(&check) == 1 { 1162 | return false 1163 | } 1164 | FeMul(&p.X, &p.X, &SqrtM1) 1165 | 1166 | FeToBytes(&tmpX, &p.X) 1167 | for i, v := range tmpX { 1168 | tmp2[31-i] = v 1169 | } 1170 | } 1171 | 1172 | if FeIsNegative(&p.X) != bit { 1173 | FeNeg(&p.X, &p.X) 1174 | } 1175 | 1176 | FeMul(&p.T, &p.X, &p.Y) 1177 | return true 1178 | } 1179 | 1180 | func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) { 1181 | FeMul(&r.X, &p.X, &p.T) 1182 | FeMul(&r.Y, &p.Y, &p.Z) 1183 | FeMul(&r.Z, &p.Z, &p.T) 1184 | } 1185 | 1186 | func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) { 1187 | FeMul(&r.X, &p.X, &p.T) 1188 | FeMul(&r.Y, &p.Y, &p.Z) 1189 | FeMul(&r.Z, &p.Z, &p.T) 1190 | FeMul(&r.T, &p.X, &p.Y) 1191 | } 1192 | 1193 | func (p *PreComputedGroupElement) Zero() { 1194 | FeOne(&p.yPlusX) 1195 | FeOne(&p.yMinusX) 1196 | FeZero(&p.xy2d) 1197 | } 1198 | 1199 | func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { 1200 | var t0 FieldElement 1201 | 1202 | FeAdd(&r.X, &p.Y, &p.X) 1203 | FeSub(&r.Y, &p.Y, &p.X) 1204 | FeMul(&r.Z, &r.X, &q.yPlusX) 1205 | FeMul(&r.Y, &r.Y, &q.yMinusX) 1206 | FeMul(&r.T, &q.T2d, &p.T) 1207 | FeMul(&r.X, &p.Z, &q.Z) 1208 | FeAdd(&t0, &r.X, &r.X) 1209 | FeSub(&r.X, &r.Z, &r.Y) 1210 | FeAdd(&r.Y, &r.Z, &r.Y) 1211 | FeAdd(&r.Z, &t0, &r.T) 1212 | FeSub(&r.T, &t0, &r.T) 1213 | } 1214 | 1215 | func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { 1216 | var t0 FieldElement 1217 | 1218 | FeAdd(&r.X, &p.Y, &p.X) 1219 | FeSub(&r.Y, &p.Y, &p.X) 1220 | FeMul(&r.Z, &r.X, &q.yMinusX) 1221 | FeMul(&r.Y, &r.Y, &q.yPlusX) 1222 | FeMul(&r.T, &q.T2d, &p.T) 1223 | FeMul(&r.X, &p.Z, &q.Z) 1224 | FeAdd(&t0, &r.X, &r.X) 1225 | FeSub(&r.X, &r.Z, &r.Y) 1226 | FeAdd(&r.Y, &r.Z, &r.Y) 1227 | FeSub(&r.Z, &t0, &r.T) 1228 | FeAdd(&r.T, &t0, &r.T) 1229 | } 1230 | 1231 | func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { 1232 | var t0 FieldElement 1233 | 1234 | FeAdd(&r.X, &p.Y, &p.X) 1235 | FeSub(&r.Y, &p.Y, &p.X) 1236 | FeMul(&r.Z, &r.X, &q.yPlusX) 1237 | FeMul(&r.Y, &r.Y, &q.yMinusX) 1238 | FeMul(&r.T, &q.xy2d, &p.T) 1239 | FeAdd(&t0, &p.Z, &p.Z) 1240 | FeSub(&r.X, &r.Z, &r.Y) 1241 | FeAdd(&r.Y, &r.Z, &r.Y) 1242 | FeAdd(&r.Z, &t0, &r.T) 1243 | FeSub(&r.T, &t0, &r.T) 1244 | } 1245 | 1246 | func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { 1247 | var t0 FieldElement 1248 | 1249 | FeAdd(&r.X, &p.Y, &p.X) 1250 | FeSub(&r.Y, &p.Y, &p.X) 1251 | FeMul(&r.Z, &r.X, &q.yMinusX) 1252 | FeMul(&r.Y, &r.Y, &q.yPlusX) 1253 | FeMul(&r.T, &q.xy2d, &p.T) 1254 | FeAdd(&t0, &p.Z, &p.Z) 1255 | FeSub(&r.X, &r.Z, &r.Y) 1256 | FeAdd(&r.Y, &r.Z, &r.Y) 1257 | FeSub(&r.Z, &t0, &r.T) 1258 | FeAdd(&r.T, &t0, &r.T) 1259 | } 1260 | 1261 | func slide(r *[256]int8, a *[32]byte) { 1262 | for i := range r { 1263 | r[i] = int8(1 & (a[i>>3] >> uint(i&7))) 1264 | } 1265 | 1266 | for i := range r { 1267 | if r[i] != 0 { 1268 | for b := 1; b <= 6 && i+b < 256; b++ { 1269 | if r[i+b] != 0 { 1270 | if r[i]+(r[i+b]<= -15 { 1274 | r[i] -= r[i+b] << uint(b) 1275 | for k := i + b; k < 256; k++ { 1276 | if r[k] == 0 { 1277 | r[k] = 1 1278 | break 1279 | } 1280 | r[k] = 0 1281 | } 1282 | } else { 1283 | break 1284 | } 1285 | } 1286 | } 1287 | } 1288 | } 1289 | } 1290 | 1291 | // GeAdd sets r = a+b. r may overlaop with a and b. 1292 | func GeAdd(r, a, b *ExtendedGroupElement) { 1293 | var bca CachedGroupElement 1294 | b.ToCached(&bca) 1295 | var rc CompletedGroupElement 1296 | geAdd(&rc, a, &bca) 1297 | rc.ToExtended(r) 1298 | } 1299 | 1300 | func ExtendedGroupElementCopy(t, u *ExtendedGroupElement) { 1301 | FeCopy(&t.X, &u.X) 1302 | FeCopy(&t.Y, &u.Y) 1303 | FeCopy(&t.Z, &u.Z) 1304 | FeCopy(&t.T, &u.T) 1305 | } 1306 | 1307 | func ExtendedGroupElementCMove(t, u *ExtendedGroupElement, b int32) { 1308 | FeCMove(&t.X, &u.X, b) 1309 | FeCMove(&t.Y, &u.Y, b) 1310 | FeCMove(&t.Z, &u.Z, b) 1311 | FeCMove(&t.T, &u.T, b) 1312 | } 1313 | 1314 | // GeScalarMult sets r = a*A 1315 | // where a = a[0]+256*a[1]+...+256^31 a[31]. 1316 | func GeScalarMult(r *ExtendedGroupElement, a *[32]byte, A *ExtendedGroupElement) { 1317 | var p, q ExtendedGroupElement 1318 | q.Zero() 1319 | ExtendedGroupElementCopy(&p, A) 1320 | for i := uint(0); i < 256; i++ { 1321 | bit := int32(a[i>>3]>>(i&7)) & 1 1322 | var t ExtendedGroupElement 1323 | GeAdd(&t, &q, &p) 1324 | ExtendedGroupElementCMove(&q, &t, bit) 1325 | GeDouble(&p, &p) 1326 | } 1327 | ExtendedGroupElementCopy(r, &q) 1328 | } 1329 | 1330 | // GeDoubleScalarMultVartime sets r = a*A + b*B 1331 | // where a = a[0]+256*a[1]+...+256^31 a[31]. 1332 | // and b = b[0]+256*b[1]+...+256^31 b[31]. 1333 | // B is the Ed25519 base point (x,4/5) with x positive. 1334 | func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { 1335 | var aSlide, bSlide [256]int8 1336 | var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A 1337 | var t CompletedGroupElement 1338 | var u, A2 ExtendedGroupElement 1339 | var i int 1340 | 1341 | slide(&aSlide, a) 1342 | slide(&bSlide, b) 1343 | 1344 | A.ToCached(&Ai[0]) 1345 | A.Double(&t) 1346 | t.ToExtended(&A2) 1347 | 1348 | for i := 0; i < 7; i++ { 1349 | geAdd(&t, &A2, &Ai[i]) 1350 | t.ToExtended(&u) 1351 | u.ToCached(&Ai[i+1]) 1352 | } 1353 | 1354 | r.Zero() 1355 | 1356 | for i = 255; i >= 0; i-- { 1357 | if aSlide[i] != 0 || bSlide[i] != 0 { 1358 | break 1359 | } 1360 | } 1361 | 1362 | for ; i >= 0; i-- { 1363 | r.Double(&t) 1364 | 1365 | if aSlide[i] > 0 { 1366 | t.ToExtended(&u) 1367 | geAdd(&t, &u, &Ai[aSlide[i]/2]) 1368 | } else if aSlide[i] < 0 { 1369 | t.ToExtended(&u) 1370 | geSub(&t, &u, &Ai[(-aSlide[i])/2]) 1371 | } 1372 | 1373 | if bSlide[i] > 0 { 1374 | t.ToExtended(&u) 1375 | geMixedAdd(&t, &u, &bi[bSlide[i]/2]) 1376 | } else if bSlide[i] < 0 { 1377 | t.ToExtended(&u) 1378 | geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) 1379 | } 1380 | 1381 | t.ToProjective(r) 1382 | } 1383 | } 1384 | 1385 | // equal returns 1 if b == c and 0 otherwise. 1386 | func equal(b, c int32) int32 { 1387 | x := uint32(b ^ c) 1388 | x-- 1389 | return int32(x >> 31) 1390 | } 1391 | 1392 | // negative returns 1 if b < 0 and 0 otherwise. 1393 | func negative(b int32) int32 { 1394 | return (b >> 31) & 1 1395 | } 1396 | 1397 | func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { 1398 | FeCMove(&t.yPlusX, &u.yPlusX, b) 1399 | FeCMove(&t.yMinusX, &u.yMinusX, b) 1400 | FeCMove(&t.xy2d, &u.xy2d, b) 1401 | } 1402 | 1403 | func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { 1404 | var minusT PreComputedGroupElement 1405 | bNegative := negative(b) 1406 | bAbs := b - (((-bNegative) & b) << 1) 1407 | 1408 | t.Zero() 1409 | for i := int32(0); i < 8; i++ { 1410 | PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) 1411 | } 1412 | FeCopy(&minusT.yPlusX, &t.yMinusX) 1413 | FeCopy(&minusT.yMinusX, &t.yPlusX) 1414 | FeNeg(&minusT.xy2d, &t.xy2d) 1415 | PreComputedGroupElementCMove(t, &minusT, bNegative) 1416 | } 1417 | 1418 | // GeScalarMultBase computes h = a*B, where 1419 | // a = a[0]+256*a[1]+...+256^31 a[31] 1420 | // B is the Ed25519 base point (x,4/5) with x positive. 1421 | // 1422 | // Preconditions: 1423 | // a[31] <= 127 1424 | func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { 1425 | var e [64]int8 1426 | 1427 | for i, v := range a { 1428 | e[2*i] = int8(v & 15) 1429 | e[2*i+1] = int8((v >> 4) & 15) 1430 | } 1431 | 1432 | // each e[i] is between 0 and 15 and e[63] is between 0 and 7. 1433 | 1434 | carry := int8(0) 1435 | for i := 0; i < 63; i++ { 1436 | e[i] += carry 1437 | carry = (e[i] + 8) >> 4 1438 | e[i] -= carry << 4 1439 | } 1440 | e[63] += carry 1441 | // each e[i] is between -8 and 8. 1442 | 1443 | h.Zero() 1444 | var t PreComputedGroupElement 1445 | var r CompletedGroupElement 1446 | for i := int32(1); i < 64; i += 2 { 1447 | selectPoint(&t, i/2, int32(e[i])) 1448 | geMixedAdd(&r, h, &t) 1449 | r.ToExtended(h) 1450 | } 1451 | 1452 | var s ProjectiveGroupElement 1453 | 1454 | h.Double(&r) 1455 | r.ToProjective(&s) 1456 | s.Double(&r) 1457 | r.ToProjective(&s) 1458 | s.Double(&r) 1459 | r.ToProjective(&s) 1460 | s.Double(&r) 1461 | r.ToExtended(h) 1462 | 1463 | for i := int32(0); i < 64; i += 2 { 1464 | selectPoint(&t, i/2, int32(e[i])) 1465 | geMixedAdd(&r, h, &t) 1466 | r.ToExtended(h) 1467 | } 1468 | } 1469 | 1470 | // The scalars are GF(2^252 + 27742317777372353535851937790883648493). 1471 | 1472 | // Input: 1473 | // a[0]+256*a[1]+...+256^31*a[31] = a 1474 | // b[0]+256*b[1]+...+256^31*b[31] = b 1475 | // c[0]+256*c[1]+...+256^31*c[31] = c 1476 | // 1477 | // Output: 1478 | // s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l 1479 | // where l = 2^252 + 27742317777372353535851937790883648493. 1480 | func ScMulAdd(s, a, b, c *[32]byte) { 1481 | a0 := 2097151 & load3(a[:]) 1482 | a1 := 2097151 & (load4(a[2:]) >> 5) 1483 | a2 := 2097151 & (load3(a[5:]) >> 2) 1484 | a3 := 2097151 & (load4(a[7:]) >> 7) 1485 | a4 := 2097151 & (load4(a[10:]) >> 4) 1486 | a5 := 2097151 & (load3(a[13:]) >> 1) 1487 | a6 := 2097151 & (load4(a[15:]) >> 6) 1488 | a7 := 2097151 & (load3(a[18:]) >> 3) 1489 | a8 := 2097151 & load3(a[21:]) 1490 | a9 := 2097151 & (load4(a[23:]) >> 5) 1491 | a10 := 2097151 & (load3(a[26:]) >> 2) 1492 | a11 := (load4(a[28:]) >> 7) 1493 | b0 := 2097151 & load3(b[:]) 1494 | b1 := 2097151 & (load4(b[2:]) >> 5) 1495 | b2 := 2097151 & (load3(b[5:]) >> 2) 1496 | b3 := 2097151 & (load4(b[7:]) >> 7) 1497 | b4 := 2097151 & (load4(b[10:]) >> 4) 1498 | b5 := 2097151 & (load3(b[13:]) >> 1) 1499 | b6 := 2097151 & (load4(b[15:]) >> 6) 1500 | b7 := 2097151 & (load3(b[18:]) >> 3) 1501 | b8 := 2097151 & load3(b[21:]) 1502 | b9 := 2097151 & (load4(b[23:]) >> 5) 1503 | b10 := 2097151 & (load3(b[26:]) >> 2) 1504 | b11 := (load4(b[28:]) >> 7) 1505 | c0 := 2097151 & load3(c[:]) 1506 | c1 := 2097151 & (load4(c[2:]) >> 5) 1507 | c2 := 2097151 & (load3(c[5:]) >> 2) 1508 | c3 := 2097151 & (load4(c[7:]) >> 7) 1509 | c4 := 2097151 & (load4(c[10:]) >> 4) 1510 | c5 := 2097151 & (load3(c[13:]) >> 1) 1511 | c6 := 2097151 & (load4(c[15:]) >> 6) 1512 | c7 := 2097151 & (load3(c[18:]) >> 3) 1513 | c8 := 2097151 & load3(c[21:]) 1514 | c9 := 2097151 & (load4(c[23:]) >> 5) 1515 | c10 := 2097151 & (load3(c[26:]) >> 2) 1516 | c11 := (load4(c[28:]) >> 7) 1517 | var carry [23]int64 1518 | 1519 | s0 := c0 + a0*b0 1520 | s1 := c1 + a0*b1 + a1*b0 1521 | s2 := c2 + a0*b2 + a1*b1 + a2*b0 1522 | s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 1523 | s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 1524 | s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 1525 | s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 1526 | s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 1527 | s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 1528 | s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 1529 | s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 1530 | 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 1531 | s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 1532 | s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 1533 | s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 1534 | s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 1535 | s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 1536 | s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 1537 | s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 1538 | s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 1539 | s20 := a9*b11 + a10*b10 + a11*b9 1540 | s21 := a10*b11 + a11*b10 1541 | s22 := a11 * b11 1542 | s23 := int64(0) 1543 | 1544 | carry[0] = (s0 + (1 << 20)) >> 21 1545 | s1 += carry[0] 1546 | s0 -= carry[0] << 21 1547 | carry[2] = (s2 + (1 << 20)) >> 21 1548 | s3 += carry[2] 1549 | s2 -= carry[2] << 21 1550 | carry[4] = (s4 + (1 << 20)) >> 21 1551 | s5 += carry[4] 1552 | s4 -= carry[4] << 21 1553 | carry[6] = (s6 + (1 << 20)) >> 21 1554 | s7 += carry[6] 1555 | s6 -= carry[6] << 21 1556 | carry[8] = (s8 + (1 << 20)) >> 21 1557 | s9 += carry[8] 1558 | s8 -= carry[8] << 21 1559 | carry[10] = (s10 + (1 << 20)) >> 21 1560 | s11 += carry[10] 1561 | s10 -= carry[10] << 21 1562 | carry[12] = (s12 + (1 << 20)) >> 21 1563 | s13 += carry[12] 1564 | s12 -= carry[12] << 21 1565 | carry[14] = (s14 + (1 << 20)) >> 21 1566 | s15 += carry[14] 1567 | s14 -= carry[14] << 21 1568 | carry[16] = (s16 + (1 << 20)) >> 21 1569 | s17 += carry[16] 1570 | s16 -= carry[16] << 21 1571 | carry[18] = (s18 + (1 << 20)) >> 21 1572 | s19 += carry[18] 1573 | s18 -= carry[18] << 21 1574 | carry[20] = (s20 + (1 << 20)) >> 21 1575 | s21 += carry[20] 1576 | s20 -= carry[20] << 21 1577 | carry[22] = (s22 + (1 << 20)) >> 21 1578 | s23 += carry[22] 1579 | s22 -= carry[22] << 21 1580 | 1581 | carry[1] = (s1 + (1 << 20)) >> 21 1582 | s2 += carry[1] 1583 | s1 -= carry[1] << 21 1584 | carry[3] = (s3 + (1 << 20)) >> 21 1585 | s4 += carry[3] 1586 | s3 -= carry[3] << 21 1587 | carry[5] = (s5 + (1 << 20)) >> 21 1588 | s6 += carry[5] 1589 | s5 -= carry[5] << 21 1590 | carry[7] = (s7 + (1 << 20)) >> 21 1591 | s8 += carry[7] 1592 | s7 -= carry[7] << 21 1593 | carry[9] = (s9 + (1 << 20)) >> 21 1594 | s10 += carry[9] 1595 | s9 -= carry[9] << 21 1596 | carry[11] = (s11 + (1 << 20)) >> 21 1597 | s12 += carry[11] 1598 | s11 -= carry[11] << 21 1599 | carry[13] = (s13 + (1 << 20)) >> 21 1600 | s14 += carry[13] 1601 | s13 -= carry[13] << 21 1602 | carry[15] = (s15 + (1 << 20)) >> 21 1603 | s16 += carry[15] 1604 | s15 -= carry[15] << 21 1605 | carry[17] = (s17 + (1 << 20)) >> 21 1606 | s18 += carry[17] 1607 | s17 -= carry[17] << 21 1608 | carry[19] = (s19 + (1 << 20)) >> 21 1609 | s20 += carry[19] 1610 | s19 -= carry[19] << 21 1611 | carry[21] = (s21 + (1 << 20)) >> 21 1612 | s22 += carry[21] 1613 | s21 -= carry[21] << 21 1614 | 1615 | s11 += s23 * 666643 1616 | s12 += s23 * 470296 1617 | s13 += s23 * 654183 1618 | s14 -= s23 * 997805 1619 | s15 += s23 * 136657 1620 | s16 -= s23 * 683901 1621 | s23 = 0 1622 | 1623 | s10 += s22 * 666643 1624 | s11 += s22 * 470296 1625 | s12 += s22 * 654183 1626 | s13 -= s22 * 997805 1627 | s14 += s22 * 136657 1628 | s15 -= s22 * 683901 1629 | s22 = 0 1630 | 1631 | s9 += s21 * 666643 1632 | s10 += s21 * 470296 1633 | s11 += s21 * 654183 1634 | s12 -= s21 * 997805 1635 | s13 += s21 * 136657 1636 | s14 -= s21 * 683901 1637 | s21 = 0 1638 | 1639 | s8 += s20 * 666643 1640 | s9 += s20 * 470296 1641 | s10 += s20 * 654183 1642 | s11 -= s20 * 997805 1643 | s12 += s20 * 136657 1644 | s13 -= s20 * 683901 1645 | s20 = 0 1646 | 1647 | s7 += s19 * 666643 1648 | s8 += s19 * 470296 1649 | s9 += s19 * 654183 1650 | s10 -= s19 * 997805 1651 | s11 += s19 * 136657 1652 | s12 -= s19 * 683901 1653 | s19 = 0 1654 | 1655 | s6 += s18 * 666643 1656 | s7 += s18 * 470296 1657 | s8 += s18 * 654183 1658 | s9 -= s18 * 997805 1659 | s10 += s18 * 136657 1660 | s11 -= s18 * 683901 1661 | s18 = 0 1662 | 1663 | carry[6] = (s6 + (1 << 20)) >> 21 1664 | s7 += carry[6] 1665 | s6 -= carry[6] << 21 1666 | carry[8] = (s8 + (1 << 20)) >> 21 1667 | s9 += carry[8] 1668 | s8 -= carry[8] << 21 1669 | carry[10] = (s10 + (1 << 20)) >> 21 1670 | s11 += carry[10] 1671 | s10 -= carry[10] << 21 1672 | carry[12] = (s12 + (1 << 20)) >> 21 1673 | s13 += carry[12] 1674 | s12 -= carry[12] << 21 1675 | carry[14] = (s14 + (1 << 20)) >> 21 1676 | s15 += carry[14] 1677 | s14 -= carry[14] << 21 1678 | carry[16] = (s16 + (1 << 20)) >> 21 1679 | s17 += carry[16] 1680 | s16 -= carry[16] << 21 1681 | 1682 | carry[7] = (s7 + (1 << 20)) >> 21 1683 | s8 += carry[7] 1684 | s7 -= carry[7] << 21 1685 | carry[9] = (s9 + (1 << 20)) >> 21 1686 | s10 += carry[9] 1687 | s9 -= carry[9] << 21 1688 | carry[11] = (s11 + (1 << 20)) >> 21 1689 | s12 += carry[11] 1690 | s11 -= carry[11] << 21 1691 | carry[13] = (s13 + (1 << 20)) >> 21 1692 | s14 += carry[13] 1693 | s13 -= carry[13] << 21 1694 | carry[15] = (s15 + (1 << 20)) >> 21 1695 | s16 += carry[15] 1696 | s15 -= carry[15] << 21 1697 | 1698 | s5 += s17 * 666643 1699 | s6 += s17 * 470296 1700 | s7 += s17 * 654183 1701 | s8 -= s17 * 997805 1702 | s9 += s17 * 136657 1703 | s10 -= s17 * 683901 1704 | s17 = 0 1705 | 1706 | s4 += s16 * 666643 1707 | s5 += s16 * 470296 1708 | s6 += s16 * 654183 1709 | s7 -= s16 * 997805 1710 | s8 += s16 * 136657 1711 | s9 -= s16 * 683901 1712 | s16 = 0 1713 | 1714 | s3 += s15 * 666643 1715 | s4 += s15 * 470296 1716 | s5 += s15 * 654183 1717 | s6 -= s15 * 997805 1718 | s7 += s15 * 136657 1719 | s8 -= s15 * 683901 1720 | s15 = 0 1721 | 1722 | s2 += s14 * 666643 1723 | s3 += s14 * 470296 1724 | s4 += s14 * 654183 1725 | s5 -= s14 * 997805 1726 | s6 += s14 * 136657 1727 | s7 -= s14 * 683901 1728 | s14 = 0 1729 | 1730 | s1 += s13 * 666643 1731 | s2 += s13 * 470296 1732 | s3 += s13 * 654183 1733 | s4 -= s13 * 997805 1734 | s5 += s13 * 136657 1735 | s6 -= s13 * 683901 1736 | s13 = 0 1737 | 1738 | s0 += s12 * 666643 1739 | s1 += s12 * 470296 1740 | s2 += s12 * 654183 1741 | s3 -= s12 * 997805 1742 | s4 += s12 * 136657 1743 | s5 -= s12 * 683901 1744 | s12 = 0 1745 | 1746 | carry[0] = (s0 + (1 << 20)) >> 21 1747 | s1 += carry[0] 1748 | s0 -= carry[0] << 21 1749 | carry[2] = (s2 + (1 << 20)) >> 21 1750 | s3 += carry[2] 1751 | s2 -= carry[2] << 21 1752 | carry[4] = (s4 + (1 << 20)) >> 21 1753 | s5 += carry[4] 1754 | s4 -= carry[4] << 21 1755 | carry[6] = (s6 + (1 << 20)) >> 21 1756 | s7 += carry[6] 1757 | s6 -= carry[6] << 21 1758 | carry[8] = (s8 + (1 << 20)) >> 21 1759 | s9 += carry[8] 1760 | s8 -= carry[8] << 21 1761 | carry[10] = (s10 + (1 << 20)) >> 21 1762 | s11 += carry[10] 1763 | s10 -= carry[10] << 21 1764 | 1765 | carry[1] = (s1 + (1 << 20)) >> 21 1766 | s2 += carry[1] 1767 | s1 -= carry[1] << 21 1768 | carry[3] = (s3 + (1 << 20)) >> 21 1769 | s4 += carry[3] 1770 | s3 -= carry[3] << 21 1771 | carry[5] = (s5 + (1 << 20)) >> 21 1772 | s6 += carry[5] 1773 | s5 -= carry[5] << 21 1774 | carry[7] = (s7 + (1 << 20)) >> 21 1775 | s8 += carry[7] 1776 | s7 -= carry[7] << 21 1777 | carry[9] = (s9 + (1 << 20)) >> 21 1778 | s10 += carry[9] 1779 | s9 -= carry[9] << 21 1780 | carry[11] = (s11 + (1 << 20)) >> 21 1781 | s12 += carry[11] 1782 | s11 -= carry[11] << 21 1783 | 1784 | s0 += s12 * 666643 1785 | s1 += s12 * 470296 1786 | s2 += s12 * 654183 1787 | s3 -= s12 * 997805 1788 | s4 += s12 * 136657 1789 | s5 -= s12 * 683901 1790 | s12 = 0 1791 | 1792 | carry[0] = s0 >> 21 1793 | s1 += carry[0] 1794 | s0 -= carry[0] << 21 1795 | carry[1] = s1 >> 21 1796 | s2 += carry[1] 1797 | s1 -= carry[1] << 21 1798 | carry[2] = s2 >> 21 1799 | s3 += carry[2] 1800 | s2 -= carry[2] << 21 1801 | carry[3] = s3 >> 21 1802 | s4 += carry[3] 1803 | s3 -= carry[3] << 21 1804 | carry[4] = s4 >> 21 1805 | s5 += carry[4] 1806 | s4 -= carry[4] << 21 1807 | carry[5] = s5 >> 21 1808 | s6 += carry[5] 1809 | s5 -= carry[5] << 21 1810 | carry[6] = s6 >> 21 1811 | s7 += carry[6] 1812 | s6 -= carry[6] << 21 1813 | carry[7] = s7 >> 21 1814 | s8 += carry[7] 1815 | s7 -= carry[7] << 21 1816 | carry[8] = s8 >> 21 1817 | s9 += carry[8] 1818 | s8 -= carry[8] << 21 1819 | carry[9] = s9 >> 21 1820 | s10 += carry[9] 1821 | s9 -= carry[9] << 21 1822 | carry[10] = s10 >> 21 1823 | s11 += carry[10] 1824 | s10 -= carry[10] << 21 1825 | carry[11] = s11 >> 21 1826 | s12 += carry[11] 1827 | s11 -= carry[11] << 21 1828 | 1829 | s0 += s12 * 666643 1830 | s1 += s12 * 470296 1831 | s2 += s12 * 654183 1832 | s3 -= s12 * 997805 1833 | s4 += s12 * 136657 1834 | s5 -= s12 * 683901 1835 | s12 = 0 1836 | 1837 | carry[0] = s0 >> 21 1838 | s1 += carry[0] 1839 | s0 -= carry[0] << 21 1840 | carry[1] = s1 >> 21 1841 | s2 += carry[1] 1842 | s1 -= carry[1] << 21 1843 | carry[2] = s2 >> 21 1844 | s3 += carry[2] 1845 | s2 -= carry[2] << 21 1846 | carry[3] = s3 >> 21 1847 | s4 += carry[3] 1848 | s3 -= carry[3] << 21 1849 | carry[4] = s4 >> 21 1850 | s5 += carry[4] 1851 | s4 -= carry[4] << 21 1852 | carry[5] = s5 >> 21 1853 | s6 += carry[5] 1854 | s5 -= carry[5] << 21 1855 | carry[6] = s6 >> 21 1856 | s7 += carry[6] 1857 | s6 -= carry[6] << 21 1858 | carry[7] = s7 >> 21 1859 | s8 += carry[7] 1860 | s7 -= carry[7] << 21 1861 | carry[8] = s8 >> 21 1862 | s9 += carry[8] 1863 | s8 -= carry[8] << 21 1864 | carry[9] = s9 >> 21 1865 | s10 += carry[9] 1866 | s9 -= carry[9] << 21 1867 | carry[10] = s10 >> 21 1868 | s11 += carry[10] 1869 | s10 -= carry[10] << 21 1870 | 1871 | s[0] = byte(s0 >> 0) 1872 | s[1] = byte(s0 >> 8) 1873 | s[2] = byte((s0 >> 16) | (s1 << 5)) 1874 | s[3] = byte(s1 >> 3) 1875 | s[4] = byte(s1 >> 11) 1876 | s[5] = byte((s1 >> 19) | (s2 << 2)) 1877 | s[6] = byte(s2 >> 6) 1878 | s[7] = byte((s2 >> 14) | (s3 << 7)) 1879 | s[8] = byte(s3 >> 1) 1880 | s[9] = byte(s3 >> 9) 1881 | s[10] = byte((s3 >> 17) | (s4 << 4)) 1882 | s[11] = byte(s4 >> 4) 1883 | s[12] = byte(s4 >> 12) 1884 | s[13] = byte((s4 >> 20) | (s5 << 1)) 1885 | s[14] = byte(s5 >> 7) 1886 | s[15] = byte((s5 >> 15) | (s6 << 6)) 1887 | s[16] = byte(s6 >> 2) 1888 | s[17] = byte(s6 >> 10) 1889 | s[18] = byte((s6 >> 18) | (s7 << 3)) 1890 | s[19] = byte(s7 >> 5) 1891 | s[20] = byte(s7 >> 13) 1892 | s[21] = byte(s8 >> 0) 1893 | s[22] = byte(s8 >> 8) 1894 | s[23] = byte((s8 >> 16) | (s9 << 5)) 1895 | s[24] = byte(s9 >> 3) 1896 | s[25] = byte(s9 >> 11) 1897 | s[26] = byte((s9 >> 19) | (s10 << 2)) 1898 | s[27] = byte(s10 >> 6) 1899 | s[28] = byte((s10 >> 14) | (s11 << 7)) 1900 | s[29] = byte(s11 >> 1) 1901 | s[30] = byte(s11 >> 9) 1902 | s[31] = byte(s11 >> 17) 1903 | } 1904 | 1905 | // Input: 1906 | // s[0]+256*s[1]+...+256^63*s[63] = s 1907 | // s <= l 1908 | // 1909 | // Output: 1910 | // s[0]+256*s[1]+...+256^31*s[31] = l - s 1911 | // where l = 2^252 + 27742317777372353535851937790883648493. 1912 | func ScNeg(r, s *[32]byte) { 1913 | l := [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} 1914 | var carry int32 1915 | for i := 0; i < 32; i++ { 1916 | carry = carry + int32(l[i]) - int32(s[i]) 1917 | negative := carry & math.MinInt32 // extract the sign bit (min=0b100...) 1918 | negative |= negative >> 16 1919 | negative |= negative >> 8 1920 | negative |= negative >> 4 1921 | negative |= negative >> 2 1922 | negative |= negative >> 1 1923 | carry += negative & 256 // +=256 if negative, unmodified otherwise 1924 | r[i] = byte(carry) 1925 | // carry for next iteration 1926 | carry = negative & (-1) // -1 if negative, 0 otherwise 1927 | } 1928 | } 1929 | 1930 | // Input: 1931 | // s[0]+256*s[1]+...+256^63*s[63] = s 1932 | // 1933 | // Output: 1934 | // s[0]+256*s[1]+...+256^31*s[31] = s mod l 1935 | // where l = 2^252 + 27742317777372353535851937790883648493. 1936 | func ScReduce(out *[32]byte, s *[64]byte) { 1937 | s0 := 2097151 & load3(s[:]) 1938 | s1 := 2097151 & (load4(s[2:]) >> 5) 1939 | s2 := 2097151 & (load3(s[5:]) >> 2) 1940 | s3 := 2097151 & (load4(s[7:]) >> 7) 1941 | s4 := 2097151 & (load4(s[10:]) >> 4) 1942 | s5 := 2097151 & (load3(s[13:]) >> 1) 1943 | s6 := 2097151 & (load4(s[15:]) >> 6) 1944 | s7 := 2097151 & (load3(s[18:]) >> 3) 1945 | s8 := 2097151 & load3(s[21:]) 1946 | s9 := 2097151 & (load4(s[23:]) >> 5) 1947 | s10 := 2097151 & (load3(s[26:]) >> 2) 1948 | s11 := 2097151 & (load4(s[28:]) >> 7) 1949 | s12 := 2097151 & (load4(s[31:]) >> 4) 1950 | s13 := 2097151 & (load3(s[34:]) >> 1) 1951 | s14 := 2097151 & (load4(s[36:]) >> 6) 1952 | s15 := 2097151 & (load3(s[39:]) >> 3) 1953 | s16 := 2097151 & load3(s[42:]) 1954 | s17 := 2097151 & (load4(s[44:]) >> 5) 1955 | s18 := 2097151 & (load3(s[47:]) >> 2) 1956 | s19 := 2097151 & (load4(s[49:]) >> 7) 1957 | s20 := 2097151 & (load4(s[52:]) >> 4) 1958 | s21 := 2097151 & (load3(s[55:]) >> 1) 1959 | s22 := 2097151 & (load4(s[57:]) >> 6) 1960 | s23 := (load4(s[60:]) >> 3) 1961 | 1962 | s11 += s23 * 666643 1963 | s12 += s23 * 470296 1964 | s13 += s23 * 654183 1965 | s14 -= s23 * 997805 1966 | s15 += s23 * 136657 1967 | s16 -= s23 * 683901 1968 | s23 = 0 1969 | 1970 | s10 += s22 * 666643 1971 | s11 += s22 * 470296 1972 | s12 += s22 * 654183 1973 | s13 -= s22 * 997805 1974 | s14 += s22 * 136657 1975 | s15 -= s22 * 683901 1976 | s22 = 0 1977 | 1978 | s9 += s21 * 666643 1979 | s10 += s21 * 470296 1980 | s11 += s21 * 654183 1981 | s12 -= s21 * 997805 1982 | s13 += s21 * 136657 1983 | s14 -= s21 * 683901 1984 | s21 = 0 1985 | 1986 | s8 += s20 * 666643 1987 | s9 += s20 * 470296 1988 | s10 += s20 * 654183 1989 | s11 -= s20 * 997805 1990 | s12 += s20 * 136657 1991 | s13 -= s20 * 683901 1992 | s20 = 0 1993 | 1994 | s7 += s19 * 666643 1995 | s8 += s19 * 470296 1996 | s9 += s19 * 654183 1997 | s10 -= s19 * 997805 1998 | s11 += s19 * 136657 1999 | s12 -= s19 * 683901 2000 | s19 = 0 2001 | 2002 | s6 += s18 * 666643 2003 | s7 += s18 * 470296 2004 | s8 += s18 * 654183 2005 | s9 -= s18 * 997805 2006 | s10 += s18 * 136657 2007 | s11 -= s18 * 683901 2008 | s18 = 0 2009 | 2010 | var carry [17]int64 2011 | 2012 | carry[6] = (s6 + (1 << 20)) >> 21 2013 | s7 += carry[6] 2014 | s6 -= carry[6] << 21 2015 | carry[8] = (s8 + (1 << 20)) >> 21 2016 | s9 += carry[8] 2017 | s8 -= carry[8] << 21 2018 | carry[10] = (s10 + (1 << 20)) >> 21 2019 | s11 += carry[10] 2020 | s10 -= carry[10] << 21 2021 | carry[12] = (s12 + (1 << 20)) >> 21 2022 | s13 += carry[12] 2023 | s12 -= carry[12] << 21 2024 | carry[14] = (s14 + (1 << 20)) >> 21 2025 | s15 += carry[14] 2026 | s14 -= carry[14] << 21 2027 | carry[16] = (s16 + (1 << 20)) >> 21 2028 | s17 += carry[16] 2029 | s16 -= carry[16] << 21 2030 | 2031 | carry[7] = (s7 + (1 << 20)) >> 21 2032 | s8 += carry[7] 2033 | s7 -= carry[7] << 21 2034 | carry[9] = (s9 + (1 << 20)) >> 21 2035 | s10 += carry[9] 2036 | s9 -= carry[9] << 21 2037 | carry[11] = (s11 + (1 << 20)) >> 21 2038 | s12 += carry[11] 2039 | s11 -= carry[11] << 21 2040 | carry[13] = (s13 + (1 << 20)) >> 21 2041 | s14 += carry[13] 2042 | s13 -= carry[13] << 21 2043 | carry[15] = (s15 + (1 << 20)) >> 21 2044 | s16 += carry[15] 2045 | s15 -= carry[15] << 21 2046 | 2047 | s5 += s17 * 666643 2048 | s6 += s17 * 470296 2049 | s7 += s17 * 654183 2050 | s8 -= s17 * 997805 2051 | s9 += s17 * 136657 2052 | s10 -= s17 * 683901 2053 | s17 = 0 2054 | 2055 | s4 += s16 * 666643 2056 | s5 += s16 * 470296 2057 | s6 += s16 * 654183 2058 | s7 -= s16 * 997805 2059 | s8 += s16 * 136657 2060 | s9 -= s16 * 683901 2061 | s16 = 0 2062 | 2063 | s3 += s15 * 666643 2064 | s4 += s15 * 470296 2065 | s5 += s15 * 654183 2066 | s6 -= s15 * 997805 2067 | s7 += s15 * 136657 2068 | s8 -= s15 * 683901 2069 | s15 = 0 2070 | 2071 | s2 += s14 * 666643 2072 | s3 += s14 * 470296 2073 | s4 += s14 * 654183 2074 | s5 -= s14 * 997805 2075 | s6 += s14 * 136657 2076 | s7 -= s14 * 683901 2077 | s14 = 0 2078 | 2079 | s1 += s13 * 666643 2080 | s2 += s13 * 470296 2081 | s3 += s13 * 654183 2082 | s4 -= s13 * 997805 2083 | s5 += s13 * 136657 2084 | s6 -= s13 * 683901 2085 | s13 = 0 2086 | 2087 | s0 += s12 * 666643 2088 | s1 += s12 * 470296 2089 | s2 += s12 * 654183 2090 | s3 -= s12 * 997805 2091 | s4 += s12 * 136657 2092 | s5 -= s12 * 683901 2093 | s12 = 0 2094 | 2095 | carry[0] = (s0 + (1 << 20)) >> 21 2096 | s1 += carry[0] 2097 | s0 -= carry[0] << 21 2098 | carry[2] = (s2 + (1 << 20)) >> 21 2099 | s3 += carry[2] 2100 | s2 -= carry[2] << 21 2101 | carry[4] = (s4 + (1 << 20)) >> 21 2102 | s5 += carry[4] 2103 | s4 -= carry[4] << 21 2104 | carry[6] = (s6 + (1 << 20)) >> 21 2105 | s7 += carry[6] 2106 | s6 -= carry[6] << 21 2107 | carry[8] = (s8 + (1 << 20)) >> 21 2108 | s9 += carry[8] 2109 | s8 -= carry[8] << 21 2110 | carry[10] = (s10 + (1 << 20)) >> 21 2111 | s11 += carry[10] 2112 | s10 -= carry[10] << 21 2113 | 2114 | carry[1] = (s1 + (1 << 20)) >> 21 2115 | s2 += carry[1] 2116 | s1 -= carry[1] << 21 2117 | carry[3] = (s3 + (1 << 20)) >> 21 2118 | s4 += carry[3] 2119 | s3 -= carry[3] << 21 2120 | carry[5] = (s5 + (1 << 20)) >> 21 2121 | s6 += carry[5] 2122 | s5 -= carry[5] << 21 2123 | carry[7] = (s7 + (1 << 20)) >> 21 2124 | s8 += carry[7] 2125 | s7 -= carry[7] << 21 2126 | carry[9] = (s9 + (1 << 20)) >> 21 2127 | s10 += carry[9] 2128 | s9 -= carry[9] << 21 2129 | carry[11] = (s11 + (1 << 20)) >> 21 2130 | s12 += carry[11] 2131 | s11 -= carry[11] << 21 2132 | 2133 | s0 += s12 * 666643 2134 | s1 += s12 * 470296 2135 | s2 += s12 * 654183 2136 | s3 -= s12 * 997805 2137 | s4 += s12 * 136657 2138 | s5 -= s12 * 683901 2139 | s12 = 0 2140 | 2141 | carry[0] = s0 >> 21 2142 | s1 += carry[0] 2143 | s0 -= carry[0] << 21 2144 | carry[1] = s1 >> 21 2145 | s2 += carry[1] 2146 | s1 -= carry[1] << 21 2147 | carry[2] = s2 >> 21 2148 | s3 += carry[2] 2149 | s2 -= carry[2] << 21 2150 | carry[3] = s3 >> 21 2151 | s4 += carry[3] 2152 | s3 -= carry[3] << 21 2153 | carry[4] = s4 >> 21 2154 | s5 += carry[4] 2155 | s4 -= carry[4] << 21 2156 | carry[5] = s5 >> 21 2157 | s6 += carry[5] 2158 | s5 -= carry[5] << 21 2159 | carry[6] = s6 >> 21 2160 | s7 += carry[6] 2161 | s6 -= carry[6] << 21 2162 | carry[7] = s7 >> 21 2163 | s8 += carry[7] 2164 | s7 -= carry[7] << 21 2165 | carry[8] = s8 >> 21 2166 | s9 += carry[8] 2167 | s8 -= carry[8] << 21 2168 | carry[9] = s9 >> 21 2169 | s10 += carry[9] 2170 | s9 -= carry[9] << 21 2171 | carry[10] = s10 >> 21 2172 | s11 += carry[10] 2173 | s10 -= carry[10] << 21 2174 | carry[11] = s11 >> 21 2175 | s12 += carry[11] 2176 | s11 -= carry[11] << 21 2177 | 2178 | s0 += s12 * 666643 2179 | s1 += s12 * 470296 2180 | s2 += s12 * 654183 2181 | s3 -= s12 * 997805 2182 | s4 += s12 * 136657 2183 | s5 -= s12 * 683901 2184 | s12 = 0 2185 | 2186 | carry[0] = s0 >> 21 2187 | s1 += carry[0] 2188 | s0 -= carry[0] << 21 2189 | carry[1] = s1 >> 21 2190 | s2 += carry[1] 2191 | s1 -= carry[1] << 21 2192 | carry[2] = s2 >> 21 2193 | s3 += carry[2] 2194 | s2 -= carry[2] << 21 2195 | carry[3] = s3 >> 21 2196 | s4 += carry[3] 2197 | s3 -= carry[3] << 21 2198 | carry[4] = s4 >> 21 2199 | s5 += carry[4] 2200 | s4 -= carry[4] << 21 2201 | carry[5] = s5 >> 21 2202 | s6 += carry[5] 2203 | s5 -= carry[5] << 21 2204 | carry[6] = s6 >> 21 2205 | s7 += carry[6] 2206 | s6 -= carry[6] << 21 2207 | carry[7] = s7 >> 21 2208 | s8 += carry[7] 2209 | s7 -= carry[7] << 21 2210 | carry[8] = s8 >> 21 2211 | s9 += carry[8] 2212 | s8 -= carry[8] << 21 2213 | carry[9] = s9 >> 21 2214 | s10 += carry[9] 2215 | s9 -= carry[9] << 21 2216 | carry[10] = s10 >> 21 2217 | s11 += carry[10] 2218 | s10 -= carry[10] << 21 2219 | 2220 | out[0] = byte(s0 >> 0) 2221 | out[1] = byte(s0 >> 8) 2222 | out[2] = byte((s0 >> 16) | (s1 << 5)) 2223 | out[3] = byte(s1 >> 3) 2224 | out[4] = byte(s1 >> 11) 2225 | out[5] = byte((s1 >> 19) | (s2 << 2)) 2226 | out[6] = byte(s2 >> 6) 2227 | out[7] = byte((s2 >> 14) | (s3 << 7)) 2228 | out[8] = byte(s3 >> 1) 2229 | out[9] = byte(s3 >> 9) 2230 | out[10] = byte((s3 >> 17) | (s4 << 4)) 2231 | out[11] = byte(s4 >> 4) 2232 | out[12] = byte(s4 >> 12) 2233 | out[13] = byte((s4 >> 20) | (s5 << 1)) 2234 | out[14] = byte(s5 >> 7) 2235 | out[15] = byte((s5 >> 15) | (s6 << 6)) 2236 | out[16] = byte(s6 >> 2) 2237 | out[17] = byte(s6 >> 10) 2238 | out[18] = byte((s6 >> 18) | (s7 << 3)) 2239 | out[19] = byte(s7 >> 5) 2240 | out[20] = byte(s7 >> 13) 2241 | out[21] = byte(s8 >> 0) 2242 | out[22] = byte(s8 >> 8) 2243 | out[23] = byte((s8 >> 16) | (s9 << 5)) 2244 | out[24] = byte(s9 >> 3) 2245 | out[25] = byte(s9 >> 11) 2246 | out[26] = byte((s9 >> 19) | (s10 << 2)) 2247 | out[27] = byte(s10 >> 6) 2248 | out[28] = byte((s10 >> 14) | (s11 << 7)) 2249 | out[29] = byte(s11 >> 1) 2250 | out[30] = byte(s11 >> 9) 2251 | out[31] = byte(s11 >> 17) 2252 | } 2253 | -------------------------------------------------------------------------------- /crypto/internal/ed25519/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 extendedBaseEl = ExtendedGroupElement{ 27 | FieldElement{25485296, 5318399, 8791791, -8299916, -14349720, 6939349, -3324311, -7717049, 7287234, -6577708}, 28 | FieldElement{-758052, -1832720, 13046421, -4857925, 6576754, 14371947, -13139572, 6845540, -2198883, -4003719}, 29 | FieldElement{-947565, 6097708, -469190, 10704810, -8556274, -15589498, -16424464, -16608899, 14028613, -5004649}, 30 | FieldElement{6966464, -2456167, 7033433, 6781840, 28785542, 12262365, -2659449, 13959020, -21013759, -5262166}, 31 | } 32 | 33 | // BaseBytes can be used to hash the base point if necessary 34 | var BaseBytes [32]byte 35 | 36 | func init() { 37 | extendedBaseEl.ToBytes(&BaseBytes) 38 | } 39 | 40 | var bi = [8]PreComputedGroupElement{ 41 | { 42 | FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, 43 | FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, 44 | FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, 45 | }, 46 | { 47 | FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, 48 | FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, 49 | FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, 50 | }, 51 | { 52 | FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, 53 | FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, 54 | FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, 55 | }, 56 | { 57 | FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, 58 | FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, 59 | FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, 60 | }, 61 | { 62 | FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877}, 63 | FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951}, 64 | FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784}, 65 | }, 66 | { 67 | FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436}, 68 | FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918}, 69 | FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877}, 70 | }, 71 | { 72 | FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800}, 73 | FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305}, 74 | FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300}, 75 | }, 76 | { 77 | FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876}, 78 | FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619}, 79 | FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683}, 80 | }, 81 | } 82 | 83 | var base = [32][8]PreComputedGroupElement{ 84 | { 85 | { 86 | FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, 87 | FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, 88 | FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, 89 | }, 90 | { 91 | FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303}, 92 | FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081}, 93 | FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697}, 94 | }, 95 | { 96 | FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, 97 | FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, 98 | FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, 99 | }, 100 | { 101 | FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540}, 102 | FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397}, 103 | FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325}, 104 | }, 105 | { 106 | FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, 107 | FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, 108 | FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, 109 | }, 110 | { 111 | FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777}, 112 | FieldElement{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737}, 113 | FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652}, 114 | }, 115 | { 116 | FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, 117 | FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, 118 | FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, 119 | }, 120 | { 121 | FieldElement{14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726}, 122 | FieldElement{-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955}, 123 | FieldElement{27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425}, 124 | }, 125 | }, 126 | { 127 | { 128 | FieldElement{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171}, 129 | FieldElement{27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510}, 130 | FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660}, 131 | }, 132 | { 133 | FieldElement{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639}, 134 | FieldElement{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963}, 135 | FieldElement{5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950}, 136 | }, 137 | { 138 | FieldElement{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568}, 139 | FieldElement{12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335}, 140 | FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628}, 141 | }, 142 | { 143 | FieldElement{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007}, 144 | FieldElement{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772}, 145 | FieldElement{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653}, 146 | }, 147 | { 148 | FieldElement{2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, -7236665, 24316168, -5253567}, 149 | FieldElement{13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 33040651, -13424532, -20729456, 8321686}, 150 | FieldElement{21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 23845965, -11874838, -9984458, 608372}, 151 | }, 152 | { 153 | FieldElement{-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 1123968, -6780577, 27229399, 23887}, 154 | FieldElement{-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 12797024, -6440308, -1633405, 16678954}, 155 | FieldElement{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, 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