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update btcec to v2 and use the latest btcutil (#9250)
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* secp256k1: upgrade to latest btcec v2 and btcutil

* Update CHANGELOG_PENDING.md

Co-authored-by: Thane Thomson <connect@thanethomson.com>

* fix rebase

Co-authored-by: Thane Thomson <connect@thanethomson.com>
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@wcsiu @thanethomson
wcsiu and thanethomson committed Sep 9, 2022
1 parent 5013bc3 commit 0f45086
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Showing 6 changed files with 81 additions and 60 deletions.
1 change: 1 addition & 0 deletions CHANGELOG_PENDING.md
View file
Expand Up @@ -41,6 +41,7 @@ Friendly reminder, we have a [bug bounty program](https://hackerone.com/tendermi
- [abci] \#9301 New ABCI methods `PrepareProposal` and `ProcessProposal` which give the app control over transactions proposed and allows for verification of proposed blocks.

### IMPROVEMENTS
- [crypto] \#9250 Update to use btcec v2 and the latest btcutil. (@wcsiu)

- [proto] \#9356 Migrate from `gogo/protobuf` to `cosmos/gogoproto` (@julienrbrt)
- [rpc] \#9276 Added `header` and `header_by_hash` queries to the RPC client (@samricotta)
Expand Down
52 changes: 21 additions & 31 deletions crypto/secp256k1/secp256k1.go
View file
Expand Up @@ -8,7 +8,8 @@ import (
"io"
"math/big"

secp256k1 "github.com/btcsuite/btcd/btcec"
secp256k1 "github.com/btcsuite/btcd/btcec/v2"
"github.com/btcsuite/btcd/btcec/v2/ecdsa"
"golang.org/x/crypto/ripemd160" //nolint: staticcheck // necessary for Bitcoin address format

"github.com/tendermint/tendermint/crypto"
Expand Down Expand Up @@ -42,7 +43,7 @@ func (privKey PrivKey) Bytes() []byte {
// PubKey performs the point-scalar multiplication from the privKey on the
// generator point to get the pubkey.
func (privKey PrivKey) PubKey() crypto.PubKey {
_, pubkeyObject := secp256k1.PrivKeyFromBytes(secp256k1.S256(), privKey)
_, pubkeyObject := secp256k1.PrivKeyFromBytes(privKey)

pk := pubkeyObject.SerializeCompressed()

Expand Down Expand Up @@ -122,24 +123,18 @@ func GenPrivKeySecp256k1(secret []byte) PrivKey {
return PrivKey(privKey32)
}

// used to reject malleable signatures
// see:
// - https://github.com/ethereum/go-ethereum/blob/f9401ae011ddf7f8d2d95020b7446c17f8d98dc1/crypto/signature_nocgo.go#L90-L93
// - https://github.com/ethereum/go-ethereum/blob/f9401ae011ddf7f8d2d95020b7446c17f8d98dc1/crypto/crypto.go#L39
var secp256k1halfN = new(big.Int).Rsh(secp256k1.S256().N, 1)

// Sign creates an ECDSA signature on curve Secp256k1, using SHA256 on the msg.
// The returned signature will be of the form R || S (in lower-S form).
func (privKey PrivKey) Sign(msg []byte) ([]byte, error) {
priv, _ := secp256k1.PrivKeyFromBytes(secp256k1.S256(), privKey)
priv, _ := secp256k1.PrivKeyFromBytes(privKey)

sig, err := priv.Sign(crypto.Sha256(msg))
sig, err := ecdsa.SignCompact(priv, crypto.Sha256(msg), false)
if err != nil {
return nil, err
}

sigBytes := serializeSig(sig)
return sigBytes, nil
// remove the first byte which is compactSigRecoveryCode
return sig[1:], nil
}

//-------------------------------------
Expand Down Expand Up @@ -199,7 +194,7 @@ func (pubKey PubKey) VerifySignature(msg []byte, sigStr []byte) bool {
return false
}

pub, err := secp256k1.ParsePubKey(pubKey, secp256k1.S256())
pub, err := secp256k1.ParsePubKey(pubKey)
if err != nil {
return false
}
Expand All @@ -208,7 +203,13 @@ func (pubKey PubKey) VerifySignature(msg []byte, sigStr []byte) bool {
signature := signatureFromBytes(sigStr)
// Reject malleable signatures. libsecp256k1 does this check but btcec doesn't.
// see: https://github.com/ethereum/go-ethereum/blob/f9401ae011ddf7f8d2d95020b7446c17f8d98dc1/crypto/signature_nocgo.go#L90-L93
if signature.S.Cmp(secp256k1halfN) > 0 {
// Serialize() would negate S value if it is over half order.
// Hence, if the signature is different after Serialize() if should be rejected.
var modifiedSignature, parseErr = ecdsa.ParseDERSignature(signature.Serialize())
if parseErr != nil {
return false
}
if !signature.IsEqual(modifiedSignature) {
return false
}

Expand All @@ -217,21 +218,10 @@ func (pubKey PubKey) VerifySignature(msg []byte, sigStr []byte) bool {

// Read Signature struct from R || S. Caller needs to ensure
// that len(sigStr) == 64.
func signatureFromBytes(sigStr []byte) *secp256k1.Signature {
return &secp256k1.Signature{
R: new(big.Int).SetBytes(sigStr[:32]),
S: new(big.Int).SetBytes(sigStr[32:64]),
}
}

// Serialize signature to R || S.
// R, S are padded to 32 bytes respectively.
func serializeSig(sig *secp256k1.Signature) []byte {
rBytes := sig.R.Bytes()
sBytes := sig.S.Bytes()
sigBytes := make([]byte, 64)
// 0 pad the byte arrays from the left if they aren't big enough.
copy(sigBytes[32-len(rBytes):32], rBytes)
copy(sigBytes[64-len(sBytes):64], sBytes)
return sigBytes
func signatureFromBytes(sigStr []byte) *ecdsa.Signature {
var r secp256k1.ModNScalar
r.SetByteSlice(sigStr[:32])
var s secp256k1.ModNScalar
s.SetByteSlice(sigStr[32:64])
return ecdsa.NewSignature(&r, &s)
}
24 changes: 17 additions & 7 deletions crypto/secp256k1/secp256k1_internal_test.go
View file
Expand Up @@ -7,7 +7,7 @@ import (

"github.com/stretchr/testify/require"

secp256k1 "github.com/btcsuite/btcd/btcec"
secp256k1 "github.com/btcsuite/btcd/btcec/v2"
)

func Test_genPrivKey(t *testing.T) {
Expand Down Expand Up @@ -54,20 +54,30 @@ func TestSignatureVerificationAndRejectUpperS(t *testing.T) {
priv := GenPrivKey()
sigStr, err := priv.Sign(msg)
require.NoError(t, err)
sig := signatureFromBytes(sigStr)
require.False(t, sig.S.Cmp(secp256k1halfN) > 0)
var r secp256k1.ModNScalar
r.SetByteSlice(sigStr[:32])
var s secp256k1.ModNScalar
s.SetByteSlice(sigStr[32:64])
require.False(t, s.IsOverHalfOrder())

pub := priv.PubKey()
require.True(t, pub.VerifySignature(msg, sigStr))

// malleate:
sig.S.Sub(secp256k1.S256().CurveParams.N, sig.S)
require.True(t, sig.S.Cmp(secp256k1halfN) > 0)
malSigStr := serializeSig(sig)
var S256 secp256k1.ModNScalar
S256.SetByteSlice(secp256k1.S256().N.Bytes())
s.Negate().Add(&S256)
require.True(t, s.IsOverHalfOrder())

rBytes := r.Bytes()
sBytes := s.Bytes()
malSigStr := make([]byte, 64)
copy(malSigStr[32-len(rBytes):32], rBytes[:])
copy(malSigStr[64-len(sBytes):64], sBytes[:])

require.False(t, pub.VerifySignature(msg, malSigStr),
"VerifyBytes incorrect with malleated & invalid S. sig=%v, key=%v",
sig,
malSigStr,
priv,
)
}
Expand Down
6 changes: 3 additions & 3 deletions crypto/secp256k1/secp256k1_test.go
View file
Expand Up @@ -5,14 +5,14 @@ import (
"math/big"
"testing"

"github.com/btcsuite/btcutil/base58"
"github.com/btcsuite/btcd/btcutil/base58"
"github.com/stretchr/testify/assert"
"github.com/stretchr/testify/require"

"github.com/tendermint/tendermint/crypto"
"github.com/tendermint/tendermint/crypto/secp256k1"

underlyingSecp256k1 "github.com/btcsuite/btcd/btcec"
underlyingSecp256k1 "github.com/btcsuite/btcd/btcec/v2"
)

type keyData struct {
Expand Down Expand Up @@ -75,7 +75,7 @@ func TestSecp256k1LoadPrivkeyAndSerializeIsIdentity(t *testing.T) {

// This function creates a private and public key in the underlying libraries format.
// The private key is basically calling new(big.Int).SetBytes(pk), which removes leading zero bytes
priv, _ := underlyingSecp256k1.PrivKeyFromBytes(underlyingSecp256k1.S256(), privKeyBytes[:])
priv, _ := underlyingSecp256k1.PrivKeyFromBytes(privKeyBytes[:])
// this takes the bytes returned by `(big int).Bytes()`, and if the length is less than 32 bytes,
// pads the bytes from the left with zero bytes. Therefore these two functions composed
// result in the identity function on privKeyBytes, hence the following equality check
Expand Down
28 changes: 16 additions & 12 deletions go.mod
View file
Expand Up @@ -6,12 +6,7 @@ require (
github.com/BurntSushi/toml v1.2.0
github.com/ChainSafe/go-schnorrkel v0.0.0-20200405005733-88cbf1b4c40d
github.com/adlio/schema v1.3.3
github.com/btcsuite/btcd v0.22.1
github.com/btcsuite/btcutil v1.0.3-0.20201208143702-a53e38424cce
github.com/bufbuild/buf v1.7.0
github.com/cenkalti/backoff v2.2.1+incompatible // indirect
github.com/cosmos/gogoproto v1.4.1
github.com/creachadair/taskgroup v0.3.2
github.com/fortytw2/leaktest v1.3.0
github.com/go-kit/kit v0.12.0
github.com/go-kit/log v0.2.1
Expand All @@ -38,18 +33,26 @@ require (
github.com/spf13/cobra v1.5.0
github.com/spf13/viper v1.12.0
github.com/stretchr/testify v1.8.0
github.com/syndtr/goleveldb v1.0.1-0.20200815110645-5c35d600f0ca
github.com/tendermint/tm-db v0.6.6
golang.org/x/crypto v0.0.0-20220722155217-630584e8d5aa
golang.org/x/net v0.0.0-20220726230323-06994584191e
google.golang.org/grpc v1.49.0
)

require (
github.com/bufbuild/buf v1.7.0
github.com/creachadair/taskgroup v0.3.2
github.com/syndtr/goleveldb v1.0.1-0.20210819022825-2ae1ddf74ef7
)

require (
github.com/btcsuite/btcd/btcec/v2 v2.2.1
github.com/btcsuite/btcd/btcutil v1.1.2
github.com/cosmos/gogoproto v1.4.1
github.com/gofrs/uuid v4.2.0+incompatible
github.com/google/uuid v1.3.0
github.com/tendermint/tm-db v0.6.6
github.com/vektra/mockery/v2 v2.14.0
golang.org/x/crypto v0.0.0-20220722155217-630584e8d5aa
golang.org/x/net v0.0.0-20220726230323-06994584191e
gonum.org/v1/gonum v0.12.0
google.golang.org/grpc v1.49.0
gonum.org/v1/gonum v0.8.2
google.golang.org/protobuf v1.28.1
)

Expand Down Expand Up @@ -89,6 +92,7 @@ require (
github.com/curioswitch/go-reassign v0.1.2 // indirect
github.com/daixiang0/gci v0.6.3 // indirect
github.com/davecgh/go-spew v1.1.1 // indirect
github.com/decred/dcrd/dcrec/secp256k1/v4 v4.0.1 // indirect
github.com/denis-tingaikin/go-header v0.4.3 // indirect
github.com/dgraph-io/badger/v2 v2.2007.2 // indirect
github.com/dgraph-io/ristretto v0.0.3-0.20200630154024-f66de99634de // indirect
Expand Down Expand Up @@ -121,7 +125,7 @@ require (
github.com/gofrs/flock v0.8.1 // indirect
github.com/gogo/protobuf v1.3.2 // indirect
github.com/golang/groupcache v0.0.0-20210331224755-41bb18bfe9da // indirect
github.com/golang/snappy v0.0.3 // indirect
github.com/golang/snappy v0.0.4 // indirect
github.com/golangci/check v0.0.0-20180506172741-cfe4005ccda2 // indirect
github.com/golangci/dupl v0.0.0-20180902072040-3e9179ac440a // indirect
github.com/golangci/go-misc v0.0.0-20220329215616-d24fe342adfe // indirect
Expand Down
30 changes: 23 additions & 7 deletions go.sum
View file
Expand Up @@ -145,13 +145,23 @@ github.com/breml/errchkjson v0.3.0/go.mod h1:9Cogkyv9gcT8HREpzi3TiqBxCqDzo8awa92
github.com/btcsuite/btcd v0.20.1-beta/go.mod h1:wVuoA8VJLEcwgqHBwHmzLRazpKxTv13Px/pDuV7OomQ=
github.com/btcsuite/btcd v0.21.0-beta/go.mod h1:ZSWyehm27aAuS9bvkATT+Xte3hjHZ+MRgMY/8NJ7K94=
github.com/btcsuite/btcd v0.22.0-beta/go.mod h1:9n5ntfhhHQBIhUvlhDvD3Qg6fRUj4jkN0VB8L8svzOA=
github.com/btcsuite/btcd v0.22.1 h1:CnwP9LM/M9xuRrGSCGeMVs9iv09uMqwsVX7EeIpgV2c=
github.com/btcsuite/btcd v0.22.1/go.mod h1:wqgTSL29+50LRkmOVknEdmt8ZojIzhuWvgu/iptuN7Y=
github.com/btcsuite/btcd v0.22.0-beta.0.20220111032746-97732e52810c/go.mod h1:tjmYdS6MLJ5/s0Fj4DbLgSbDHbEqLJrtnHecBFkdz5M=
github.com/btcsuite/btcd v0.23.0 h1:V2/ZgjfDFIygAX3ZapeigkVBoVUtOJKSwrhZdlpSvaA=
github.com/btcsuite/btcd v0.23.0/go.mod h1:0QJIIN1wwIXF/3G/m87gIwGniDMDQqjVn4SZgnFpsYY=
github.com/btcsuite/btcd/btcec/v2 v2.1.0/go.mod h1:2VzYrv4Gm4apmbVVsSq5bqf1Ec8v56E48Vt0Y/umPgA=
github.com/btcsuite/btcd/btcec/v2 v2.1.3/go.mod h1:ctjw4H1kknNJmRN4iP1R7bTQ+v3GJkZBd6mui8ZsAZE=
github.com/btcsuite/btcd/btcec/v2 v2.2.1 h1:xP60mv8fvp+0khmrN0zTdPC3cNm24rfeE6lh2R/Yv3E=
github.com/btcsuite/btcd/btcec/v2 v2.2.1/go.mod h1:9/CSmJxmuvqzX9Wh2fXMWToLOHhPd11lSPuIupwTkI8=
github.com/btcsuite/btcd/btcutil v1.0.0/go.mod h1:Uoxwv0pqYWhD//tfTiipkxNfdhG9UrLwaeswfjfdF0A=
github.com/btcsuite/btcd/btcutil v1.1.0/go.mod h1:5OapHB7A2hBBWLm48mmw4MOHNJCcUBTwmWH/0Jn8VHE=
github.com/btcsuite/btcd/btcutil v1.1.2 h1:XLMbX8JQEiwMcYft2EGi8zPUkoa0abKIU6/BJSRsjzQ=
github.com/btcsuite/btcd/btcutil v1.1.2/go.mod h1:UR7dsSJzJUfMmFiiLlIrMq1lS9jh9EdCV7FStZSnpi0=
github.com/btcsuite/btcd/chaincfg/chainhash v1.0.0/go.mod h1:7SFka0XMvUgj3hfZtydOrQY2mwhPclbT2snogU7SQQc=
github.com/btcsuite/btcd/chaincfg/chainhash v1.0.1 h1:q0rUy8C/TYNBQS1+CGKw68tLOFYSNEs0TFnxxnS9+4U=
github.com/btcsuite/btcd/chaincfg/chainhash v1.0.1/go.mod h1:7SFka0XMvUgj3hfZtydOrQY2mwhPclbT2snogU7SQQc=
github.com/btcsuite/btclog v0.0.0-20170628155309-84c8d2346e9f/go.mod h1:TdznJufoqS23FtqVCzL0ZqgP5MqXbb4fg/WgDys70nA=
github.com/btcsuite/btcutil v0.0.0-20190425235716-9e5f4b9a998d/go.mod h1:+5NJ2+qvTyV9exUAL/rxXi3DcLg2Ts+ymUAY5y4NvMg=
github.com/btcsuite/btcutil v1.0.2/go.mod h1:j9HUFwoQRsZL3V4n+qG+CUnEGHOarIxfC3Le2Yhbcts=
github.com/btcsuite/btcutil v1.0.3-0.20201208143702-a53e38424cce h1:YtWJF7RHm2pYCvA5t0RPmAaLUhREsKuKd+SLhxFbFeQ=
github.com/btcsuite/btcutil v1.0.3-0.20201208143702-a53e38424cce/go.mod h1:0DVlHczLPewLcPGEIeUEzfOJhqGPQ0mJJRDBtD307+o=
github.com/btcsuite/go-socks v0.0.0-20170105172521-4720035b7bfd/go.mod h1:HHNXQzUsZCxOoE+CPiyCTO6x34Zs86zZUiwtpXoGdtg=
github.com/btcsuite/goleveldb v0.0.0-20160330041536-7834afc9e8cd/go.mod h1:F+uVaaLLH7j4eDXPRvw78tMflu7Ie2bzYOH4Y8rRKBY=
Expand Down Expand Up @@ -252,6 +262,10 @@ github.com/davecgh/go-spew v0.0.0-20171005155431-ecdeabc65495/go.mod h1:J7Y8YcW2
github.com/davecgh/go-spew v1.1.0/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c=
github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
github.com/decred/dcrd/crypto/blake256 v1.0.0 h1:/8DMNYp9SGi5f0w7uCm6d6M4OU2rGFK09Y2A4Xv7EE0=
github.com/decred/dcrd/crypto/blake256 v1.0.0/go.mod h1:sQl2p6Y26YV+ZOcSTP6thNdn47hh8kt6rqSlvmrXFAc=
github.com/decred/dcrd/dcrec/secp256k1/v4 v4.0.1 h1:YLtO71vCjJRCBcrPMtQ9nqBsqpA1m5sE92cU+pd5Mcc=
github.com/decred/dcrd/dcrec/secp256k1/v4 v4.0.1/go.mod h1:hyedUtir6IdtD/7lIxGeCxkaw7y45JueMRL4DIyJDKs=
github.com/decred/dcrd/lru v1.0.0/go.mod h1:mxKOwFd7lFjN2GZYsiz/ecgqR6kkYAl+0pz0tEMk218=
github.com/denis-tingaikin/go-header v0.4.3 h1:tEaZKAlqql6SKCY++utLmkPLd6K8IBM20Ha7UVm+mtU=
github.com/denis-tingaikin/go-header v0.4.3/go.mod h1:0wOCWuN71D5qIgE2nz9KrKmuYBAC2Mra5RassOIQ2/c=
Expand Down Expand Up @@ -430,8 +444,9 @@ github.com/golang/protobuf v1.5.2 h1:ROPKBNFfQgOUMifHyP+KYbvpjbdoFNs+aK7DXlji0Tw
github.com/golang/protobuf v1.5.2/go.mod h1:XVQd3VNwM+JqD3oG2Ue2ip4fOMUkwXdXDdiuN0vRsmY=
github.com/golang/snappy v0.0.0-20180518054509-2e65f85255db/go.mod h1:/XxbfmMg8lxefKM7IXC3fBNl/7bRcc72aCRzEWrmP2Q=
github.com/golang/snappy v0.0.1/go.mod h1:/XxbfmMg8lxefKM7IXC3fBNl/7bRcc72aCRzEWrmP2Q=
github.com/golang/snappy v0.0.3 h1:fHPg5GQYlCeLIPB9BZqMVR5nR9A+IM5zcgeTdjMYmLA=
github.com/golang/snappy v0.0.3/go.mod h1:/XxbfmMg8lxefKM7IXC3fBNl/7bRcc72aCRzEWrmP2Q=
github.com/golang/snappy v0.0.4 h1:yAGX7huGHXlcLOEtBnF4w7FQwA26wojNCwOYAEhLjQM=
github.com/golang/snappy v0.0.4/go.mod h1:/XxbfmMg8lxefKM7IXC3fBNl/7bRcc72aCRzEWrmP2Q=
github.com/golangci/check v0.0.0-20180506172741-cfe4005ccda2 h1:23T5iq8rbUYlhpt5DB4XJkc6BU31uODLD1o1gKvZmD0=
github.com/golangci/check v0.0.0-20180506172741-cfe4005ccda2/go.mod h1:k9Qvh+8juN+UKMCS/3jFtGICgW8O96FVaZsaxdzDkR4=
github.com/golangci/dupl v0.0.0-20180902072040-3e9179ac440a h1:w8hkcTqaFpzKqonE9uMCefW1WDie15eSP/4MssdenaM=
Expand Down Expand Up @@ -1075,8 +1090,9 @@ github.com/subosito/gotenv v1.4.0/go.mod h1:mZd6rFysKEcUhUHXJk0C/08wAgyDBFuwEYL7
github.com/sylvia7788/contextcheck v1.0.6 h1:o2EZgVPyMKE/Mtoqym61DInKEjwEbsmyoxg3VrmjNO4=
github.com/sylvia7788/contextcheck v1.0.6/go.mod h1:9XDxwvxyuKD+8N+a7Gs7bfWLityh5t70g/GjdEt2N2M=
github.com/syndtr/gocapability v0.0.0-20200815063812-42c35b437635/go.mod h1:hkRG7XYTFWNJGYcbNJQlaLq0fg1yr4J4t/NcTQtrfww=
github.com/syndtr/goleveldb v1.0.1-0.20200815110645-5c35d600f0ca h1:Ld/zXl5t4+D69SiV4JoN7kkfvJdOWlPpfxrzxpLMoUk=
github.com/syndtr/goleveldb v1.0.1-0.20200815110645-5c35d600f0ca/go.mod h1:u2MKkTVTVJWe5D1rCvame8WqhBd88EuIwODJZ1VHCPM=
github.com/syndtr/goleveldb v1.0.1-0.20210819022825-2ae1ddf74ef7 h1:epCh84lMvA70Z7CTTCmYQn2CKbY8j86K7/FAIr141uY=
github.com/syndtr/goleveldb v1.0.1-0.20210819022825-2ae1ddf74ef7/go.mod h1:q4W45IWZaF22tdD+VEXcAWRA037jwmWEB5VWYORlTpc=
github.com/tdakkota/asciicheck v0.1.1 h1:PKzG7JUTUmVspQTDqtkX9eSiLGossXTybutHwTXuO0A=
github.com/tdakkota/asciicheck v0.1.1/go.mod h1:yHp0ai0Z9gUljN3o0xMhYJnH/IcvkdTBOX2fmJ93JEM=
github.com/tecbot/gorocksdb v0.0.0-20191217155057-f0fad39f321c h1:g+WoO5jjkqGAzHWCjJB1zZfXPIAaDpzXIEJ0eS6B5Ok=
Expand Down Expand Up @@ -1566,9 +1582,9 @@ golang.org/x/xerrors v0.0.0-20191011141410-1b5146add898/go.mod h1:I/5z698sn9Ka8T
golang.org/x/xerrors v0.0.0-20191204190536-9bdfabe68543/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0=
golang.org/x/xerrors v0.0.0-20200804184101-5ec99f83aff1/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0=
gonum.org/v1/gonum v0.0.0-20180816165407-929014505bf4/go.mod h1:Y+Yx5eoAFn32cQvJDxZx5Dpnq+c3wtXuadVZAcxbbBo=
gonum.org/v1/gonum v0.8.2 h1:CCXrcPKiGGotvnN6jfUsKk4rRqm7q09/YbKb5xCEvtM=
gonum.org/v1/gonum v0.8.2/go.mod h1:oe/vMfY3deqTw+1EZJhuvEW2iwGF1bW9wwu7XCu0+v0=
gonum.org/v1/gonum v0.12.0 h1:xKuo6hzt+gMav00meVPUlXwSdoEJP46BR+wdxQEFK2o=
gonum.org/v1/gonum v0.12.0/go.mod h1:73TDxJfAAHeA8Mk9mf8NlIppyhQNo5GLTcYeqgo2lvY=
gonum.org/v1/netlib v0.0.0-20190313105609-8cb42192e0e0 h1:OE9mWmgKkjJyEmDAAtGMPjXu+YNeGvK9VTSHY6+Qihc=
gonum.org/v1/netlib v0.0.0-20190313105609-8cb42192e0e0/go.mod h1:wa6Ws7BG/ESfp6dHfk7C6KdzKA7wR7u/rKwOGE66zvw=
gonum.org/v1/plot v0.0.0-20190515093506-e2840ee46a6b/go.mod h1:Wt8AAjI+ypCyYX3nZBvf6cAIx93T+c/OS2HFAYskSZc=
google.golang.org/api v0.3.1/go.mod h1:6wY9I6uQWHQ8EM57III9mq/AjF+i8G65rmVagqKMtkk=
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