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kzg-commitments-study
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Add batch proofs of KZG Commitments

master
arnaucube 2 years ago
parent
2c4a5c6089
commit
2e62793fd6
7 changed files with 166 additions and 62 deletions
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  1. +18
    -0
      README.md
  2. +5
    -3
      arithmetic.go
  3. +4
    -5
      arithmetic_test.go
  4. +0
    -1
      go.mod
  5. +0
    -41
      go.sum
  6. +73
    -10
      kzg.go
  7. +66
    -2
      kzg_test.go

+ 18
- 0
README.md
View File

@ -4,7 +4,10 @@ Doing this to study and learn [KZG commitments](http://cacr.uwaterloo.ca/techrep
Thanks to [Dankrad Feist](https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html), [Alin Tomescu](https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html), [Tom Walton-Pocock](https://hackmd.io/@tompocock/Hk2A7BD6U) for their articles, which helped me understand a bit the KZG Commitments.
It uses the [ethereum bn256](https://github.com/ethereum/go-ethereum/tree/master/crypto/bn256/cloudflare).
### Usage
```go
// p(x) = x^3 + x + 5
p := []*big.Int{
@ -34,3 +37,18 @@ assert.Nil(t, err)
v := Verify(ts, c, proof, z, y)
assert.True(t, v)
```
Batch Proofs:
```go
// zs & ys contain the f(z_i)=y_i values that will be proved inside a batch proof
zs := []*big.Int{z0, z1, z2}
ys := []*big.Int{y0, y1, y2}
// prove an evaluation of the multiple z_i & y_i
proof, err := EvaluationBatchProof(ts, p, zs, ys)
assert.Nil(t, err)
// batch proof verification
v := VerifyBatchProof(ts, c, proof, zs, ys)
assert.True(t, v)
```

+ 5
- 3
arithmetic.go
View File

@ -223,9 +223,9 @@ func newPolZeroAt(pointPos, totalPoints int, height *big.Int) []*big.Int {
// zeroPolynomial returns the zero polynomial:
// z(x) = (x - z_0) (x - z_1) ... (x - z_{k-1})
func zeroPolynomial(zs []*big.Int) []*big.Int {
z := []*big.Int{fNeg(zs[0]), big.NewInt(1)}
z := []*big.Int{fNeg(zs[0]), big.NewInt(1)} // (x - z0)
for i := 1; i < len(zs); i++ {
z = polynomialMul(z, []*big.Int{fNeg(zs[i]), big.NewInt(1)})
z = polynomialMul(z, []*big.Int{fNeg(zs[i]), big.NewInt(1)}) // (x - zi)
}
return z
}
@ -257,7 +257,9 @@ func intToSNum(n int) string {
func PolynomialToString(p []*big.Int) string {
s := ""
for i := len(p) - 1; i >= 1; i-- {
if !bytes.Equal(p[i].Bytes(), big.NewInt(0).Bytes()) {
if bytes.Equal(p[i].Bytes(), big.NewInt(1).Bytes()) {
s += fmt.Sprintf("x%s + ", intToSNum(i))
} else if !bytes.Equal(p[i].Bytes(), big.NewInt(0).Bytes()) {
s += fmt.Sprintf("%sx%s + ", p[i], intToSNum(i))
}
}

+ 4
- 5
arithmetic_test.go
View File

@ -6,7 +6,6 @@ import (
"math/big"
"testing"
cryptoConstants "github.com/iden3/go-iden3-crypto/constants"
"github.com/stretchr/testify/assert"
)
@ -18,7 +17,7 @@ func randBI() *big.Int {
panic(err)
}
r := new(big.Int).SetBytes(b)
return new(big.Int).Mod(r, cryptoConstants.Q)
return new(big.Int).Mod(r, Q)
}
func neg(a *big.Int) *big.Int {
@ -89,7 +88,7 @@ func TestPolynomial(t *testing.T) {
big.NewInt(0), // x^2
big.NewInt(1), // x^3
}
assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p))
assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p))
assert.Equal(t, "35", polynomialEval(p, big.NewInt(3)).String())
assert.Equal(t, "1015", polynomialEval(p, big.NewInt(10)).String())
assert.Equal(t, "16777477", polynomialEval(p, big.NewInt(256)).String())
@ -140,7 +139,7 @@ func TestLagrangeInterpolation(t *testing.T) {
p, err := LagrangeInterpolation(xs, ys)
assert.Nil(t, err)
assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p))
assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p))
assert.Equal(t, y0, polynomialEval(p, x0))
assert.Equal(t, y1, polynomialEval(p, x1))
@ -154,7 +153,7 @@ func TestZeroPolynomial(t *testing.T) {
xs := []*big.Int{x0, x1, x2}
z := zeroPolynomial(xs)
assert.Equal(t, "1x³ "+
assert.Equal(t, "x³ "+
"+ 21888242871839275222246405745257275088548364400416034343698204186575808495064x² "+
"+ 21032x¹ + 21888242871839275222246405745257275088548364400416034343698204186575808475137",
PolynomialToString(z))

+ 0
- 1
go.mod
View File

@ -4,6 +4,5 @@ go 1.16
require (
github.com/ethereum/go-ethereum v1.10.4
github.com/iden3/go-iden3-crypto v0.0.5
github.com/stretchr/testify v1.7.0
)

+ 0
- 41
go.sum
View File

@ -35,9 +35,7 @@ github.com/BurntSushi/toml v0.3.1/go.mod h1:xHWCNGjB5oqiDr8zfno3MHue2Ht5sIBksp03
github.com/BurntSushi/xgb v0.0.0-20160522181843-27f122750802/go.mod h1:IVnqGOEym/WlBOVXweHU+Q+/VP0lqqI8lqeDx9IjBqo=
github.com/DATA-DOG/go-sqlmock v1.3.3/go.mod h1:f/Ixk793poVmq4qj/V1dPUg2JEAKC73Q5eFN3EC/SaM=
github.com/OneOfOne/xxhash v1.2.2/go.mod h1:HSdplMjZKSmBqAxg5vPj2TmRDmfkzw+cTzAElWljhcU=
github.com/OneOfOne/xxhash v1.2.5/go.mod h1:eZbhyaAYD41SGSSsnmcpxVoRiQ/MPUTjUdIIOT9Um7Q=
github.com/StackExchange/wmi v0.0.0-20180116203802-5d049714c4a6/go.mod h1:3eOhrUMpNV+6aFIbp5/iudMxNCF27Vw2OZgy4xEx0Fg=
github.com/VictoriaMetrics/fastcache v1.5.3/go.mod h1:+jv9Ckb+za/P1ZRg/sulP5Ni1v49daAVERr0H3CuscE=
github.com/VictoriaMetrics/fastcache v1.6.0/go.mod h1:0qHz5QP0GMX4pfmMA/zt5RgfNuXJrTP0zS7DqpHGGTw=
github.com/aead/siphash v1.0.1/go.mod h1:Nywa3cDsYNNK3gaciGTWPwHt0wlpNV15vwmswBAUSII=
github.com/ajstarks/svgo v0.0.0-20180226025133-644b8db467af/go.mod h1:K08gAheRH3/J6wwsYMMT4xOr94bZjxIelGM0+d/wbFw=
@ -46,8 +44,6 @@ github.com/alecthomas/units v0.0.0-20151022065526-2efee857e7cf/go.mod h1:ybxpYRF
github.com/allegro/bigcache v1.2.1-0.20190218064605-e24eb225f156/go.mod h1:Cb/ax3seSYIx7SuZdm2G2xzfwmv3TPSk2ucNfQESPXM=
github.com/andreyvit/diff v0.0.0-20170406064948-c7f18ee00883/go.mod h1:rCTlJbsFo29Kk6CurOXKm700vrz8f0KW0JNfpkRJY/8=
github.com/apache/arrow/go/arrow v0.0.0-20191024131854-af6fa24be0db/go.mod h1:VTxUBvSJ3s3eHAg65PNgrsn5BtqCRPdmyXh6rAfdxN0=
github.com/aristanetworks/goarista v0.0.0-20170210015632-ea17b1a17847/go.mod h1:D/tb0zPVXnP7fmsLZjtdUhSsumbK/ij54UXjjVgMGxQ=
github.com/aws/aws-sdk-go v1.25.48/go.mod h1:KmX6BPdI08NWTb3/sm4ZGu5ShLoqVDhKgpiN924inxo=
github.com/aws/aws-sdk-go-v2 v1.2.0/go.mod h1:zEQs02YRBw1DjK0PoJv3ygDYOFTre1ejlJWl8FwAuQo=
github.com/aws/aws-sdk-go-v2/config v1.1.1/go.mod h1:0XsVy9lBI/BCXm+2Tuvt39YmdHwS5unDQmxZOYe8F5Y=
github.com/aws/aws-sdk-go-v2/credentials v1.1.1/go.mod h1:mM2iIjwl7LULWtS6JCACyInboHirisUUdkBPoTHMOUo=
@ -61,7 +57,6 @@ github.com/beorn7/perks v0.0.0-20180321164747-3a771d992973/go.mod h1:Dwedo/Wpr24
github.com/beorn7/perks v1.0.0/go.mod h1:KWe93zE9D1o94FZ5RNwFwVgaQK1VOXiVxmqh+CedLV8=
github.com/bmizerany/pat v0.0.0-20170815010413-6226ea591a40/go.mod h1:8rLXio+WjiTceGBHIoTvn60HIbs7Hm7bcHjyrSqYB9c=
github.com/boltdb/bolt v1.3.1/go.mod h1:clJnj/oiGkjum5o1McbSZDSLxVThjynRyGBgiAx27Ps=
github.com/btcsuite/btcd v0.0.0-20171128150713-2e60448ffcc6/go.mod h1:Dmm/EzmjnCiweXmzRIAiUWCInVmPgjkzgv5k4tVyXiQ=
github.com/btcsuite/btcd v0.20.1-beta/go.mod h1:wVuoA8VJLEcwgqHBwHmzLRazpKxTv13Px/pDuV7OomQ=
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=
@ -74,13 +69,11 @@ github.com/c-bata/go-prompt v0.2.2/go.mod h1:VzqtzE2ksDBcdln8G7mk2RX9QyGjH+OVqOC
github.com/census-instrumentation/opencensus-proto v0.2.1/go.mod h1:f6KPmirojxKA12rnyqOA5BBL4O983OfeGPqjHWSTneU=
github.com/cespare/cp v0.1.0/go.mod h1:SOGHArjBr4JWaSDEVpWpo/hNg6RoKrls6Oh40hiwW+s=
github.com/cespare/xxhash v1.1.0/go.mod h1:XrSqR1VqqWfGrhpAt58auRo0WTKS1nRRg3ghfAqPWnc=
github.com/cespare/xxhash/v2 v2.0.1-0.20190104013014-3767db7a7e18/go.mod h1:HD5P3vAIAh+Y2GAxg0PrPN1P8WkepXGpjbUPDHJqqKM=
github.com/cespare/xxhash/v2 v2.1.1/go.mod h1:VGX0DQ3Q6kWi7AoAeZDth3/j3BFtOZR5XLFGgcrjCOs=
github.com/chzyer/logex v1.1.10/go.mod h1:+Ywpsq7O8HXn0nuIou7OrIPyXbp3wmkHB+jjWRnGsAI=
github.com/chzyer/readline v0.0.0-20180603132655-2972be24d48e/go.mod h1:nSuG5e5PlCu98SY8svDHJxuZscDgtXS6KTTbou5AhLI=
github.com/chzyer/test v0.0.0-20180213035817-a1ea475d72b1/go.mod h1:Q3SI9o4m/ZMnBNeIyt5eFwwo7qiLfzFZmjNmxjkiQlU=
github.com/client9/misspell v0.3.4/go.mod h1:qj6jICC3Q7zFZvVWo7KLAzC3yx5G7kyvSDkc90ppPyw=
github.com/cloudflare/cloudflare-go v0.10.2-0.20190916151808-a80f83b9add9/go.mod h1:1MxXX1Ux4x6mqPmjkUgTP1CdXIBXKX7T+Jk9Gxrmx+U=
github.com/cloudflare/cloudflare-go v0.14.0/go.mod h1:EnwdgGMaFOruiPZRFSgn+TsQ3hQ7C/YWzIGLeu5c304=
github.com/consensys/bavard v0.1.8-0.20210406032232-f3452dc9b572/go.mod h1:Bpd0/3mZuaj6Sj+PqrmIquiOKy397AKGThQPaGzNXAQ=
github.com/consensys/gnark-crypto v0.4.1-0.20210426202927-39ac3d4b3f1f/go.mod h1:815PAHg3wvysy0SyIqanF8gZ0Y1wjk/hrDHD/iT88+Q=
@ -90,27 +83,20 @@ 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/dchest/blake512 v1.0.0/go.mod h1:FV1x7xPPLWukZlpDpWQ88rF/SFwZ5qbskrzhLMB92JI=
github.com/deckarep/golang-set v0.0.0-20180603214616-504e848d77ea/go.mod h1:93vsz/8Wt4joVM7c2AVqh+YRMiUSc14yDtF28KmMOgQ=
github.com/dgrijalva/jwt-go v3.2.0+incompatible/go.mod h1:E3ru+11k8xSBh+hMPgOLZmtrrCbhqsmaPHjLKYnJCaQ=
github.com/dgryski/go-bitstream v0.0.0-20180413035011-3522498ce2c8/go.mod h1:VMaSuZ+SZcx/wljOQKvp5srsbCiKDEb6K2wC4+PiBmQ=
github.com/dgryski/go-sip13 v0.0.0-20181026042036-e10d5fee7954/go.mod h1:vAd38F8PWV+bWy6jNmig1y/TA+kYO4g3RSRF0IAv0no=
github.com/dlclark/regexp2 v1.2.0/go.mod h1:2pZnwuY/m+8K6iRw6wQdMtk+rH5tNGR1i55kozfMjCc=
github.com/docker/docker v1.4.2-0.20180625184442-8e610b2b55bf/go.mod h1:eEKB0N0r5NX/I1kEveEz05bcu8tLC/8azJZsviup8Sk=
github.com/dop251/goja v0.0.0-20200219165308-d1232e640a87/go.mod h1:Mw6PkjjMXWbTj+nnj4s3QPXq1jaT0s5pC0iFD4+BOAA=
github.com/dop251/goja v0.0.0-20200721192441-a695b0cdd498/go.mod h1:Mw6PkjjMXWbTj+nnj4s3QPXq1jaT0s5pC0iFD4+BOAA=
github.com/eclipse/paho.mqtt.golang v1.2.0/go.mod h1:H9keYFcgq3Qr5OUJm/JZI/i6U7joQ8SYLhZwfeOo6Ts=
github.com/edsrzf/mmap-go v0.0.0-20160512033002-935e0e8a636c/go.mod h1:YO35OhQPt3KJa3ryjFM5Bs14WD66h8eGKpfaBNrHW5M=
github.com/edsrzf/mmap-go v1.0.0/go.mod h1:YO35OhQPt3KJa3ryjFM5Bs14WD66h8eGKpfaBNrHW5M=
github.com/elastic/gosigar v0.8.1-0.20180330100440-37f05ff46ffa/go.mod h1:cdorVVzy1fhmEqmtgqkoE3bYtCfSCkVyjTyCIo22xvs=
github.com/envoyproxy/go-control-plane v0.9.1-0.20191026205805-5f8ba28d4473/go.mod h1:YTl/9mNaCwkRvm6d1a2C3ymFceY/DCBVvsKhRF0iEA4=
github.com/envoyproxy/protoc-gen-validate v0.1.0/go.mod h1:iSmxcyjqTsJpI2R4NaDN7+kN2VEUnK/pcBlmesArF7c=
github.com/ethereum/go-ethereum v1.9.12/go.mod h1:PvsVkQmhZFx92Y+h2ylythYlheEDt/uBgFbl61Js/jo=
github.com/ethereum/go-ethereum v1.10.4 h1:JPZPL2MHbegfFStcaOrrggMVIcf57OQHQ0J3UhjQ+xQ=
github.com/ethereum/go-ethereum v1.10.4/go.mod h1:nEE0TP5MtxGzOMd7egIrbPJMQBnhVU3ELNxhBglIzhg=
github.com/fatih/color v1.3.0/go.mod h1:Zm6kSWBoL9eyXnKyktHP6abPY2pDugNf5KwzbycvMj4=
github.com/fatih/color v1.7.0/go.mod h1:Zm6kSWBoL9eyXnKyktHP6abPY2pDugNf5KwzbycvMj4=
github.com/fjl/memsize v0.0.0-20180418122429-ca190fb6ffbc/go.mod h1:VvhXpOYNQvB+uIk2RvXzuaQtkQJzzIx6lSBe1xv7hi0=
github.com/fjl/memsize v0.0.0-20190710130421-bcb5799ab5e5/go.mod h1:VvhXpOYNQvB+uIk2RvXzuaQtkQJzzIx6lSBe1xv7hi0=
github.com/fogleman/gg v1.2.1-0.20190220221249-0403632d5b90/go.mod h1:R/bRT+9gY/C5z7JzPU0zXsXHKM4/ayA+zqcVNZzPa1k=
github.com/fsnotify/fsnotify v1.4.7/go.mod h1:jwhsz4b93w/PPRr/qN1Yymfu8t87LnFCMoQvtojpjFo=
@ -140,7 +126,6 @@ github.com/golang/mock v1.2.0/go.mod h1:oTYuIxOrZwtPieC+H1uAHpcLFnEyAGVDL/k47Jfb
github.com/golang/mock v1.3.1/go.mod h1:sBzyDLLjw3U8JLTeZvSv8jJB+tU5PVekmnlKIyFUx0Y=
github.com/golang/protobuf v1.2.0/go.mod h1:6lQm79b+lXiMfvg/cZm0SGofjICqVBUtrP5yJMmIC1U=
github.com/golang/protobuf v1.3.1/go.mod h1:6lQm79b+lXiMfvg/cZm0SGofjICqVBUtrP5yJMmIC1U=
github.com/golang/protobuf v1.3.2-0.20190517061210-b285ee9cfc6c/go.mod h1:6lQm79b+lXiMfvg/cZm0SGofjICqVBUtrP5yJMmIC1U=
github.com/golang/protobuf v1.3.2/go.mod h1:6lQm79b+lXiMfvg/cZm0SGofjICqVBUtrP5yJMmIC1U=
github.com/golang/protobuf v1.4.0-rc.1/go.mod h1:ceaxUfeHdC40wWswd/P6IGgMaK3YpKi5j83Wpe3EHw8=
github.com/golang/protobuf v1.4.0-rc.1.0.20200221234624-67d41d38c208/go.mod h1:xKAWHe0F5eneWXFV3EuXVDTCmh+JuBKY0li0aMyXATA=
@ -171,26 +156,19 @@ github.com/google/uuid v1.1.5/go.mod h1:TIyPZe4MgqvfeYDBFedMoGGpEw/LqOeaOT+nhxU+
github.com/googleapis/gax-go/v2 v2.0.4/go.mod h1:0Wqv26UfaUD9n4G6kQubkQ+KchISgw+vpHVxEJEs9eg=
github.com/googleapis/gax-go/v2 v2.0.5/go.mod h1:DWXyrwAJ9X0FpwwEdw+IPEYBICEFu5mhpdKc/us6bOk=
github.com/gopherjs/gopherjs v0.0.0-20181017120253-0766667cb4d1/go.mod h1:wJfORRmW1u3UXTncJ5qlYoELFm8eSnnEO6hX4iZ3EWY=
github.com/gorilla/websocket v1.4.1-0.20190629185528-ae1634f6a989/go.mod h1:E7qHFY5m1UJ88s3WnNqhKjPHQ0heANvMoAMk2YaljkQ=
github.com/gorilla/websocket v1.4.2/go.mod h1:YR8l580nyteQvAITg2hZ9XVh4b55+EU/adAjf1fMHhE=
github.com/graph-gophers/graphql-go v0.0.0-20191115155744-f33e81362277/go.mod h1:9CQHMSxwO4MprSdzoIEobiHpoLtHm77vfxsvsIN5Vuc=
github.com/graph-gophers/graphql-go v0.0.0-20201113091052-beb923fada29/go.mod h1:9CQHMSxwO4MprSdzoIEobiHpoLtHm77vfxsvsIN5Vuc=
github.com/hashicorp/golang-lru v0.0.0-20160813221303-0a025b7e63ad/go.mod h1:/m3WP610KZHVQ1SGc6re/UDhFvYD7pJ4Ao+sR/qLZy8=
github.com/hashicorp/golang-lru v0.5.0/go.mod h1:/m3WP610KZHVQ1SGc6re/UDhFvYD7pJ4Ao+sR/qLZy8=
github.com/hashicorp/golang-lru v0.5.1/go.mod h1:/m3WP610KZHVQ1SGc6re/UDhFvYD7pJ4Ao+sR/qLZy8=
github.com/hashicorp/golang-lru v0.5.5-0.20210104140557-80c98217689d/go.mod h1:iADmTwqILo4mZ8BN3D2Q6+9jd8WM5uGBxy+E8yxSoD4=
github.com/holiman/bloomfilter/v2 v2.0.3/go.mod h1:zpoh+gs7qcpqrHr3dB55AMiJwo0iURXE7ZOP9L9hSkA=
github.com/holiman/uint256 v1.2.0/go.mod h1:y4ga/t+u+Xwd7CpDgZESaRcWy0I7XMlTMA25ApIH5Jw=
github.com/hpcloud/tail v1.0.0/go.mod h1:ab1qPbhIpdTxEkNHXyeSf5vhxWSCs/tWer42PpOxQnU=
github.com/huin/goupnp v0.0.0-20161224104101-679507af18f3/go.mod h1:MZ2ZmwcBpvOoJ22IJsc7va19ZwoheaBk43rKg12SKag=
github.com/huin/goupnp v1.0.1-0.20210310174557-0ca763054c88/go.mod h1:nNs7wvRfN1eKaMknBydLNQU6146XQim8t4h+q90biWo=
github.com/huin/goutil v0.0.0-20170803182201-1ca381bf3150/go.mod h1:PpLOETDnJ0o3iZrZfqZzyLl6l7F3c6L1oWn7OICBi6o=
github.com/ianlancetaylor/demangle v0.0.0-20181102032728-5e5cf60278f6/go.mod h1:aSSvb/t6k1mPoxDqO4vJh6VOCGPwU4O0C2/Eqndh1Sc=
github.com/iden3/go-iden3-crypto v0.0.5 h1:inCSm5a+ry+nbpVTL/9+m6UcIwSv6nhUm0tnIxEbcps=
github.com/iden3/go-iden3-crypto v0.0.5/go.mod h1:XKw1oDwYn2CIxKOtr7m/mL5jMn4mLOxAxtZBRxQBev8=
github.com/inconshreveable/mousetrap v1.0.0/go.mod h1:PxqpIevigyE2G7u3NXJIT2ANytuPF1OarO4DADm73n8=
github.com/influxdata/flux v0.65.1/go.mod h1:J754/zds0vvpfwuq7Gc2wRdVwEodfpCFM7mYlOw2LqY=
github.com/influxdata/influxdb v1.2.3-0.20180221223340-01288bdb0883/go.mod h1:qZna6X/4elxqT3yI9iZYdZrWWdeFOOprn86kgg4+IzY=
github.com/influxdata/influxdb v1.8.3/go.mod h1:JugdFhsvvI8gadxOI6noqNeeBHvWNTbfYGtiAn+2jhI=
github.com/influxdata/influxql v1.1.1-0.20200828144457-65d3ef77d385/go.mod h1:gHp9y86a/pxhjJ+zMjNXiQAA197Xk9wLxaz+fGG+kWk=
github.com/influxdata/line-protocol v0.0.0-20180522152040-32c6aa80de5e/go.mod h1:4kt73NQhadE3daL3WhR5EJ/J2ocX0PZzwxQ0gXJ7oFE=
@ -201,7 +179,6 @@ github.com/influxdata/usage-client v0.0.0-20160829180054-6d3895376368/go.mod h1:
github.com/jackpal/go-nat-pmp v1.0.2-0.20160603034137-1fa385a6f458/go.mod h1:QPH045xvCAeXUZOxsnwmrtiCoxIr9eob+4orBN1SBKc=
github.com/jedisct1/go-minisign v0.0.0-20190909160543-45766022959e/go.mod h1:G1CVv03EnqU1wYL2dFwXxW2An0az9JTl/ZsqXQeBlkU=
github.com/jessevdk/go-flags v0.0.0-20141203071132-1679536dcc89/go.mod h1:4FA24M0QyGHXBuZZK/XkWh8h0e1EYbRYJSGM75WSRxI=
github.com/jmespath/go-jmespath v0.0.0-20180206201540-c2b33e8439af/go.mod h1:Nht3zPeWKUH0NzdCt2Blrr5ys8VGpn0CEB0cQHVjt7k=
github.com/jmespath/go-jmespath v0.4.0/go.mod h1:T8mJZnbsbmF+m6zOOFylbeCJqk5+pHWvzYPziyZiYoo=
github.com/jmespath/go-jmespath/internal/testify v1.5.1/go.mod h1:L3OGu8Wl2/fWfCI6z80xFu9LTZmf1ZRjMHUOPmWr69U=
github.com/jrick/logrotate v1.0.0/go.mod h1:LNinyqDIJnpAur+b8yyulnQw/wDuN1+BYKlTRt3OuAQ=
@ -210,7 +187,6 @@ github.com/jstemmer/go-junit-report v0.0.0-20190106144839-af01ea7f8024/go.mod h1
github.com/jstemmer/go-junit-report v0.9.1/go.mod h1:Brl9GWCQeLvo8nXZwPNNblvFj/XSXhF0NWZEnDohbsk=
github.com/jsternberg/zap-logfmt v1.0.0/go.mod h1:uvPs/4X51zdkcm5jXl5SYoN+4RK21K8mysFmDaM/h+o=
github.com/jtolds/gls v4.20.0+incompatible/go.mod h1:QJZ7F/aHp+rZTRtaJ1ow/lLfFfVYBRgL+9YlvaHOwJU=
github.com/julienschmidt/httprouter v1.1.1-0.20170430222011-975b5c4c7c21/go.mod h1:SYymIcj16QtmaHHD7aYtjjsJG7VTCxuUUipMqKk8s4w=
github.com/julienschmidt/httprouter v1.2.0/go.mod h1:SYymIcj16QtmaHHD7aYtjjsJG7VTCxuUUipMqKk8s4w=
github.com/jung-kurt/gofpdf v1.0.3-0.20190309125859-24315acbbda5/go.mod h1:7Id9E/uU8ce6rXgefFLlgrJj/GYY22cpxn+r32jIOes=
github.com/jwilder/encoding v0.0.0-20170811194829-b4e1701a28ef/go.mod h1:Ct9fl0F6iIOGgxJ5npU/IUOhOhqlVrGjyIZc8/MagT0=
@ -239,7 +215,6 @@ github.com/mattn/go-ieproxy v0.0.0-20190702010315-6dee0af9227d/go.mod h1:31jz6HN
github.com/mattn/go-isatty v0.0.4/go.mod h1:M+lRXTBqGeGNdLjl/ufCoiOlB5xdOkqRJdNxMWT7Zi4=
github.com/mattn/go-isatty v0.0.5-0.20180830101745-3fb116b82035/go.mod h1:M+lRXTBqGeGNdLjl/ufCoiOlB5xdOkqRJdNxMWT7Zi4=
github.com/mattn/go-runewidth v0.0.3/go.mod h1:LwmH8dsx7+W8Uxz3IHJYH5QSwggIsqBzpuz5H//U1FU=
github.com/mattn/go-runewidth v0.0.4/go.mod h1:LwmH8dsx7+W8Uxz3IHJYH5QSwggIsqBzpuz5H//U1FU=
github.com/mattn/go-runewidth v0.0.9/go.mod h1:H031xJmbD/WCDINGzjvQ9THkh0rPKHF+m2gUSrubnMI=
github.com/mattn/go-sqlite3 v1.11.0/go.mod h1:FPy6KqzDD04eiIsT53CuJW3U88zkxoIYsOqkbpncsNc=
github.com/mattn/go-tty v0.0.0-20180907095812-13ff1204f104/go.mod h1:XPvLUNfbS4fJH25nqRHfWLMa1ONC8Amw+mIA639KxkE=
@ -252,8 +227,6 @@ github.com/naoina/go-stringutil v0.1.0/go.mod h1:XJ2SJL9jCtBh+P9q5btrd/Ylo8XwT/h
github.com/naoina/toml v0.1.2-0.20170918210437-9fafd6967416/go.mod h1:NBIhNtsFMo3G2szEBne+bO4gS192HuIYRqfvOWb4i1E=
github.com/nxadm/tail v1.4.4/go.mod h1:kenIhsEOeOJmVchQTgglprH7qJGnHDVpk1VPCcaMI8A=
github.com/oklog/ulid v1.3.1/go.mod h1:CirwcVhetQ6Lv90oh/F+FBtV6XMibvdAFo93nm5qn4U=
github.com/olekukonko/tablewriter v0.0.1/go.mod h1:vsDQFd/mU46D+Z4whnwzcISnGGzXWMclvtLoiIKAKIo=
github.com/olekukonko/tablewriter v0.0.2-0.20190409134802-7e037d187b0c/go.mod h1:vsDQFd/mU46D+Z4whnwzcISnGGzXWMclvtLoiIKAKIo=
github.com/olekukonko/tablewriter v0.0.5/go.mod h1:hPp6KlRPjbx+hW8ykQs1w3UBbZlj6HuIJcUGPhkA7kY=
github.com/onsi/ginkgo v1.6.0/go.mod h1:lLunBs/Ym6LB5Z9jYTR76FiuTmxDTDusOGeTQH+WWjE=
github.com/onsi/ginkgo v1.7.0/go.mod h1:lLunBs/Ym6LB5Z9jYTR76FiuTmxDTDusOGeTQH+WWjE=
@ -266,7 +239,6 @@ github.com/opentracing/opentracing-go v1.0.2/go.mod h1:UkNAQd3GIcIGf0SeVgPpRdFSt
github.com/opentracing/opentracing-go v1.0.3-0.20180606204148-bd9c31933947/go.mod h1:UkNAQd3GIcIGf0SeVgPpRdFStlNbqXla1AfSYxPUl2o=
github.com/opentracing/opentracing-go v1.1.0/go.mod h1:UkNAQd3GIcIGf0SeVgPpRdFStlNbqXla1AfSYxPUl2o=
github.com/paulbellamy/ratecounter v0.2.0/go.mod h1:Hfx1hDpSGoqxkVVpBi/IlYD7kChlfo5C6hzIHwPqfFE=
github.com/pborman/uuid v0.0.0-20170112150404-1b00554d8222/go.mod h1:VyrYX9gd7irzKovcSS6BIIEwPRkP2Wm2m9ufcdFSJ34=
github.com/peterh/liner v1.0.1-0.20180619022028-8c1271fcf47f/go.mod h1:xIteQHvHuaLYG9IFj6mSxM0fCKrs34IrEQUhOYuGPHc=
github.com/peterh/liner v1.1.1-0.20190123174540-a2c9a5303de7/go.mod h1:CRroGNssyjTd/qIG2FyxByd2S8JEAZXBl4qUrZf8GS0=
github.com/philhofer/fwd v1.0.0/go.mod h1:gk3iGcWd9+svBvR0sR+KPcfE+RNWozjowpeBVG3ZVNU=
@ -287,14 +259,11 @@ github.com/prometheus/common v0.4.1/go.mod h1:TNfzLD0ON7rHzMJeJkieUDPYmFC7Snx/y8
github.com/prometheus/common v0.6.0/go.mod h1:eBmuwkDJBwy6iBfxCBob6t6dR6ENT/y+J+Zk0j9GMYc=
github.com/prometheus/procfs v0.0.0-20181005140218-185b4288413d/go.mod h1:c3At6R/oaqEKCNdg8wHV1ftS6bRYblBhIjjI8uT2IGk=
github.com/prometheus/procfs v0.0.2/go.mod h1:TjEm7ze935MbeOT/UhFTIMYKhuLP4wbCsTZCD3I8kEA=
github.com/prometheus/tsdb v0.6.2-0.20190402121629-4f204dcbc150/go.mod h1:qhTCs0VvXwvX/y3TZrWD7rabWM+ijKTux40TwIPHuXU=
github.com/prometheus/tsdb v0.7.1/go.mod h1:qhTCs0VvXwvX/y3TZrWD7rabWM+ijKTux40TwIPHuXU=
github.com/retailnext/hllpp v1.0.1-0.20180308014038-101a6d2f8b52/go.mod h1:RDpi1RftBQPUCDRw6SmxeaREsAaRKnOclghuzp/WRzc=
github.com/rjeczalik/notify v0.9.1/go.mod h1:rKwnCoCGeuQnwBtTSPL9Dad03Vh2n40ePRrjvIXnJho=
github.com/rogpeppe/go-internal v1.3.0/go.mod h1:M8bDsm7K2OlrFYOpmOWEs/qY81heoFRclV5y23lUDJ4=
github.com/rs/cors v0.0.0-20160617231935-a62a804a8a00/go.mod h1:gFx+x8UowdsKA9AchylcLynDq+nNFfI8FkUZdN/jGCU=
github.com/rs/cors v1.7.0/go.mod h1:gFx+x8UowdsKA9AchylcLynDq+nNFfI8FkUZdN/jGCU=
github.com/rs/xhandler v0.0.0-20160618193221-ed27b6fd6521/go.mod h1:RvLn4FgxWubrpZHtQLnOf6EwhN2hEMusxZOhcW9H3UQ=
github.com/russross/blackfriday/v2 v2.0.1/go.mod h1:+Rmxgy9KzJVeS9/2gXHxylqXiyQDYRxCVz55jmeOWTM=
github.com/segmentio/kafka-go v0.1.0/go.mod h1:X6itGqS9L4jDletMsxZ7Dz+JFWxM6JHfPOCvTvk+EJo=
github.com/segmentio/kafka-go v0.2.0/go.mod h1:X6itGqS9L4jDletMsxZ7Dz+JFWxM6JHfPOCvTvk+EJo=
@ -305,13 +274,10 @@ github.com/sirupsen/logrus v1.2.0/go.mod h1:LxeOpSwHxABJmUn/MG1IvRgCAasNZTLOkJPx
github.com/smartystreets/assertions v0.0.0-20180927180507-b2de0cb4f26d/go.mod h1:OnSkiWE9lh6wB0YB77sQom3nweQdgAjqCqsofrRNTgc=
github.com/smartystreets/goconvey v1.6.4/go.mod h1:syvi0/a8iFYH4r/RixwvyeAJjdLS9QV7WQ/tjFTllLA=
github.com/spaolacci/murmur3 v0.0.0-20180118202830-f09979ecbc72/go.mod h1:JwIasOWyU6f++ZhiEuf87xNszmSA2myDM2Kzu9HwQUA=
github.com/spaolacci/murmur3 v1.0.1-0.20190317074736-539464a789e9/go.mod h1:JwIasOWyU6f++ZhiEuf87xNszmSA2myDM2Kzu9HwQUA=
github.com/spf13/cast v1.3.0/go.mod h1:Qx5cxh0v+4UWYiBimWS+eyWzqEqokIECu5etghLkUJE=
github.com/spf13/cobra v0.0.3/go.mod h1:1l0Ry5zgKvJasoi3XT1TypsSe7PqH0Sj9dhYf7v3XqQ=
github.com/spf13/pflag v1.0.3/go.mod h1:DYY7MBk1bdzusC3SYhjObp+wFpr4gzcvqqNjLnInEg4=
github.com/status-im/keycard-go v0.0.0-20190316090335-8537d3370df4/go.mod h1:RZLeN1LMWmRsyYjvAu+I6Dm9QmlDaIIt+Y+4Kd7Tp+Q=
github.com/steakknife/bloomfilter v0.0.0-20180922174646-6819c0d2a570/go.mod h1:8OR4w3TdeIHIh1g6EMY5p0gVNOovcWC+1vpc7naMuAw=
github.com/steakknife/hamming v0.0.0-20180906055917-c99c65617cd3/go.mod h1:hpGUWaI9xL8pRQCTXQgocU38Qw1g0Us7n5PxxTwTCYU=
github.com/stretchr/objx v0.1.0/go.mod h1:HFkY916IF+rwdDfMAkV7OtwuqBVzrE8GR6GFx+wExME=
github.com/stretchr/objx v0.1.1/go.mod h1:HFkY916IF+rwdDfMAkV7OtwuqBVzrE8GR6GFx+wExME=
github.com/stretchr/testify v1.2.0/go.mod h1:a8OnRcib4nhh0OaRAV+Yts87kKdq0PP7pXfy6kDkUVs=
@ -320,16 +286,13 @@ github.com/stretchr/testify v1.3.0/go.mod h1:M5WIy9Dh21IEIfnGCwXGc5bZfKNJtfHm1UV
github.com/stretchr/testify v1.4.0/go.mod h1:j7eGeouHqKxXV5pUuKE4zz7dFj8WfuZ+81PSLYec5m4=
github.com/stretchr/testify v1.7.0 h1:nwc3DEeHmmLAfoZucVR881uASk0Mfjw8xYJ99tb5CcY=
github.com/stretchr/testify v1.7.0/go.mod h1:6Fq8oRcR53rry900zMqJjRRixrwX3KX962/h/Wwjteg=
github.com/syndtr/goleveldb v1.0.1-0.20190923125748-758128399b1d/go.mod h1:9OrXJhf154huy1nPWmuSrkgjPUtUNhA+Zmy+6AESzuA=
github.com/syndtr/goleveldb v1.0.1-0.20210305035536-64b5b1c73954/go.mod h1:u2MKkTVTVJWe5D1rCvame8WqhBd88EuIwODJZ1VHCPM=
github.com/tinylib/msgp v1.0.2/go.mod h1:+d+yLhGm8mzTaHzB+wgMYrodPfmZrzkirds8fDWklFE=
github.com/tklauser/go-sysconf v0.3.5/go.mod h1:MkWzOF4RMCshBAMXuhXJs64Rte09mITnppBXY/rYEFI=
github.com/tklauser/numcpus v0.2.2/go.mod h1:x3qojaO3uyYt0i56EW/VUYs7uBvdl2fkfZFu0T9wgjM=
github.com/tyler-smith/go-bip39 v1.0.1-0.20181017060643-dbb3b84ba2ef/go.mod h1:sJ5fKU0s6JVwZjjcUEX2zFOnvq0ASQ2K9Zr6cf67kNs=
github.com/urfave/cli v1.22.1/go.mod h1:Gos4lmkARVdJ6EkW0WaNv/tZAAMe9V7XWyB60NtXRu0=
github.com/urfave/cli/v2 v2.3.0/go.mod h1:LJmUH05zAU44vOAcrfzZQKsZbVcdbOG8rtL3/XcUArI=
github.com/willf/bitset v1.1.3/go.mod h1:RjeCKbqT1RxIR/KWY6phxZiaY1IyutSBfGjNPySAYV4=
github.com/wsddn/go-ecdh v0.0.0-20161211032359-48726bab9208/go.mod h1:IotVbo4F+mw0EzQ08zFqg7pK3FebNXpaMsRy2RT+Ees=
github.com/xlab/treeprint v0.0.0-20180616005107-d6fb6747feb6/go.mod h1:ce1O1j6UtZfjr22oyGxGLbauSBp2YVXpARAosm7dHBg=
github.com/yuin/goldmark v1.2.1/go.mod h1:3hX8gzYuyVAZsxl0MRgGTJEmQBFcNTphYh9decYSb74=
go.opencensus.io v0.21.0/go.mod h1:mSImk1erAIZhrmZN+AvHh14ztQfjbGwt4TtuofqLduU=
@ -345,7 +308,6 @@ golang.org/x/crypto v0.0.0-20190510104115-cbcb75029529/go.mod h1:yigFU9vqHzYiE8U
golang.org/x/crypto v0.0.0-20190605123033-f99c8df09eb5/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI=
golang.org/x/crypto v0.0.0-20190909091759-094676da4a83/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI=
golang.org/x/crypto v0.0.0-20191011191535-87dc89f01550/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI=
golang.org/x/crypto v0.0.0-20200311171314-f7b00557c8c4/go.mod h1:LzIPMQfyMNhhGPhUkYOs5KpL4U8rLKemX1yGLhDgUto=
golang.org/x/crypto v0.0.0-20200622213623-75b288015ac9/go.mod h1:LzIPMQfyMNhhGPhUkYOs5KpL4U8rLKemX1yGLhDgUto=
golang.org/x/crypto v0.0.0-20210322153248-0c34fe9e7dc2/go.mod h1:T9bdIzuCu7OtxOm1hfPfRQxPLYneinmdGuTeoZ9dtd4=
golang.org/x/exp v0.0.0-20180321215751-8460e604b9de/go.mod h1:CJ0aWSM057203Lf6IL+f9T1iT9GByDxfZKAQTCR3kQA=
@ -392,7 +354,6 @@ golang.org/x/net v0.0.0-20190613194153-d28f0bde5980/go.mod h1:z5CRVTTTmAJ677TzLL
golang.org/x/net v0.0.0-20190620200207-3b0461eec859/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s=
golang.org/x/net v0.0.0-20190724013045-ca1201d0de80/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s=
golang.org/x/net v0.0.0-20191209160850-c0dbc17a3553/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s=
golang.org/x/net v0.0.0-20200301022130-244492dfa37a/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s=
golang.org/x/net v0.0.0-20200520004742-59133d7f0dd7/go.mod h1:qpuaurCH72eLCgpAm/N6yyVIVM9cpaDIP3A8BGJEC5A=
golang.org/x/net v0.0.0-20200813134508-3edf25e44fcc/go.mod h1:/O7V0waA8r7cgGh81Ro3o1hOxt32SMVPicZroKQ2sZA=
golang.org/x/net v0.0.0-20201021035429-f5854403a974/go.mod h1:sp8m0HH+o8qH0wwXwYZr8TS3Oi6o0r6Gce1SSxlDquU=
@ -431,7 +392,6 @@ golang.org/x/sys v0.0.0-20191120155948-bd437916bb0e/go.mod h1:h1NjWce9XRLGQEsW7w
golang.org/x/sys v0.0.0-20191204072324-ce4227a45e2e/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20191228213918-04cbcbbfeed8/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20200107162124-548cf772de50/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20200302150141-5c8b2ff67527/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20200323222414-85ca7c5b95cd/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20200519105757-fe76b779f299/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
golang.org/x/sys v0.0.0-20200814200057-3d37ad5750ed/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
@ -532,7 +492,6 @@ gopkg.in/check.v1 v1.0.0-20180628173108-788fd7840127/go.mod h1:Co6ibVJAznAaIkqp8
gopkg.in/errgo.v2 v2.1.0/go.mod h1:hNsd1EY+bozCKY1Ytp96fpM3vjJbqLJn88ws8XvfDNI=
gopkg.in/fsnotify.v1 v1.4.7/go.mod h1:Tz8NjZHkW78fSQdbUxIjBTcgA1z1m8ZHf0WmKUhAMys=
gopkg.in/natefinch/npipe.v2 v2.0.0-20160621034901-c1b8fa8bdcce/go.mod h1:5AcXVHNjg+BDxry382+8OKon8SEWiKktQR07RKPsv1c=
gopkg.in/olebedev/go-duktape.v3 v3.0.0-20190213234257-ec84240a7772/go.mod h1:uAJfkITjFhyEEuUfm7bsmCZRbW5WRq8s9EY8HZ6hCns=
gopkg.in/olebedev/go-duktape.v3 v3.0.0-20200619000410-60c24ae608a6/go.mod h1:uAJfkITjFhyEEuUfm7bsmCZRbW5WRq8s9EY8HZ6hCns=
gopkg.in/tomb.v1 v1.0.0-20141024135613-dd632973f1e7/go.mod h1:dt/ZhP58zS4L8KSrWDmTeBkI65Dw0HsyUHuEVlX15mw=
gopkg.in/urfave/cli.v1 v1.20.0/go.mod h1:vuBzUtMdQeixQj8LVd+/98pzhxNGQoyuPBlsXHOQNO0=

+ 73
- 10
kzg.go
View File

@ -15,7 +15,7 @@ type TrustedSetup struct {
// NewTrustedSetup returns a new trusted setup. This step should be done in a
// secure & distributed way
func NewTrustedSetup(p []*big.Int) (*TrustedSetup, error) {
func NewTrustedSetup(l int) (*TrustedSetup, error) {
// compute random s
s, err := randBigInt()
if err != nil {
@ -26,10 +26,10 @@ func NewTrustedSetup(p []*big.Int) (*TrustedSetup, error) {
// τ₁: [x₀]₁, [x₁]₁, [x₂]₁, ..., [x n₋₁]₁
// τ₂: [x₀]₂, [x₁]₂, [x₂]₂, ..., [x n₋₁]₂
// sPow := make([]*big.Int, len(p))
tauG1 := make([]*bn256.G1, len(p))
tauG2 := make([]*bn256.G2, len(p))
for i := 0; i < len(p); i++ {
// sPow := make([]*big.Int, l)
tauG1 := make([]*bn256.G1, l)
tauG2 := make([]*bn256.G2, l)
for i := 0; i < l; i++ {
sPow := fExp(s, big.NewInt(int64(i)))
tauG1[i] = new(bn256.G1).ScalarBaseMult(sPow)
tauG2[i] = new(bn256.G2).ScalarBaseMult(sPow)
@ -73,19 +73,18 @@ func EvaluationProof(ts *TrustedSetup, p []*big.Int, z, y *big.Int) (*bn256.G1,
return nil,
fmt.Errorf("remainder should be 0, instead is %d", rem)
}
fmt.Println("q(x):", PolynomialToString(q)) // TMP DBG
// proof: e = [q(s)]₁
// proof: e = [q(t)]₁
e := evaluateG1(ts, q)
return e, nil
}
// Verify computes the KZG commitment verification
func Verify(ts *TrustedSetup, c, proof *bn256.G1, z, y *big.Int) bool {
s2 := ts.Tau2[1] // [s]₂ = sG ∈ 𝔾₂ = Tau2[1]
s2 := ts.Tau2[1] // [t]₂ = sG ∈ 𝔾₂ = Tau2[1]
zG2Neg := new(bn256.G2).Neg(
new(bn256.G2).ScalarBaseMult(z)) // [z]₂ = zG ∈ 𝔾₂
// [s]₂ - [z]₂
// [t]₂ - [z]₂
sz := new(bn256.G2).Add(s2, zG2Neg)
yG1Neg := new(bn256.G1).Neg(
@ -95,8 +94,72 @@ func Verify(ts *TrustedSetup, c, proof *bn256.G1, z, y *big.Int) bool {
h := new(bn256.G2).ScalarBaseMult(big.NewInt(1)) // H ∈ 𝔾₂
// e(proof, [s]₂ - [z]₂) == e(c - [y]₁, H)
// e(proof, [t]₂ - [z]₂) == e(c - [y]₁, H)
e1 := bn256.Pair(proof, sz)
e2 := bn256.Pair(cy, h)
return e1.String() == e2.String()
}
//
// Batch proofs
//
// EvaluationBatchProof generates the evalutation proof for the given list of points
func EvaluationBatchProof(ts *TrustedSetup, p []*big.Int, zs, ys []*big.Int) (*bn256.G1, error) {
if len(zs) != len(ys) {
return nil, fmt.Errorf("len(zs)!=len(ys), %d!=%d", len(zs), len(ys))
}
if len(p) <= len(zs)+1 {
return nil, fmt.Errorf("polynomial p(x) can not be of degree"+
" equal or smaller than the number of given points+1."+
" Polynomial p(x) degree: %d, number of points: %d",
len(p), len(zs))
}
// z(x) = (x-z0)(x-z1)...(x-zn)
z := zeroPolynomial(zs)
// I(x) = Lagrange interpolation through (z0, y0), (z1, y1), ...
i, err := LagrangeInterpolation(zs, ys)
if err != nil {
return nil, err
}
// q(x) = ( p(x) - I(x) ) / z(x)
pMinusI := polynomialSub(p, i)
q, rem := polynomialDiv(pMinusI, z)
if compareBigIntArray(rem, arrayOfZeroes(len(rem))) {
return nil,
fmt.Errorf("remainder should be 0, instead is %d", rem)
}
// proof: e = [q(t)]₁
e := evaluateG1(ts, q)
return e, nil
}
// VerifyBatchProof computes the KZG batch proof commitment verification
func VerifyBatchProof(ts *TrustedSetup, c, proof *bn256.G1, zs, ys []*big.Int) bool {
// [z(s)]₂
z := zeroPolynomial(zs)
zG2 := evaluateG2(ts, z) // [z(t)]₂ = z(t) G ∈ 𝔾₂
// I(x) = Lagrange interpolation through (z0, y0), (z1, y1), ...
i, err := LagrangeInterpolation(zs, ys)
if err != nil {
return false
}
// [i(t)]₁
iG1 := evaluateG1(ts, i) // [i(t)]₁ = i(t) G ∈ 𝔾₁
// c - [i(t)]₁
iG1Neg := new(bn256.G1).Neg(iG1)
ciG1 := new(bn256.G1).Add(c, iG1Neg)
h := new(bn256.G2).ScalarBaseMult(big.NewInt(1)) // H ∈ 𝔾₂
// e(proof, [z(t)]₂) == e(c - [I(t)]₁, H)
e1 := bn256.Pair(proof, zG2)
e2 := bn256.Pair(ciG1, h)
return e1.String() == e2.String()
}

+ 66
- 2
kzg_test.go
View File

@ -15,10 +15,10 @@ func TestSimpleFlow(t *testing.T) {
big.NewInt(0), // x^2
big.NewInt(1), // x^3
}
assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p))
assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p))
// TrustedSetup
ts, err := NewTrustedSetup(p)
ts, err := NewTrustedSetup(len(p))
assert.Nil(t, err)
// Commit
@ -38,3 +38,67 @@ func TestSimpleFlow(t *testing.T) {
v = Verify(ts, c, proof, big.NewInt(4), y)
assert.False(t, v)
}
func TestBatchProof(t *testing.T) {
// p(x) = x^3 + x + 5
p := []*big.Int{
big.NewInt(5),
big.NewInt(1), // x^1
big.NewInt(0), // x^2
big.NewInt(1), // x^3
big.NewInt(10), // x^4
}
assert.Equal(t, "10x⁴ + x³ + x¹ + 5", PolynomialToString(p))
// TrustedSetup
ts, err := NewTrustedSetup(len(p))
assert.Nil(t, err)
// Commit
c := Commit(ts, p)
// 1st point: p(z)=y --> p(3)=35
z0 := big.NewInt(3)
y0 := polynomialEval(p, z0)
// 2nd point: p(10)=1015
z1 := big.NewInt(10)
y1 := polynomialEval(p, z1)
// 3nd point: p(256)=16777477
z2 := big.NewInt(256)
y2 := polynomialEval(p, z2)
zs := []*big.Int{z0, z1, z2}
ys := []*big.Int{y0, y1, y2}
// prove an evaluation of the multiple z_i & y_i
proof, err := EvaluationBatchProof(ts, p, zs, ys)
assert.Nil(t, err)
// batch proof verification
v := VerifyBatchProof(ts, c, proof, zs, ys)
assert.True(t, v)
// changing order of the points to be verified
zs[0], zs[1], zs[2] = zs[1], zs[2], zs[0]
ys[0], ys[1], ys[2] = ys[1], ys[2], ys[0]
v = VerifyBatchProof(ts, c, proof, zs, ys)
assert.True(t, v)
// change a value of zs and check that verification fails
zs[0] = big.NewInt(2)
v = VerifyBatchProof(ts, c, proof, zs, ys)
assert.False(t, v)
// using a value that is not in the evaluation proof should generate a
// proof that will not correctly be verified
zs = []*big.Int{z0, z1, z2}
ys = []*big.Int{y0, y1, y2}
proof, err = EvaluationBatchProof(ts, p, zs, ys)
assert.Nil(t, err)
zs[2] = big.NewInt(2500)
ys[2] = polynomialEval(p, zs[2])
v = VerifyBatchProof(ts, c, proof, zs, ys)
assert.False(t, v)
}

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