go-snark/snark.go
snark.goRaw
1// implementation of https://eprint.iacr.org/2013/879.pdf
2
3package snark
4
5import (
6 "fmt"
7 "math/big"
8 "os"
9
10 "github.com/arnaucube/go-snark/bn128"
11 "github.com/arnaucube/go-snark/circuitcompiler"
12 "github.com/arnaucube/go-snark/fields"
13 "github.com/arnaucube/go-snark/r1csqap"
14)
15
16type Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH)
17 G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H
18 A [][3]*big.Int
19 B [][3][2]*big.Int
20 C [][3]*big.Int
21 Kp [][3]*big.Int
22 Ap [][3]*big.Int
23 Bp [][3]*big.Int
24 Cp [][3]*big.Int
25 Z []*big.Int
26}
27
28type Vk struct {
29 Vka [3][2]*big.Int
30 Vkb [3]*big.Int
31 Vkc [3][2]*big.Int
32 IC [][3]*big.Int
33 G1Kbg [3]*big.Int // g1 * Kbeta * Kgamma
34 G2Kbg [3][2]*big.Int // g2 * Kbeta * Kgamma
35 G2Kg [3][2]*big.Int // g2 * Kgamma
36 Vkz [3][2]*big.Int
37}
38
39// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed
40type Setup struct {
41 Toxic struct {
42 T *big.Int // trusted setup secret
43 Ka *big.Int // prover
44 Kb *big.Int // prover
45 Kc *big.Int // prover
46 Kbeta *big.Int
47 Kgamma *big.Int
48 RhoA *big.Int
49 RhoB *big.Int
50 RhoC *big.Int
51 }
52
53 // public
54 Pk Pk
55 Vk Vk
56}
57
58// Proof contains the parameters to proof the zkSNARK
59type Proof struct {
60 PiA [3]*big.Int
61 PiAp [3]*big.Int
62 PiB [3][2]*big.Int
63 PiBp [3]*big.Int
64 PiC [3]*big.Int
65 PiCp [3]*big.Int
66 PiH [3]*big.Int
67 PiKp [3]*big.Int
68 // PublicSignals []*big.Int
69}
70
71type utils struct {
72 Bn bn128.Bn128
73 FqR fields.Fq
74 PF r1csqap.PolynomialField
75}
76
77// Utils is the data structure holding the BN128, FqR Finite Field over R, PolynomialField, that will be used inside the snarks operations
78var Utils = prepareUtils()
79
80func prepareUtils() utils {
81 bn, err := bn128.NewBn128()
82 if err != nil {
83 panic(err)
84 }
85 // new Finite Field
86 fqR := fields.NewFq(bn.R)
87 // new Polynomial Field
88 pf := r1csqap.NewPolynomialField(fqR)
89
90 return utils{
91 Bn: bn,
92 FqR: fqR,
93 PF: pf,
94 }
95}
96
97// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
98func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int) (Setup, error) {
99 var setup Setup
100 var err error
101
102 // input soundness
103 // for i := 0; i < len(alphas); i++ {
104 // for j := 0; j < len(alphas[i]); j++ {
105 // if j <= circuit.NPublic {
106 // if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
107 // alphas[i][j] = Utils.FqR.One()
108 // }
109 // }
110 // }
111 // }
112
113 // generate random t value
114 setup.Toxic.T, err = Utils.FqR.Rand()
115 if err != nil {
116 return Setup{}, err
117 }
118
119 // k for calculating pi' and Vk
120 setup.Toxic.Ka, err = Utils.FqR.Rand()
121 if err != nil {
122 return Setup{}, err
123 }
124 setup.Toxic.Kb, err = Utils.FqR.Rand()
125 if err != nil {
126 return Setup{}, err
127 }
128 setup.Toxic.Kc, err = Utils.FqR.Rand()
129 if err != nil {
130 return Setup{}, err
131 }
132
133 // generate Kβ (Kbeta) and Kγ (Kgamma)
134 setup.Toxic.Kbeta, err = Utils.FqR.Rand()
135 if err != nil {
136 return Setup{}, err
137 }
138 setup.Toxic.Kgamma, err = Utils.FqR.Rand()
139 if err != nil {
140 return Setup{}, err
141 }
142
143 // generate ρ (Rho): ρA, ρB, ρC
144 setup.Toxic.RhoA, err = Utils.FqR.Rand()
145 if err != nil {
146 return Setup{}, err
147 }
148 setup.Toxic.RhoB, err = Utils.FqR.Rand()
149 if err != nil {
150 return Setup{}, err
151 }
152 setup.Toxic.RhoC = Utils.FqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB)
153
154 // calculated more down
155 // for i := 0; i < witnessLength; i++ {
156 // tPow := Utils.FqR.Exp(setup.Toxic.T, big.NewInt(int64(i)))
157 // tEncr1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tPow)
158 // gt1 = append(gt1, tEncr1)
159 // tEncr2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, tPow)
160 // gt2 = append(gt2, tEncr2)
161 // }
162 // gt1: g1, g1*t, g1*t^2, g1*t^3, ...
163 // gt2: g2, g2*t, g2*t^2, ...
164
165 setup.Vk.Vka = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Ka)
166 setup.Vk.Vkb = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kb)
167 setup.Vk.Vkc = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kc)
168
169 /*
170 Verification keys:
171 - Vk_betagamma1: setup.G1Kbg = g1 * Kbeta*Kgamma
172 - Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma
173 - Vk_gamma: setup.G2Kg = g2 * Kgamma
174 */
175 kbg := Utils.FqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma)
176 setup.Vk.G1Kbg = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kbg)
177 setup.Vk.G2Kbg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, kbg)
178 setup.Vk.G2Kg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma)
179
180 // for i := 0; i < circuit.NVars; i++ {
181 for i := 0; i < len(circuit.Signals); i++ {
182 at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
183 // rhoAat := Utils.Bn.Fq1.Mul(setup.Toxic.RhoA, at)
184 rhoAat := Utils.FqR.Mul(setup.Toxic.RhoA, at)
185 a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoAat)
186 setup.Pk.A = append(setup.Pk.A, a)
187 if i <= circuit.NPublic {
188 setup.Vk.IC = append(setup.Vk.IC, a)
189 }
190
191 bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
192 // rhoBbt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoB, bt)
193 rhoBbt := Utils.FqR.Mul(setup.Toxic.RhoB, bt)
194 bg1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoBbt)
195 bg2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoBbt)
196 setup.Pk.B = append(setup.Pk.B, bg2)
197
198 ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
199 // rhoCct := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, ct)
200 rhoCct := Utils.FqR.Mul(setup.Toxic.RhoC, ct)
201 c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoCct)
202 setup.Pk.C = append(setup.Pk.C, c)
203
204 kt := Utils.FqR.Add(Utils.FqR.Add(rhoAat, rhoBbt), rhoCct)
205 k := Utils.Bn.G1.Affine(Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt))
206
207 ktest := Utils.Bn.G1.Affine(Utils.Bn.G1.Add(Utils.Bn.G1.Add(a, bg1), c))
208 if !Utils.Bn.Fq2.Equal(k, ktest) {
209 os.Exit(1)
210 return setup, err
211 }
212
213 setup.Pk.Ap = append(setup.Pk.Ap, Utils.Bn.G1.MulScalar(a, setup.Toxic.Ka))
214 setup.Pk.Bp = append(setup.Pk.Bp, Utils.Bn.G1.MulScalar(bg1, setup.Toxic.Kb))
215 setup.Pk.Cp = append(setup.Pk.Cp, Utils.Bn.G1.MulScalar(c, setup.Toxic.Kc))
216 k_ := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt)
217 setup.Pk.Kp = append(setup.Pk.Kp, Utils.Bn.G1.MulScalar(k_, setup.Toxic.Kbeta))
218 }
219
220 // z pol
221 zpol := []*big.Int{big.NewInt(int64(1))}
222 // for i := 0; i < len(circuit.Constraints); i++ {
223 for i := 1; i < len(alphas)-1; i++ {
224 zpol = Utils.PF.Mul(
225 zpol,
226 []*big.Int{
227 Utils.FqR.Neg( // neg over R
228 big.NewInt(int64(i))),
229 big.NewInt(int64(1)),
230 })
231 }
232 setup.Pk.Z = zpol
233
234 zt := Utils.PF.Eval(zpol, setup.Toxic.T)
235 // rhoCzt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, zt)
236 rhoCzt := Utils.FqR.Mul(setup.Toxic.RhoC, zt)
237 setup.Vk.Vkz = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoCzt)
238
239 // encrypt t values with curve generators
240 var gt1 [][3]*big.Int
241 gt1 = append(gt1, Utils.Bn.G1.G) // the first is t**0 * G1 = 1 * G1 = G1
242 tEncr := setup.Toxic.T
243 for i := 1; i < len(zpol); i++ { //should be G1T = pkH = (tau**i * G1) from i=0 to d, where d is degree of pol Z(x)
244 gt1 = append(gt1, Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tEncr))
245 // tEncr = Utils.Bn.Fq1.Mul(tEncr, setup.Toxic.T)
246 tEncr = Utils.FqR.Mul(tEncr, setup.Toxic.T)
247 }
248 setup.Pk.G1T = gt1
249
250 return setup, nil
251}
252
253// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
254func GenerateProofs(circuit circuitcompiler.Circuit, pk Pk, w []*big.Int, px []*big.Int) (Proof, error) {
255 var proof Proof
256 proof.PiA = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
257 proof.PiAp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
258 proof.PiB = Utils.Bn.Fq6.Zero()
259 proof.PiBp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
260 proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
261 proof.PiCp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
262 proof.PiH = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
263 proof.PiKp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
264
265 for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
266 proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(pk.A[i], w[i]))
267 proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(pk.Ap[i], w[i]))
268 }
269
270 for i := 0; i < circuit.NVars; i++ {
271 proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(pk.B[i], w[i]))
272 proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(pk.Bp[i], w[i]))
273
274 proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(pk.C[i], w[i]))
275 proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(pk.Cp[i], w[i]))
276
277 proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(pk.Kp[i], w[i]))
278 }
279
280 hx := Utils.PF.DivisorPolynomial(px, pk.Z) // maybe move this calculation to a previous step
281
282 // piH = pkH,0 + sum ( hi * pk H,i ), where pkH = G1T, hi=hx
283 // proof.PiH = Utils.Bn.G1.Add(proof.PiH, pk.G1T[0])
284 for i := 0; i < len(hx); i++ {
285 proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(pk.G1T[i], hx[i]))
286 }
287
288 return proof, nil
289}
290
291// VerifyProof verifies over the BN128 the Pairings of the Proof
292func VerifyProof(vk Vk, proof Proof, publicSignals []*big.Int, debug bool) bool {
293 // e(piA, Va) == e(piA', g2)
294 pairingPiaVa := Utils.Bn.Pairing(proof.PiA, vk.Vka)
295 pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G)
296 if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) {
297 if debug {
298 fmt.Println("❌ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
299 }
300 return false
301 }
302 if debug {
303 fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
304 }
305
306 // e(Vb, piB) == e(piB', g2)
307 pairingVbPib := Utils.Bn.Pairing(vk.Vkb, proof.PiB)
308 pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G)
309 if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) {
310 if debug {
311 fmt.Println("❌ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
312 }
313 return false
314 }
315 if debug {
316 fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
317 }
318
319 // e(piC, Vc) == e(piC', g2)
320 pairingPicVc := Utils.Bn.Pairing(proof.PiC, vk.Vkc)
321 pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G)
322 if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) {
323 if debug {
324 fmt.Println("❌ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
325 }
326 return false
327 }
328 if debug {
329 fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
330 }
331
332 // Vkx, to then calculate Vkx+piA
333 vkxpia := vk.IC[0]
334 for i := 0; i < len(publicSignals); i++ {
335 vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(vk.IC[i+1], publicSignals[i]))
336 }
337
338 // e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
339 if !Utils.Bn.Fq12.Equal(
340 Utils.Bn.Pairing(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiB), // TODO Add(vkxpia, proof.PiA) can go outside in order to save computation, as is reused later
341 Utils.Bn.Fq12.Mul(
342 Utils.Bn.Pairing(proof.PiH, vk.Vkz),
343 Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) {
344 if debug {
345 fmt.Println("❌ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
346 }
347 return false
348 }
349 if debug {
350 fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
351 }
352
353 // e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
354 // == e(piK, g2Kgamma)
355 piApiC := Utils.Bn.G1.Add(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiC)
356 pairingPiACG2Kbg := Utils.Bn.Pairing(piApiC, vk.G2Kbg)
357 pairingG1KbgPiB := Utils.Bn.Pairing(vk.G1Kbg, proof.PiB)
358 pairingL := Utils.Bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB)
359 pairingR := Utils.Bn.Pairing(proof.PiKp, vk.G2Kg)
360 if !Utils.Bn.Fq12.Equal(pairingL, pairingR) {
361 fmt.Println("❌ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
362 return false
363 }
364 if debug {
365 fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
366 }
367
368 return true
369}