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
16// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed
17type Setup struct {
18 Toxic struct {
19 T *big.Int // trusted setup secret
20 Ka *big.Int // prover
21 Kb *big.Int // prover
22 Kc *big.Int // prover
23 Kbeta *big.Int
24 Kgamma *big.Int
25 RhoA *big.Int
26 RhoB *big.Int
27 RhoC *big.Int
28 }
29
30 // public
31 G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H
32 G2T [][3][2]*big.Int // t encrypted in G2 curve
33 Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH)
34 A [][3]*big.Int
35 B [][3][2]*big.Int
36 C [][3]*big.Int
37 Kp [][3]*big.Int
38 Ap [][3]*big.Int
39 Bp [][3]*big.Int
40 Cp [][3]*big.Int
41 Z []*big.Int
42 }
43 Vk struct {
44 Vka [3][2]*big.Int
45 Vkb [3]*big.Int
46 Vkc [3][2]*big.Int
47 IC [][3]*big.Int
48 G1Kbg [3]*big.Int // g1 * Kbeta * Kgamma
49 G2Kbg [3][2]*big.Int // g2 * Kbeta * Kgamma
50 G2Kg [3][2]*big.Int // g2 * Kgamma
51 Vkz [3][2]*big.Int
52 }
53}
54
55// Proof contains the parameters to proof the zkSNARK
56type Proof struct {
57 PiA [3]*big.Int
58 PiAp [3]*big.Int
59 PiB [3][2]*big.Int
60 PiBp [3]*big.Int
61 PiC [3]*big.Int
62 PiCp [3]*big.Int
63 PiH [3]*big.Int
64 PiKp [3]*big.Int
65 // PublicSignals []*big.Int
66}
67
68type utils struct {
69 Bn bn128.Bn128
70 FqR fields.Fq
71 PF r1csqap.PolynomialField
72}
73
74// Utils is the data structure holding the BN128, FqR Finite Field over R, PolynomialField, that will be used inside the snarks operations
75var Utils = prepareUtils()
76
77func prepareUtils() utils {
78 bn, err := bn128.NewBn128()
79 if err != nil {
80 panic(err)
81 }
82 // new Finite Field
83 fqR := fields.NewFq(bn.R)
84 // new Polynomial Field
85 pf := r1csqap.NewPolynomialField(fqR)
86
87 return utils{
88 Bn: bn,
89 FqR: fqR,
90 PF: pf,
91 }
92}
93
94// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
95func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int) (Setup, error) {
96 var setup Setup
97 var err error
98
99 // input soundness
100 // for i := 0; i < len(alphas); i++ {
101 // for j := 0; j < len(alphas[i]); j++ {
102 // if j <= circuit.NPublic {
103 // if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
104 // alphas[i][j] = Utils.FqR.One()
105 // }
106 // }
107 // }
108 // }
109
110 // generate random t value
111 setup.Toxic.T, err = Utils.FqR.Rand()
112 if err != nil {
113 return Setup{}, err
114 }
115
116 // k for calculating pi' and Vk
117 setup.Toxic.Ka, err = Utils.FqR.Rand()
118 if err != nil {
119 return Setup{}, err
120 }
121 setup.Toxic.Kb, err = Utils.FqR.Rand()
122 if err != nil {
123 return Setup{}, err
124 }
125 setup.Toxic.Kc, err = Utils.FqR.Rand()
126 if err != nil {
127 return Setup{}, err
128 }
129
130 // generate Kβ (Kbeta) and Kγ (Kgamma)
131 setup.Toxic.Kbeta, err = Utils.FqR.Rand()
132 if err != nil {
133 return Setup{}, err
134 }
135 setup.Toxic.Kgamma, err = Utils.FqR.Rand()
136 if err != nil {
137 return Setup{}, err
138 }
139
140 // generate ρ (Rho): ρA, ρB, ρC
141 setup.Toxic.RhoA, err = Utils.FqR.Rand()
142 if err != nil {
143 return Setup{}, err
144 }
145 setup.Toxic.RhoB, err = Utils.FqR.Rand()
146 if err != nil {
147 return Setup{}, err
148 }
149 setup.Toxic.RhoC = Utils.FqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB)
150
151 // calculated more down
152 // for i := 0; i < witnessLength; i++ {
153 // tPow := Utils.FqR.Exp(setup.Toxic.T, big.NewInt(int64(i)))
154 // tEncr1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tPow)
155 // gt1 = append(gt1, tEncr1)
156 // tEncr2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, tPow)
157 // gt2 = append(gt2, tEncr2)
158 // }
159 // gt1: g1, g1*t, g1*t^2, g1*t^3, ...
160 // gt2: g2, g2*t, g2*t^2, ...
161
162 setup.Vk.Vka = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Ka)
163 setup.Vk.Vkb = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kb)
164 setup.Vk.Vkc = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kc)
165
166 /*
167 Verification keys:
168 - Vk_betagamma1: setup.G1Kbg = g1 * Kbeta*Kgamma
169 - Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma
170 - Vk_gamma: setup.G2Kg = g2 * Kgamma
171 */
172 kbg := Utils.FqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma)
173 setup.Vk.G1Kbg = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kbg)
174 setup.Vk.G2Kbg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, kbg)
175 setup.Vk.G2Kg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma)
176
177 // for i := 0; i < circuit.NVars; i++ {
178 for i := 0; i < len(circuit.Signals); i++ {
179 at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
180 // rhoAat := Utils.Bn.Fq1.Mul(setup.Toxic.RhoA, at)
181 rhoAat := Utils.FqR.Mul(setup.Toxic.RhoA, at)
182 a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoAat)
183 setup.Pk.A = append(setup.Pk.A, a)
184 if i <= circuit.NPublic {
185 setup.Vk.IC = append(setup.Vk.IC, a)
186 }
187
188 bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
189 // rhoBbt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoB, bt)
190 rhoBbt := Utils.FqR.Mul(setup.Toxic.RhoB, bt)
191 bg1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoBbt)
192 bg2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoBbt)
193 setup.Pk.B = append(setup.Pk.B, bg2)
194
195 ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
196 // rhoCct := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, ct)
197 rhoCct := Utils.FqR.Mul(setup.Toxic.RhoC, ct)
198 c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoCct)
199 setup.Pk.C = append(setup.Pk.C, c)
200
201 kt := Utils.FqR.Add(Utils.FqR.Add(rhoAat, rhoBbt), rhoCct)
202 k := Utils.Bn.G1.Affine(Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt))
203
204 ktest := Utils.Bn.G1.Affine(Utils.Bn.G1.Add(Utils.Bn.G1.Add(a, bg1), c))
205 if !Utils.Bn.Fq2.Equal(k, ktest) {
206 os.Exit(1)
207 return setup, err
208 }
209
210 setup.Pk.Ap = append(setup.Pk.Ap, Utils.Bn.G1.MulScalar(a, setup.Toxic.Ka))
211 setup.Pk.Bp = append(setup.Pk.Bp, Utils.Bn.G1.MulScalar(bg1, setup.Toxic.Kb))
212 setup.Pk.Cp = append(setup.Pk.Cp, Utils.Bn.G1.MulScalar(c, setup.Toxic.Kc))
213 k_ := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt)
214 setup.Pk.Kp = append(setup.Pk.Kp, Utils.Bn.G1.MulScalar(k_, setup.Toxic.Kbeta))
215 }
216
217 // z pol
218 zpol := []*big.Int{big.NewInt(int64(1))}
219 // for i := 0; i < len(circuit.Constraints); i++ {
220 for i := 1; i < len(alphas)-1; i++ {
221 zpol = Utils.PF.Mul(
222 zpol,
223 []*big.Int{
224 Utils.FqR.Neg( // neg over R
225 big.NewInt(int64(i))),
226 big.NewInt(int64(1)),
227 })
228 }
229 setup.Pk.Z = zpol
230
231 zt := Utils.PF.Eval(zpol, setup.Toxic.T)
232 // rhoCzt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, zt)
233 rhoCzt := Utils.FqR.Mul(setup.Toxic.RhoC, zt)
234 setup.Vk.Vkz = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoCzt)
235
236 // encrypt t values with curve generators
237 var gt1 [][3]*big.Int
238 gt1 = append(gt1, Utils.Bn.G1.G) // the first is t**0 * G1 = 1 * G1 = G1
239 tEncr := setup.Toxic.T
240 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)
241 gt1 = append(gt1, Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tEncr))
242 // tEncr = Utils.Bn.Fq1.Mul(tEncr, setup.Toxic.T)
243 tEncr = Utils.FqR.Mul(tEncr, setup.Toxic.T)
244 }
245 setup.G1T = gt1
246
247 return setup, nil
248}
249
250// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
251func GenerateProofs(circuit circuitcompiler.Circuit, setup Setup, w []*big.Int, px []*big.Int) (Proof, error) {
252 var proof Proof
253 proof.PiA = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
254 proof.PiAp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
255 proof.PiB = Utils.Bn.Fq6.Zero()
256 proof.PiBp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
257 proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
258 proof.PiCp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
259 proof.PiH = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
260 proof.PiKp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
261
262 for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
263 proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(setup.Pk.A[i], w[i]))
264 proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(setup.Pk.Ap[i], w[i]))
265 }
266
267 for i := 0; i < circuit.NVars; i++ {
268 proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(setup.Pk.B[i], w[i]))
269 proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(setup.Pk.Bp[i], w[i]))
270
271 proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.C[i], w[i]))
272 proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(setup.Pk.Cp[i], w[i]))
273
274 proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(setup.Pk.Kp[i], w[i]))
275 }
276
277 hx := Utils.PF.DivisorPolynomial(px, setup.Pk.Z) // maybe move this calculation to a previous step
278
279 // piH = pkH,0 + sum ( hi * pk H,i ), where pkH = G1T, hi=hx
280 // proof.PiH = Utils.Bn.G1.Add(proof.PiH, setup.G1T[0])
281 for i := 0; i < len(hx); i++ {
282 proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(setup.G1T[i], hx[i]))
283 }
284
285 return proof, nil
286}
287
288// VerifyProof verifies over the BN128 the Pairings of the Proof
289func VerifyProof(setup Setup, proof Proof, publicSignals []*big.Int, debug bool) bool {
290 // e(piA, Va) == e(piA', g2)
291 pairingPiaVa := Utils.Bn.Pairing(proof.PiA, setup.Vk.Vka)
292 pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G)
293 if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) {
294 if debug {
295 fmt.Println("❌ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
296 }
297 return false
298 }
299 if debug {
300 fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
301 }
302
303 // e(Vb, piB) == e(piB', g2)
304 pairingVbPib := Utils.Bn.Pairing(setup.Vk.Vkb, proof.PiB)
305 pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G)
306 if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) {
307 if debug {
308 fmt.Println("❌ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
309 }
310 return false
311 }
312 if debug {
313 fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
314 }
315
316 // e(piC, Vc) == e(piC', g2)
317 pairingPicVc := Utils.Bn.Pairing(proof.PiC, setup.Vk.Vkc)
318 pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G)
319 if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) {
320 if debug {
321 fmt.Println("❌ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
322 }
323 return false
324 }
325 if debug {
326 fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
327 }
328
329 // Vkx, to then calculate Vkx+piA
330 vkxpia := setup.Vk.IC[0]
331 for i := 0; i < len(publicSignals); i++ {
332 vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(setup.Vk.IC[i+1], publicSignals[i]))
333 }
334
335 // e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
336 if !Utils.Bn.Fq12.Equal(
337 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
338 Utils.Bn.Fq12.Mul(
339 Utils.Bn.Pairing(proof.PiH, setup.Vk.Vkz),
340 Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) {
341 if debug {
342 fmt.Println("❌ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
343 }
344 return false
345 }
346 if debug {
347 fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
348 }
349
350 // e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
351 // == e(piK, g2Kgamma)
352 piApiC := Utils.Bn.G1.Add(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiC)
353 pairingPiACG2Kbg := Utils.Bn.Pairing(piApiC, setup.Vk.G2Kbg)
354 pairingG1KbgPiB := Utils.Bn.Pairing(setup.Vk.G1Kbg, proof.PiB)
355 pairingL := Utils.Bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB)
356 pairingR := Utils.Bn.Pairing(proof.PiKp, setup.Vk.G2Kg)
357 if !Utils.Bn.Fq12.Equal(pairingL, pairingR) {
358 fmt.Println("❌ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
359 return false
360 }
361 if debug {
362 fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
363 }
364
365 return true
366}