go-snark/snark.go
snark.goRaw
1package snark
2
3import (
4 "fmt"
5 "math/big"
6 "os"
7
8 "github.com/arnaucube/go-snark/bn128"
9 "github.com/arnaucube/go-snark/circuitcompiler"
10 "github.com/arnaucube/go-snark/fields"
11 "github.com/arnaucube/go-snark/r1csqap"
12)
13
14// Setup is the data structure holding the Trusted Setup data. The Setup.Toxic sub struct must be destroyed after the GenerateTrustedSetup function is completed
15type Setup struct {
16 Toxic struct {
17 T *big.Int // trusted setup secret
18 Ka *big.Int // prover
19 Kb *big.Int // prover
20 Kc *big.Int // prover
21 Kbeta *big.Int
22 Kgamma *big.Int
23 RhoA *big.Int
24 RhoB *big.Int
25 RhoC *big.Int
26 }
27
28 // public
29 G1T [][3]*big.Int // t encrypted in G1 curve, G1T == Pk.H
30 G2T [][3][2]*big.Int // t encrypted in G2 curve
31 Pk struct { // Proving Key pk:=(pkA, pkB, pkC, pkH)
32 A [][3]*big.Int
33 B [][3][2]*big.Int
34 C [][3]*big.Int
35 Kp [][3]*big.Int
36 Ap [][3]*big.Int
37 Bp [][3]*big.Int
38 Cp [][3]*big.Int
39 Z []*big.Int
40 }
41 Vk struct {
42 Vka [3][2]*big.Int
43 Vkb [3]*big.Int
44 Vkc [3][2]*big.Int
45 IC [][3]*big.Int
46 G1Kbg [3]*big.Int // g1 * Kbeta * Kgamma
47 G2Kbg [3][2]*big.Int // g2 * Kbeta * Kgamma
48 G2Kg [3][2]*big.Int // g2 * Kgamma
49 Vkz [3][2]*big.Int
50 }
51}
52
53// Proof contains the parameters to proof the zkSNARK
54type Proof struct {
55 PiA [3]*big.Int
56 PiAp [3]*big.Int
57 PiB [3][2]*big.Int
58 PiBp [3]*big.Int
59 PiC [3]*big.Int
60 PiCp [3]*big.Int
61 PiH [3]*big.Int
62 PiKp [3]*big.Int
63 // PublicSignals []*big.Int
64}
65
66type utils struct {
67 Bn bn128.Bn128
68 FqR fields.Fq
69 PF r1csqap.PolynomialField
70}
71
72// Utils is the data structure holding the BN128, FqR Finite Field over R, PolynomialField, that will be used inside the snarks operations
73var Utils = prepareUtils()
74
75func prepareUtils() utils {
76 bn, err := bn128.NewBn128()
77 if err != nil {
78 panic(err)
79 }
80 // new Finite Field
81 fqR := fields.NewFq(bn.R)
82 // new Polynomial Field
83 pf := r1csqap.NewPolynomialField(fqR)
84
85 return utils{
86 Bn: bn,
87 FqR: fqR,
88 PF: pf,
89 }
90}
91
92// GenerateTrustedSetup generates the Trusted Setup from a compiled Circuit. The Setup.Toxic sub data structure must be destroyed
93func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, alphas, betas, gammas [][]*big.Int) (Setup, error) {
94 var setup Setup
95 var err error
96
97 // input soundness
98 // for i := 0; i < len(alphas); i++ {
99 // for j := 0; j < len(alphas[i]); j++ {
100 // if j <= circuit.NPublic {
101 // if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
102 // alphas[i][j] = Utils.FqR.One()
103 // }
104 // }
105 // }
106 // }
107
108 // generate random t value
109 setup.Toxic.T, err = Utils.FqR.Rand()
110 if err != nil {
111 return Setup{}, err
112 }
113
114 // k for calculating pi' and Vk
115 setup.Toxic.Ka, err = Utils.FqR.Rand()
116 if err != nil {
117 return Setup{}, err
118 }
119 setup.Toxic.Kb, err = Utils.FqR.Rand()
120 if err != nil {
121 return Setup{}, err
122 }
123 setup.Toxic.Kc, err = Utils.FqR.Rand()
124 if err != nil {
125 return Setup{}, err
126 }
127
128 // generate Kβ (Kbeta) and Kγ (Kgamma)
129 setup.Toxic.Kbeta, err = Utils.FqR.Rand()
130 if err != nil {
131 return Setup{}, err
132 }
133 setup.Toxic.Kgamma, err = Utils.FqR.Rand()
134 if err != nil {
135 return Setup{}, err
136 }
137
138 // generate ρ (Rho): ρA, ρB, ρC
139 setup.Toxic.RhoA, err = Utils.FqR.Rand()
140 if err != nil {
141 return Setup{}, err
142 }
143 setup.Toxic.RhoB, err = Utils.FqR.Rand()
144 if err != nil {
145 return Setup{}, err
146 }
147 setup.Toxic.RhoC = Utils.FqR.Mul(setup.Toxic.RhoA, setup.Toxic.RhoB)
148
149 // calculated more down
150 // for i := 0; i < witnessLength; i++ {
151 // tPow := Utils.FqR.Exp(setup.Toxic.T, big.NewInt(int64(i)))
152 // tEncr1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tPow)
153 // gt1 = append(gt1, tEncr1)
154 // tEncr2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, tPow)
155 // gt2 = append(gt2, tEncr2)
156 // }
157 // gt1: g1, g1*t, g1*t^2, g1*t^3, ...
158 // gt2: g2, g2*t, g2*t^2, ...
159
160 setup.Vk.Vka = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Ka)
161 setup.Vk.Vkb = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, setup.Toxic.Kb)
162 setup.Vk.Vkc = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kc)
163
164 /*
165 Verification keys:
166 - Vk_betagamma1: setup.G1Kbg = g1 * Kbeta*Kgamma
167 - Vk_betagamma2: setup.G2Kbg = g2 * Kbeta*Kgamma
168 - Vk_gamma: setup.G2Kg = g2 * Kgamma
169 */
170 kbg := Utils.FqR.Mul(setup.Toxic.Kbeta, setup.Toxic.Kgamma)
171 setup.Vk.G1Kbg = Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kbg)
172 setup.Vk.G2Kbg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, kbg)
173 setup.Vk.G2Kg = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, setup.Toxic.Kgamma)
174
175 // for i := 0; i < circuit.NVars; i++ {
176 for i := 0; i < len(circuit.Signals); i++ {
177 at := Utils.PF.Eval(alphas[i], setup.Toxic.T)
178 // rhoAat := Utils.Bn.Fq1.Mul(setup.Toxic.RhoA, at)
179 rhoAat := Utils.FqR.Mul(setup.Toxic.RhoA, at)
180 a := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoAat)
181 setup.Pk.A = append(setup.Pk.A, a)
182 if i <= circuit.NPublic {
183 setup.Vk.IC = append(setup.Vk.IC, a)
184 }
185
186 bt := Utils.PF.Eval(betas[i], setup.Toxic.T)
187 // rhoBbt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoB, bt)
188 rhoBbt := Utils.FqR.Mul(setup.Toxic.RhoB, bt)
189 bg1 := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoBbt)
190 bg2 := Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoBbt)
191 setup.Pk.B = append(setup.Pk.B, bg2)
192
193 ct := Utils.PF.Eval(gammas[i], setup.Toxic.T)
194 // rhoCct := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, ct)
195 rhoCct := Utils.FqR.Mul(setup.Toxic.RhoC, ct)
196 c := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, rhoCct)
197 setup.Pk.C = append(setup.Pk.C, c)
198
199 kt := Utils.FqR.Add(Utils.FqR.Add(rhoAat, rhoBbt), rhoCct)
200 k := Utils.Bn.G1.Affine(Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt))
201
202 ktest := Utils.Bn.G1.Affine(Utils.Bn.G1.Add(Utils.Bn.G1.Add(a, bg1), c))
203 if !Utils.Bn.Fq2.Equal(k, ktest) {
204 os.Exit(1)
205 return setup, err
206 }
207
208 setup.Pk.Ap = append(setup.Pk.Ap, Utils.Bn.G1.MulScalar(a, setup.Toxic.Ka))
209 setup.Pk.Bp = append(setup.Pk.Bp, Utils.Bn.G1.MulScalar(bg1, setup.Toxic.Kb))
210 setup.Pk.Cp = append(setup.Pk.Cp, Utils.Bn.G1.MulScalar(c, setup.Toxic.Kc))
211 k_ := Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, kt)
212 setup.Pk.Kp = append(setup.Pk.Kp, Utils.Bn.G1.MulScalar(k_, setup.Toxic.Kbeta))
213 }
214
215 // z pol
216 zpol := []*big.Int{big.NewInt(int64(1))}
217 // for i := 0; i < len(circuit.Constraints); i++ {
218 for i := 1; i < len(alphas)-1; i++ {
219 zpol = Utils.PF.Mul(
220 zpol,
221 []*big.Int{
222 Utils.FqR.Neg( // neg over R
223 big.NewInt(int64(i))),
224 big.NewInt(int64(1)),
225 })
226 }
227 setup.Pk.Z = zpol
228
229 zt := Utils.PF.Eval(zpol, setup.Toxic.T)
230 // rhoCzt := Utils.Bn.Fq1.Mul(setup.Toxic.RhoC, zt)
231 rhoCzt := Utils.FqR.Mul(setup.Toxic.RhoC, zt)
232 setup.Vk.Vkz = Utils.Bn.G2.MulScalar(Utils.Bn.G2.G, rhoCzt)
233
234 // encrypt t values with curve generators
235 var gt1 [][3]*big.Int
236 gt1 = append(gt1, Utils.Bn.G1.G) // the first is t**0 * G1 = 1 * G1 = G1
237 tEncr := setup.Toxic.T
238 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)
239 gt1 = append(gt1, Utils.Bn.G1.MulScalar(Utils.Bn.G1.G, tEncr))
240 // tEncr = Utils.Bn.Fq1.Mul(tEncr, setup.Toxic.T)
241 tEncr = Utils.FqR.Mul(tEncr, setup.Toxic.T)
242 }
243 setup.G1T = gt1
244
245 return setup, nil
246}
247
248// GenerateProofs generates all the parameters to proof the zkSNARK from the Circuit, Setup and the Witness
249func GenerateProofs(circuit circuitcompiler.Circuit, setup Setup, w []*big.Int, px []*big.Int) (Proof, error) {
250 var proof Proof
251 proof.PiA = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
252 proof.PiAp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
253 proof.PiB = Utils.Bn.Fq6.Zero()
254 proof.PiBp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
255 proof.PiC = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
256 proof.PiCp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
257 proof.PiH = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
258 proof.PiKp = [3]*big.Int{Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero(), Utils.Bn.G1.F.Zero()}
259
260 for i := circuit.NPublic + 1; i < circuit.NVars; i++ {
261 proof.PiA = Utils.Bn.G1.Add(proof.PiA, Utils.Bn.G1.MulScalar(setup.Pk.A[i], w[i]))
262 proof.PiAp = Utils.Bn.G1.Add(proof.PiAp, Utils.Bn.G1.MulScalar(setup.Pk.Ap[i], w[i]))
263 }
264
265 for i := 0; i < circuit.NVars; i++ {
266 proof.PiB = Utils.Bn.G2.Add(proof.PiB, Utils.Bn.G2.MulScalar(setup.Pk.B[i], w[i]))
267 proof.PiBp = Utils.Bn.G1.Add(proof.PiBp, Utils.Bn.G1.MulScalar(setup.Pk.Bp[i], w[i]))
268
269 proof.PiC = Utils.Bn.G1.Add(proof.PiC, Utils.Bn.G1.MulScalar(setup.Pk.C[i], w[i]))
270 proof.PiCp = Utils.Bn.G1.Add(proof.PiCp, Utils.Bn.G1.MulScalar(setup.Pk.Cp[i], w[i]))
271
272 proof.PiKp = Utils.Bn.G1.Add(proof.PiKp, Utils.Bn.G1.MulScalar(setup.Pk.Kp[i], w[i]))
273 }
274
275 hx := Utils.PF.DivisorPolynomial(px, setup.Pk.Z) // maybe move this calculation to a previous step
276
277 // piH = pkH,0 + sum ( hi * pk H,i ), where pkH = G1T, hi=hx
278 // proof.PiH = Utils.Bn.G1.Add(proof.PiH, setup.G1T[0])
279 for i := 0; i < len(hx); i++ {
280 proof.PiH = Utils.Bn.G1.Add(proof.PiH, Utils.Bn.G1.MulScalar(setup.G1T[i], hx[i]))
281 }
282
283 return proof, nil
284}
285
286// VerifyProof verifies over the BN128 the Pairings of the Proof
287func VerifyProof(circuit circuitcompiler.Circuit, setup Setup, proof Proof, publicSignals []*big.Int, debug bool) bool {
288 // e(piA, Va) == e(piA', g2)
289 pairingPiaVa := Utils.Bn.Pairing(proof.PiA, setup.Vk.Vka)
290 pairingPiapG2 := Utils.Bn.Pairing(proof.PiAp, Utils.Bn.G2.G)
291 if !Utils.Bn.Fq12.Equal(pairingPiaVa, pairingPiapG2) {
292 fmt.Println("❌ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
293 return false
294 }
295 if debug {
296 fmt.Println("✓ e(piA, Va) == e(piA', g2), valid knowledge commitment for A")
297 }
298
299 // e(Vb, piB) == e(piB', g2)
300 pairingVbPib := Utils.Bn.Pairing(setup.Vk.Vkb, proof.PiB)
301 pairingPibpG2 := Utils.Bn.Pairing(proof.PiBp, Utils.Bn.G2.G)
302 if !Utils.Bn.Fq12.Equal(pairingVbPib, pairingPibpG2) {
303 fmt.Println("❌ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
304 return false
305 }
306 if debug {
307 fmt.Println("✓ e(Vb, piB) == e(piB', g2), valid knowledge commitment for B")
308 }
309
310 // e(piC, Vc) == e(piC', g2)
311 pairingPicVc := Utils.Bn.Pairing(proof.PiC, setup.Vk.Vkc)
312 pairingPicpG2 := Utils.Bn.Pairing(proof.PiCp, Utils.Bn.G2.G)
313 if !Utils.Bn.Fq12.Equal(pairingPicVc, pairingPicpG2) {
314 fmt.Println("❌ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
315 return false
316 }
317 if debug {
318 fmt.Println("✓ e(piC, Vc) == e(piC', g2), valid knowledge commitment for C")
319 }
320
321 // Vkx, to then calculate Vkx+piA
322 vkxpia := setup.Vk.IC[0]
323 for i := 0; i < len(publicSignals); i++ {
324 vkxpia = Utils.Bn.G1.Add(vkxpia, Utils.Bn.G1.MulScalar(setup.Vk.IC[i+1], publicSignals[i]))
325 }
326
327 // e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2)
328 if !Utils.Bn.Fq12.Equal(
329 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
330 Utils.Bn.Fq12.Mul(
331 Utils.Bn.Pairing(proof.PiH, setup.Vk.Vkz),
332 Utils.Bn.Pairing(proof.PiC, Utils.Bn.G2.G))) {
333 fmt.Println("❌ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
334 return false
335 }
336 if debug {
337 fmt.Println("✓ e(Vkx+piA, piB) == e(piH, Vkz) * e(piC, g2), QAP disibility checked")
338 }
339
340 // e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB)
341 // == e(piK, g2Kgamma)
342 piApiC := Utils.Bn.G1.Add(Utils.Bn.G1.Add(vkxpia, proof.PiA), proof.PiC)
343 pairingPiACG2Kbg := Utils.Bn.Pairing(piApiC, setup.Vk.G2Kbg)
344 pairingG1KbgPiB := Utils.Bn.Pairing(setup.Vk.G1Kbg, proof.PiB)
345 pairingL := Utils.Bn.Fq12.Mul(pairingPiACG2Kbg, pairingG1KbgPiB)
346 pairingR := Utils.Bn.Pairing(proof.PiKp, setup.Vk.G2Kg)
347 if !Utils.Bn.Fq12.Equal(pairingL, pairingR) {
348 fmt.Println("❌ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
349 return false
350 }
351 if debug {
352 fmt.Println("✓ e(Vkx+piA+piC, g2KbetaKgamma) * e(g1KbetaKgamma, piB) == e(piK, g2Kgamma)")
353 }
354
355 return true
356}