diff --git a/README.md b/README.md
index 86e573f..354c59a 100644
--- a/README.md
+++ b/README.md
@@ -6,7 +6,7 @@ zkSNARK library implementation in Go
 - `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf
 - `Pinocchio: Nearly practical verifiable computation`, Bryan Parno, Craig Gentry, Jon Howell, Mariana Raykova https://eprint.iacr.org/2013/279.pdf
 
-## Caution, Warning, etc
+## Caution, Warning
 Implementation of the zkSNARK [Pinocchio protocol](https://eprint.iacr.org/2013/279.pdf) from scratch in Go to understand the concepts. Do not use in production.
 
 Not finished, implementing this in my free time to understand it better, so I don't have much time.
@@ -15,17 +15,15 @@ Current implementation status:
 - [x] Finite Fields (1, 2, 6, 12) operations
 - [x] G1 and G2 curve operations
 - [x] BN128 Pairing
-- [ ] circuit code compiler
+- [x] circuit code compiler
 	- [ ] code to flat code (improve circuit compiler)
 	- [x] flat code compiler
-	- [ ] private & public inputs. fix circuit compiler
 - [x] circuit to R1CS
 - [x] polynomial operations
 - [x] R1CS to QAP
 - [x] generate trusted setup
 - [x] generate proofs
 - [x] verify proofs with BN128 pairing
-	- [ ] fix 4th pairing proofs generation & verification: ê(Vkx+piA, piB) == ê(piH, Vkz) * ê(piC, G2)
 - [ ] move witness calculation outside the setup phase
 - [ ] Groth16
 - [ ] multiple optimizations
diff --git a/circuitcompiler/parser.go b/circuitcompiler/parser.go
index 51c9a20..d168669 100644
--- a/circuitcompiler/parser.go
+++ b/circuitcompiler/parser.go
@@ -103,6 +103,8 @@ func (p *Parser) parseLine() (*Constraint, error) {
 		params := strings.Split(varsString, ",")
 		fmt.Println("params", params)
 		// TODO
+		c.V1 = params[0]
+		c.V2 = params[1]
 		return c, nil
 	}
 	// if c.Literal == "out" {
@@ -163,21 +165,24 @@ func (p *Parser) Parse() (*Circuit, error) {
 		fmt.Println(constraint)
 		if constraint.Literal == "func" {
 			// one constraint for each input
-			for _, in := range constraint.PrivateInputs {
+			for _, in := range constraint.PublicInputs {
 				newConstr := &Constraint{
 					Op:  "in",
 					Out: in,
 				}
 				circuit.Constraints = append(circuit.Constraints, *newConstr)
 				nInputs++
+				circuit.Signals = addToArrayIfNotExist(circuit.Signals, in)
+				circuit.NPublic++
 			}
-			for _, in := range constraint.PublicInputs {
+			for _, in := range constraint.PrivateInputs {
 				newConstr := &Constraint{
 					Op:  "in",
 					Out: in,
 				}
 				circuit.Constraints = append(circuit.Constraints, *newConstr)
 				nInputs++
+				circuit.Signals = addToArrayIfNotExist(circuit.Signals, in)
 			}
 			circuit.PublicInputs = constraint.PublicInputs
 			circuit.PrivateInputs = constraint.PrivateInputs
@@ -186,6 +191,22 @@ func (p *Parser) Parse() (*Circuit, error) {
 		if constraint.Literal == "equals" {
 			// TODO
 			fmt.Println("circuit.Signals", circuit.Signals)
+			constr1 := &Constraint{
+				Op:      "*",
+				V1:      constraint.V2,
+				V2:      "1",
+				Out:     constraint.V1,
+				Literal: "equals(" + constraint.V1 + ", " + constraint.V2 + "): " + constraint.V1 + "==" + constraint.V2 + " * 1",
+			}
+			circuit.Constraints = append(circuit.Constraints, *constr1)
+			constr2 := &Constraint{
+				Op:      "*",
+				V1:      constraint.V1,
+				V2:      "1",
+				Out:     constraint.V2,
+				Literal: "equals(" + constraint.V1 + ", " + constraint.V2 + "): " + constraint.V2 + "==" + constraint.V1 + " * 1",
+			}
+			circuit.Constraints = append(circuit.Constraints, *constr2)
 			continue
 		}
 		circuit.Constraints = append(circuit.Constraints, *constraint)
@@ -197,31 +218,26 @@ func (p *Parser) Parse() (*Circuit, error) {
 		if !isVal {
 			circuit.Signals = addToArrayIfNotExist(circuit.Signals, constraint.V2)
 		}
-		// fmt.Println("---")
-		// fmt.Println(circuit.PublicInputs[0])
-		// fmt.Println(constraint.Out)
-		// fmt.Println(constraint.Out == circuit.PublicInputs[0])
-		// fmt.Println("---")
 
 		// if constraint.Out == "out" {
 		// if Out is "out", put it after first value (one) and before the inputs
 		// if constraint.Out == circuit.PublicInputs[0] {
-		if existInArray(circuit.PublicInputs, constraint.Out) {
-			// if Out is a public signal, put it after first value (one) and before the private inputs
-			if !existInArray(circuit.Signals, constraint.Out) {
-				// if already don't exists in signal array
-				signalsCopy := copyArray(circuit.Signals)
-				var auxSignals []string
-				auxSignals = append(auxSignals, signalsCopy[0])
-				auxSignals = append(auxSignals, constraint.Out)
-				auxSignals = append(auxSignals, signalsCopy[1:]...)
-				circuit.Signals = auxSignals
-				// circuit.PublicInputs = append(circuit.PublicInputs, constraint.Out)
-				circuit.NPublic++
-			}
-		} else {
-			circuit.Signals = addToArrayIfNotExist(circuit.Signals, constraint.Out)
-		}
+		// if existInArray(circuit.PublicInputs, constraint.Out) {
+		//         // if Out is a public signal, put it after first value (one) and before the private inputs
+		//         if !existInArray(circuit.Signals, constraint.Out) {
+		//                 // if already don't exists in signal array
+		//                 signalsCopy := copyArray(circuit.Signals)
+		//                 var auxSignals []string
+		//                 auxSignals = append(auxSignals, signalsCopy[0])
+		//                 auxSignals = append(auxSignals, constraint.Out)
+		//                 auxSignals = append(auxSignals, signalsCopy[1:]...)
+		//                 circuit.Signals = auxSignals
+		//                 // circuit.PublicInputs = append(circuit.PublicInputs, constraint.Out)
+		//                 circuit.NPublic++
+		//         }
+		// } else {
+		circuit.Signals = addToArrayIfNotExist(circuit.Signals, constraint.Out)
+		// }
 	}
 	circuit.NVars = len(circuit.Signals)
 	circuit.NSignals = len(circuit.Signals)
diff --git a/snark_test.go b/snark_test.go
index d416bfe..147cf96 100644
--- a/snark_test.go
+++ b/snark_test.go
@@ -25,8 +25,7 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 		s3 = s2 * s0
 		s4 = s3 + s0
 		s5 = s4 + 5
-		s1 = s5 * 1
-		s5 = s1 * 1
+		equals(s1, s5)
 		out = 1 * 1
 	`
 	fmt.Print("\nflat code of the circuit:")
@@ -127,6 +126,7 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 	// fmt.Println(proof)
 
 	// fmt.Println("public signals:", proof.PublicSignals)
+	fmt.Println("\n", circuit.Signals)
 	fmt.Println("\nwitness", w)
 	b35Verif := big.NewInt(int64(35))
 	publicSignalsVerif := []*big.Int{b35Verif}
@@ -140,31 +140,38 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 	assert.True(t, !VerifyProof(*circuit, setup, proof, wrongPublicSignalsVerif, true))
 }
 
-/*
 func TestZkMultiplication(t *testing.T) {
-
-	// compile circuit and get the R1CS
 	flatCode := `
-	func test(a, b):
-		out = a * b
+	func test(private a, private b, public c):
+		d = a * b
+		equals(c, d)
+		out = 1 * 1
 	`
+	fmt.Print("\nflat code of the circuit:")
+	fmt.Println(flatCode)
 
 	// parse the code
 	parser := circuitcompiler.NewParser(strings.NewReader(flatCode))
 	circuit, err := parser.Parse()
 	assert.Nil(t, err)
+	fmt.Println("\ncircuit data:", circuit)
+	circuitJson, _ := json.Marshal(circuit)
+	fmt.Println("circuit:", string(circuitJson))
 
 	b3 := big.NewInt(int64(3))
 	b4 := big.NewInt(int64(4))
-	inputs := []*big.Int{b3, b4}
+	privateInputs := []*big.Int{b3, b4}
+	b12 := big.NewInt(int64(12))
+	publicSignals := []*big.Int{b12}
+
 	// wittness
-	w, err := circuit.CalculateWitness(inputs)
+	w, err := circuit.CalculateWitness(privateInputs, publicSignals)
 	assert.Nil(t, err)
-
-	fmt.Println("circuit")
-	fmt.Println(circuit.NPublic)
+	fmt.Println("\n", circuit.Signals)
+	fmt.Println("witness", w)
 
 	// flat code to R1CS
+	fmt.Println("\ngenerating R1CS from flat code")
 	a, b, c := circuit.GenerateR1CS()
 	fmt.Println("\nR1CS:")
 	fmt.Println("a:", a)
@@ -172,43 +179,87 @@ func TestZkMultiplication(t *testing.T) {
 	fmt.Println("c:", c)
 
 	// R1CS to QAP
-	alphas, betas, gammas, zx := Utils.PF.R1CSToQAP(a, b, c)
+	// TODO zxQAP is not used and is an old impl, bad calculated. TODO remove
+	alphas, betas, gammas, zxQAP := Utils.PF.R1CSToQAP(a, b, c)
 	fmt.Println("qap")
-	fmt.Println("alphas", alphas)
-	fmt.Println("betas", betas)
-	fmt.Println("gammas", gammas)
+	fmt.Println("alphas", len(alphas))
+	fmt.Println("alphas[1]", alphas[1])
+	fmt.Println("betas", len(betas))
+	fmt.Println("gammas", len(gammas))
+	fmt.Println("zx length", len(zxQAP))
+	assert.True(t, !bytes.Equal(alphas[1][1].Bytes(), big.NewInt(int64(0)).Bytes()))
 
 	ax, bx, cx, px := Utils.PF.CombinePolynomials(w, alphas, betas, gammas)
+	fmt.Println("ax length", len(ax))
+	fmt.Println("bx length", len(bx))
+	fmt.Println("cx length", len(cx))
+	fmt.Println("px length", len(px))
+	fmt.Println("px[last]", px[0])
 
-	hx := Utils.PF.DivisorPolynomial(px, zx)
+	hxQAP := Utils.PF.DivisorPolynomial(px, zxQAP)
+	fmt.Println("hx length", len(hxQAP))
 
 	// hx==px/zx so px==hx*zx
-	assert.Equal(t, px, Utils.PF.Mul(hx, zx))
+	assert.Equal(t, px, Utils.PF.Mul(hxQAP, zxQAP))
 
 	// p(x) = a(x) * b(x) - c(x) == h(x) * z(x)
 	abc := Utils.PF.Sub(Utils.PF.Mul(ax, bx), cx)
 	assert.Equal(t, abc, px)
-	hz := Utils.PF.Mul(hx, zx)
-	assert.Equal(t, abc, hz)
+	hzQAP := Utils.PF.Mul(hxQAP, zxQAP)
+	assert.Equal(t, abc, hzQAP)
 
-	div, rem := Utils.PF.Div(px, zx)
-	assert.Equal(t, hx, div)
-	assert.Equal(t, rem, r1csqap.ArrayOfBigZeros(1))
+	div, rem := Utils.PF.Div(px, zxQAP)
+	assert.Equal(t, hxQAP, div)
+	assert.Equal(t, rem, r1csqap.ArrayOfBigZeros(4))
 
 	// calculate trusted setup
-	setup, err := GenerateTrustedSetup(len(w), *circuit, alphas, betas, gammas, zx)
+	setup, err := GenerateTrustedSetup(len(w), *circuit, alphas, betas, gammas)
 	assert.Nil(t, err)
+	fmt.Println("\nt:", setup.Toxic.T)
 
-	// piA = g1 * A(t), piB = g2 * B(t), piC = g1 * C(t), piH = g1 * H(t)
-	proof, err := GenerateProofs(*circuit, setup, hx, w)
+	// zx and setup.Pk.Z should be the same (currently not, the correct one is the calculation used inside GenerateTrustedSetup function), the calculation is repeated. TODO avoid repeating calculation
+	// assert.Equal(t, zxQAP, setup.Pk.Z)
+
+	fmt.Println("hx pk.z", hxQAP)
+	hx := Utils.PF.DivisorPolynomial(px, setup.Pk.Z)
+	fmt.Println("hx pk.z", hx)
+	// assert.Equal(t, hxQAP, hx)
+	div, rem = Utils.PF.Div(px, setup.Pk.Z)
+	assert.Equal(t, hx, div)
+	assert.Equal(t, rem, r1csqap.ArrayOfBigZeros(4))
+
+	assert.Equal(t, px, Utils.PF.Mul(hxQAP, zxQAP))
+	// hx==px/zx so px==hx*zx
+	assert.Equal(t, px, Utils.PF.Mul(hx, setup.Pk.Z))
+
+	// check length of polynomials H(x) and Z(x)
+	assert.Equal(t, len(hx), len(px)-len(setup.Pk.Z)+1)
+	assert.Equal(t, len(hxQAP), len(px)-len(zxQAP)+1)
+
+	// fmt.Println("pk.Z", len(setup.Pk.Z))
+	// fmt.Println("zxQAP", len(zxQAP))
+
+	proof, err := GenerateProofs(*circuit, setup, w, px)
 	assert.Nil(t, err)
 
-	// assert.True(t, VerifyProof(*circuit, setup, proof, false))
-	b35 := big.NewInt(int64(35))
-	publicSignals := []*big.Int{b35}
-	assert.True(t, VerifyProof(*circuit, setup, proof, publicSignals, true))
+	// fmt.Println("\n proofs:")
+	// fmt.Println(proof)
+
+	// fmt.Println("public signals:", proof.PublicSignals)
+	fmt.Println("\n", circuit.Signals)
+	fmt.Println("\nwitness", w)
+	b12Verif := big.NewInt(int64(12))
+	publicSignalsVerif := []*big.Int{b12Verif}
+	before := time.Now()
+	assert.True(t, VerifyProof(*circuit, setup, proof, publicSignalsVerif, true))
+	fmt.Println("verify proof time elapsed:", time.Since(before))
+
+	// check that with another public input the verification returns false
+	bOtherWrongPublic := big.NewInt(int64(11))
+	wrongPublicSignalsVerif := []*big.Int{bOtherWrongPublic}
+	assert.True(t, !VerifyProof(*circuit, setup, proof, wrongPublicSignalsVerif, true))
 }
-*/
+
 /*
 func TestZkFromHardcodedR1CS(t *testing.T) {
 	b0 := big.NewInt(int64(0))