diff --git a/README.md b/README.md
index d894590..86e573f 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
+## Caution, Warning, etc
 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.
@@ -27,6 +27,8 @@ Current implementation status:
 - [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
 
 
 ## Usage
diff --git a/circuitcompiler/circuit.go b/circuitcompiler/circuit.go
index 9a0338c..220e989 100644
--- a/circuitcompiler/circuit.go
+++ b/circuitcompiler/circuit.go
@@ -2,6 +2,7 @@ package circuitcompiler
 
 import (
 	"errors"
+	"fmt"
 	"math/big"
 	"strconv"
 
@@ -16,7 +17,6 @@ type Circuit struct {
 	PrivateInputs []string
 	PublicInputs  []string
 	Signals       []string
-	PublicSignals []string
 	Witness       []*big.Int
 	Constraints   []Constraint
 	R1CS          struct {
@@ -97,12 +97,12 @@ func (circ *Circuit) GenerateR1CS() ([][]*big.Int, [][]*big.Int, [][]*big.Int) {
 
 		// if existInArray(constraint.Out) {
 		if used[constraint.Out] {
-			panic(errors.New("out variable already used: " + constraint.Out))
+			// panic(errors.New("out variable already used: " + constraint.Out))
+			fmt.Println("variable already used")
 		}
 		used[constraint.Out] = true
 		if constraint.Op == "in" {
-			// TODO constraint.PublicInputs
-			for i := 0; i < len(constraint.PrivateInputs); i++ {
+			for i := 0; i <= len(circ.PublicInputs); i++ {
 				aConstraint[indexInArray(circ.Signals, constraint.Out)] = new(big.Int).Add(aConstraint[indexInArray(circ.Signals, constraint.Out)], big.NewInt(int64(1)))
 				aConstraint, used = insertVar(aConstraint, circ.Signals, constraint.Out, used)
 				bConstraint[0] = big.NewInt(int64(1))
@@ -166,8 +166,14 @@ func (circ *Circuit) CalculateWitness(privateInputs []*big.Int, publicInputs []*
 	}
 	w := r1csqap.ArrayOfBigZeros(len(circ.Signals))
 	w[0] = big.NewInt(int64(1))
+	for i, input := range publicInputs {
+		fmt.Println(i + 1)
+		fmt.Println(input)
+		w[i+1] = input
+	}
 	for i, input := range privateInputs {
-		w[i+2] = input
+		fmt.Println(i + len(publicInputs) + 1)
+		w[i+len(publicInputs)+1] = input
 	}
 	for _, constraint := range circ.Constraints {
 		if constraint.Op == "in" {
diff --git a/circuitcompiler/circuit_test.go b/circuitcompiler/circuit_test.go
index beb0625..8293d85 100644
--- a/circuitcompiler/circuit_test.go
+++ b/circuitcompiler/circuit_test.go
@@ -1,6 +1,7 @@
 package circuitcompiler
 
 import (
+	"encoding/json"
 	"fmt"
 	"math/big"
 	"strings"
@@ -21,17 +22,20 @@ func TestCircuitParser(t *testing.T) {
 			m2 = m1 * s1
 			m3 = m2 + s1
 			out = m3 + 5
+
 	*/
 
 	// flat code, where er is expected_result
+	// equals(s5, s1)
+	// s1 = s5 * 1
 	flat := `
-	func test(private x, public er):
-		aux = x*x
-		y = aux*x
-		z = x + y
-		res = z + 5
-		equals(er, res)
-		out = 1
+	func test(private s0, public s1):
+		s2 = s0*s0
+		s3 = s2*s0
+		s4 = s0 + s3
+		s5 = s4 + 5
+		s5 = s1 * one
+		out = 1 * 1
 	`
 	parser := NewParser(strings.NewReader(flat))
 	circuit, err := parser.Parse()
@@ -84,4 +88,11 @@ func TestCircuitParser(t *testing.T) {
 	w, err := circuit.CalculateWitness(privateInputs, publicInputs)
 	assert.Nil(t, err)
 	fmt.Println("w", w)
+
+	circuitJson, _ := json.Marshal(circuit)
+	fmt.Println("circuit:", string(circuitJson))
+
+	assert.Equal(t, circuit.NPublic, 1)
+	assert.Equal(t, len(circuit.PublicInputs), 1)
+	assert.Equal(t, len(circuit.PrivateInputs), 1)
 }
diff --git a/circuitcompiler/parser.go b/circuitcompiler/parser.go
index 7533bca..51c9a20 100644
--- a/circuitcompiler/parser.go
+++ b/circuitcompiler/parser.go
@@ -105,10 +105,10 @@ func (p *Parser) parseLine() (*Constraint, error) {
 		// TODO
 		return c, nil
 	}
-	if c.Literal == "out" {
-		// TODO
-		return c, nil
-	}
+	// if c.Literal == "out" {
+	//         // TODO
+	//         return c, nil
+	// }
 
 	_, lit = p.scanIgnoreWhitespace() // skip =
 	c.Literal += lit
@@ -197,16 +197,26 @@ func (p *Parser) Parse() (*Circuit, error) {
 		if !isVal {
 			circuit.Signals = addToArrayIfNotExist(circuit.Signals, constraint.V2)
 		}
-		if constraint.Out == "out" {
-			// if Out is "out", put it after first value (one) and before the inputs
+		// 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.PublicSignals = append(circuit.PublicSignals, constraint.Out)
+				// circuit.PublicInputs = append(circuit.PublicInputs, constraint.Out)
 				circuit.NPublic++
 			}
 		} else {
diff --git a/snark.go b/snark.go
index 581fc6c..e61191a 100644
--- a/snark.go
+++ b/snark.go
@@ -1,7 +1,6 @@
 package snark
 
 import (
-	"bytes"
 	"fmt"
 	"math/big"
 	"os"
@@ -96,15 +95,15 @@ func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, al
 	var err error
 
 	// input soundness
-	for i := 0; i < len(alphas); i++ {
-		for j := 0; j < len(alphas[i]); j++ {
-			if j <= circuit.NPublic {
-				if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
-					alphas[i][j] = Utils.FqR.One()
-				}
-			}
-		}
-	}
+	// for i := 0; i < len(alphas); i++ {
+	//         for j := 0; j < len(alphas[i]); j++ {
+	//                 if j <= circuit.NPublic {
+	//                         if bytes.Equal(alphas[i][j].Bytes(), Utils.FqR.Zero().Bytes()) {
+	//                                 alphas[i][j] = Utils.FqR.One()
+	//                         }
+	//                 }
+	//         }
+	// }
 
 	fmt.Println("alphas[1]", alphas[1])
 
@@ -217,7 +216,8 @@ func GenerateTrustedSetup(witnessLength int, circuit circuitcompiler.Circuit, al
 
 	// z pol
 	zpol := []*big.Int{big.NewInt(int64(1))}
-	for i := 1; i < len(circuit.Constraints); i++ {
+	// for i := 0; i < len(circuit.Constraints); i++ {
+	for i := 1; i < len(alphas)-1; i++ {
 		zpol = Utils.PF.Mul(
 			zpol,
 			[]*big.Int{
diff --git a/snark_test.go b/snark_test.go
index 0fe0de6..d416bfe 100644
--- a/snark_test.go
+++ b/snark_test.go
@@ -1,6 +1,7 @@
 package snark
 
 import (
+	"bytes"
 	"encoding/json"
 	"fmt"
 	"math/big"
@@ -14,14 +15,19 @@ import (
 )
 
 func TestZkFromFlatCircuitCode(t *testing.T) {
-
 	// compile circuit and get the R1CS
+
+	// circuit function
+	// y = x^3 + x + 5
 	flatCode := `
-	func test(x):
-		aux = x*x
-		y = aux*x
-		z = x + y
-		out = z + 5
+	func test(private s0, public s1):
+		s2 = s0 * s0
+		s3 = s2 * s0
+		s4 = s3 + s0
+		s5 = s4 + 5
+		s1 = s5 * 1
+		s5 = s1 * 1
+		out = 1 * 1
 	`
 	fmt.Print("\nflat code of the circuit:")
 	fmt.Println(flatCode)
@@ -36,10 +42,14 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 
 	b3 := big.NewInt(int64(3))
 	privateInputs := []*big.Int{b3}
+	b35 := big.NewInt(int64(35))
+	publicSignals := []*big.Int{b35}
+
 	// wittness
-	w, err := circuit.CalculateWitness(privateInputs)
+	w, err := circuit.CalculateWitness(privateInputs, publicSignals)
 	assert.Nil(t, err)
-	fmt.Println("\nwitness", w)
+	fmt.Println("\n", circuit.Signals)
+	fmt.Println("witness", w)
 
 	// flat code to R1CS
 	fmt.Println("\ngenerating R1CS from flat code")
@@ -58,6 +68,7 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 	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))
@@ -65,9 +76,6 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 	fmt.Println("cx length", len(cx))
 	fmt.Println("px length", len(px))
 	fmt.Println("px[last]", px[0])
-	px0 := Utils.PF.F.Add(px[0], big.NewInt(int64(88)))
-	fmt.Println(px0)
-	assert.Equal(t, px0.Bytes(), Utils.PF.F.Zero().Bytes())
 
 	hxQAP := Utils.PF.DivisorPolynomial(px, zxQAP)
 	fmt.Println("hx length", len(hxQAP))
@@ -83,7 +91,7 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 
 	div, rem := Utils.PF.Div(px, zxQAP)
 	assert.Equal(t, hxQAP, div)
-	assert.Equal(t, rem, r1csqap.ArrayOfBigZeros(4))
+	assert.Equal(t, rem, r1csqap.ArrayOfBigZeros(6))
 
 	// calculate trusted setup
 	setup, err := GenerateTrustedSetup(len(w), *circuit, alphas, betas, gammas)
@@ -97,6 +105,9 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 	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(6))
 
 	assert.Equal(t, px, Utils.PF.Mul(hxQAP, zxQAP))
 	// hx==px/zx so px==hx*zx
@@ -117,13 +128,16 @@ func TestZkFromFlatCircuitCode(t *testing.T) {
 
 	// fmt.Println("public signals:", proof.PublicSignals)
 	fmt.Println("\nwitness", w)
-	// b1 := big.NewInt(int64(1))
-	b35 := big.NewInt(int64(35))
-	// publicSignals := []*big.Int{b1, b35}
-	publicSignals := []*big.Int{b35}
+	b35Verif := big.NewInt(int64(35))
+	publicSignalsVerif := []*big.Int{b35Verif}
 	before := time.Now()
-	assert.True(t, VerifyProof(*circuit, setup, proof, publicSignals, true))
+	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(34))
+	wrongPublicSignalsVerif := []*big.Int{bOtherWrongPublic}
+	assert.True(t, !VerifyProof(*circuit, setup, proof, wrongPublicSignalsVerif, true))
 }
 
 /*