@@ -9,6 +9,14 @@ ### Example
 
 - Generate Proof
 ```go
+import (
+  "github.com/iden3/go-circom-prover-verifier/parsers"
+  "github.com/iden3/go-circom-prover-verifier/prover"
+  "github.com/iden3/go-circom-prover-verifier/verifier"
+)
+
+[...]
+
 // read ProvingKey & Witness files
 provingKeyJson, _ := ioutil.ReadFile("../testdata/small/proving_key.json")
 witnessJson, _ := ioutil.ReadFile("../testdata/small/witness.json")
@@ -20,7 +28,7 @@ // parse Witness
 w, _ := parsers.ParseWitness(witnessJson)
 
 // generate the proof
-proof, pubSignals, _ := GenerateProof(pk, w)
+proof, pubSignals, _ := prover.GenerateProof(pk, w)
 
 // print proof & publicSignals
 proofStr, _ := parsers.ProofToJson(proof)
@@ -42,7 +50,7 @@ proof, _ := parsers.ParseProof(proofJson)
 vk, _ := parsers.ParseVk(vkJson)
 
 // verify the proof with the given verificationKey & publicSignals
-v := Verify(vk, proof, public)
+v := verifier.Verify(vk, proof, public)
 fmt.Println(v)
 ```
 

@@ -7,6 +7,7 @@
 require (
 	github.com/ethereum/go-ethereum v1.9.12
 	github.com/iden3/go-circom-prover-verifier v0.0.0-00010101000000-000000000000
+	github.com/iden3/go-iden3-crypto v0.0.4
 	github.com/stretchr/testify v1.5.1
 	golang.org/x/crypto v0.0.0-20200311171314-f7b00557c8c4
 )

@@ -32,6 +32,7 @@ github.com/cpuguy83/go-md2man/v2 v2.0.0-20190314233015-f79a8a8ca69d/go.mod h1:maD7wRr/U5Z6m/iR4s+kqSMx2CaBsrgA7czyZG/E6dU=
 github.com/davecgh/go-spew v1.1.0/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
 github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c=
 github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
+github.com/dchest/blake512 v1.0.0/go.mod h1:FV1x7xPPLWukZlpDpWQ88rF/SFwZ5qbskrzhLMB92JI=
 github.com/deckarep/golang-set v0.0.0-20180603214616-504e848d77ea/go.mod h1:93vsz/8Wt4joVM7c2AVqh+YRMiUSc14yDtF28KmMOgQ=
 github.com/dgrijalva/jwt-go v3.2.0+incompatible/go.mod h1:E3ru+11k8xSBh+hMPgOLZmtrrCbhqsmaPHjLKYnJCaQ=
 github.com/dgryski/go-sip13 v0.0.0-20181026042036-e10d5fee7954/go.mod h1:vAd38F8PWV+bWy6jNmig1y/TA+kYO4g3RSRF0IAv0no=
@@ -40,6 +41,7 @@ github.com/docker/docker v1.4.2-0.20180625184442-8e610b2b55bf/go.mod h1:eEKB0N0r5NX/I1kEveEz05bcu8tLC/8azJZsviup8Sk=
 github.com/dop251/goja v0.0.0-20200219165308-d1232e640a87/go.mod h1:Mw6PkjjMXWbTj+nnj4s3QPXq1jaT0s5pC0iFD4+BOAA=
 github.com/edsrzf/mmap-go v0.0.0-20160512033002-935e0e8a636c/go.mod h1:YO35OhQPt3KJa3ryjFM5Bs14WD66h8eGKpfaBNrHW5M=
 github.com/elastic/gosigar v0.8.1-0.20180330100440-37f05ff46ffa/go.mod h1:cdorVVzy1fhmEqmtgqkoE3bYtCfSCkVyjTyCIo22xvs=
+github.com/ethereum/go-ethereum v1.8.27/go.mod h1:PwpWDrCLZrV+tfrhqqF6kPknbISMHaJv9Ln3kPCZLwY=
 github.com/ethereum/go-ethereum v1.9.12 h1:EPtimwsp/KGDSiXcNunzsI4kefdsMHZGJntKx3fvbaI=
 github.com/ethereum/go-ethereum v1.9.12/go.mod h1:PvsVkQmhZFx92Y+h2ylythYlheEDt/uBgFbl61Js/jo=
 github.com/fatih/color v1.3.0/go.mod h1:Zm6kSWBoL9eyXnKyktHP6abPY2pDugNf5KwzbycvMj4=
@@ -62,6 +64,8 @@ github.com/hashicorp/golang-lru v0.0.0-20160813221303-0a025b7e63ad/go.mod h1:/m3WP610KZHVQ1SGc6re/UDhFvYD7pJ4Ao+sR/qLZy8=
 github.com/hpcloud/tail v1.0.0/go.mod h1:ab1qPbhIpdTxEkNHXyeSf5vhxWSCs/tWer42PpOxQnU=
 github.com/huin/goupnp v0.0.0-20161224104101-679507af18f3/go.mod h1:MZ2ZmwcBpvOoJ22IJsc7va19ZwoheaBk43rKg12SKag=
 github.com/iden3/go-circom-prover-verifier v0.0.0-20200409092022-a439651fafb0 h1:iyMhjvq7JKtKT6kEKuKghXiwanoa4/u7JSCKqUBqvds=
+github.com/iden3/go-iden3-crypto v0.0.4 h1:rGQEFBvX6d4fDxqkQTizVq5UefB+xdZAg8j5FQ6uv6g=
+github.com/iden3/go-iden3-crypto v0.0.4/go.mod h1:LLcgB7DLWAUs+8eBSKne+ZHy5z7xtAmlYlEz0M9M8gE=
 github.com/influxdata/influxdb v1.2.3-0.20180221223340-01288bdb0883/go.mod h1:qZna6X/4elxqT3yI9iZYdZrWWdeFOOprn86kgg4+IzY=
 github.com/jackpal/go-nat-pmp v1.0.2-0.20160603034137-1fa385a6f458/go.mod h1:QPH045xvCAeXUZOxsnwmrtiCoxIr9eob+4orBN1SBKc=
 github.com/jmespath/go-jmespath v0.0.0-20180206201540-c2b33e8439af/go.mod h1:Nht3zPeWKUH0NzdCt2Blrr5ys8VGpn0CEB0cQHVjt7k=
@@ -120,6 +124,7 @@ github.com/tyler-smith/go-bip39 v1.0.1-0.20181017060643-dbb3b84ba2ef/go.mod h1:sJ5fKU0s6JVwZjjcUEX2zFOnvq0ASQ2K9Zr6cf67kNs=
 github.com/urfave/cli v1.22.1/go.mod h1:Gos4lmkARVdJ6EkW0WaNv/tZAAMe9V7XWyB60NtXRu0=
 github.com/wsddn/go-ecdh v0.0.0-20161211032359-48726bab9208/go.mod h1:IotVbo4F+mw0EzQ08zFqg7pK3FebNXpaMsRy2RT+Ees=
 golang.org/x/crypto v0.0.0-20190308221718-c2843e01d9a2/go.mod h1:djNgcEr1/C05ACkg1iLfiJU5Ep61QUkGW8qpdssI0+w=
+golang.org/x/crypto v0.0.0-20190621222207-cc06ce4a13d4/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI=
 golang.org/x/crypto v0.0.0-20200311171314-f7b00557c8c4 h1:QmwruyY+bKbDDL0BaglrbZABEali68eoMFhTZpCjYVA=
 golang.org/x/crypto v0.0.0-20200311171314-f7b00557c8c4/go.mod h1:LzIPMQfyMNhhGPhUkYOs5KpL4U8rLKemX1yGLhDgUto=
 golang.org/x/net v0.0.0-20180906233101-161cd47e91fd/go.mod h1:mL1N/T3taQHkDXs73rZJwtUhF3w3ftmwwsq0BUmARs4=

@@ -3,12 +3,21 @@
 import (
 	"bytes"
 	"math/big"
+
+	"github.com/iden3/go-iden3-crypto/ff"
 )
 
 func arrayOfZeroes(n int) []*big.Int {
 	var r []*big.Int
 	for i := 0; i < n; i++ {
 		r = append(r, new(big.Int).SetInt64(0))
+	}
+	return r
+}
+func arrayOfZeroesE(n int) []*ff.Element {
+	var r []*ff.Element
+	for i := 0; i < n; i++ {
+		r = append(r, ff.NewElement())
 	}
 	return r
 }
@@ -75,6 +84,17 @@ }
 	return r
 }
 
+func polynomialSubE(a, b []*ff.Element) []*ff.Element {
+	r := arrayOfZeroesE(max(len(a), len(b)))
+	for i := 0; i < len(a); i++ {
+		r[i].Add(r[i], a[i])
+	}
+	for i := 0; i < len(b); i++ {
+		r[i].Sub(r[i], b[i])
+	}
+	return r
+}
+
 func polynomialMul(a, b []*big.Int) []*big.Int {
 	r := arrayOfZeroes(len(a) + len(b) - 1)
 	for i := 0; i < len(a); i++ {
@@ -85,6 +105,16 @@ }
 	return r
 }
 
+func polynomialMulE(a, b []*ff.Element) []*ff.Element {
+	r := arrayOfZeroesE(len(a) + len(b) - 1)
+	for i := 0; i < len(a); i++ {
+		for j := 0; j < len(b); j++ {
+			r[i+j].Add(r[i+j], ff.NewElement().Mul(a[i], b[j]))
+		}
+	}
+	return r
+}
+
 func polynomialDiv(a, b []*big.Int) ([]*big.Int, []*big.Int) {
 	// https://en.wikipedia.org/wiki/Division_algorithm
 	r := arrayOfZeroes(len(a) - len(b) + 1)
@@ -100,3 +130,31 @@ rem = aux2[:len(aux2)-1]
 	}
 	return r, rem
 }
+
+func polynomialDivE(a, b []*ff.Element) ([]*ff.Element, []*ff.Element) {
+	// https://en.wikipedia.org/wiki/Division_algorithm
+	r := arrayOfZeroesE(len(a) - len(b) + 1)
+	rem := a
+	for len(rem) >= len(b) {
+		l := ff.NewElement().Div(rem[len(rem)-1], b[len(b)-1])
+		pos := len(rem) - len(b)
+		r[pos] = l
+		aux := arrayOfZeroesE(pos)
+		aux1 := append(aux, l)
+		aux2 := polynomialSubE(rem, polynomialMulE(b, aux1))
+		rem = aux2[:len(aux2)-1]
+	}
+	return r, rem
+}
+
+// once https://github.com/iden3/go-iden3-crypto/pull/22 is merged, use the fucntion from there
+func ElementArrayToBigIntArray(e []*ff.Element) []*big.Int {
+	var o []*big.Int
+	for i := range e {
+		ei := e[i]
+		bi := big.NewInt(0)
+		ei.ToBigIntRegular(bi)
+		o = append(o, bi)
+	}
+	return o
+}

@@ -0,0 +1,66 @@
+package prover
+
+import (
+	"crypto/rand"
+	"math/big"
+	"testing"
+
+	cryptoConstants "github.com/iden3/go-iden3-crypto/constants"
+	"github.com/iden3/go-iden3-crypto/utils"
+)
+
+func randBI() *big.Int {
+	maxbits := 253
+	b := make([]byte, (maxbits/8)-1)
+	_, err := rand.Read(b)
+	if err != nil {
+		panic(err)
+	}
+	r := new(big.Int).SetBytes(b)
+	return new(big.Int).Mod(r, cryptoConstants.Q)
+}
+
+func BenchmarkArithmetic(b *testing.B) {
+	// generate arrays with bigint
+	var p, q []*big.Int
+	for i := 0; i < 1000; i++ {
+		pi := randBI()
+		p = append(p, pi)
+	}
+	for i := 1000 - 1; i >= 0; i-- {
+		q = append(q, p[i])
+	}
+	pe := utils.BigIntArrayToElementArray(p)
+	qe := utils.BigIntArrayToElementArray(q)
+
+	b.Run("polynomialSub", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialSub(p, q)
+		}
+	})
+	b.Run("polynomialSubE", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialSubE(pe, qe)
+		}
+	})
+	b.Run("polynomialMul", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialMul(p, q)
+		}
+	})
+	b.Run("polynomialMulE", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialMulE(pe, qe)
+		}
+	})
+	b.Run("polynomialDiv", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialDiv(p, q)
+		}
+	})
+	b.Run("polynomialDivE", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialDivE(pe, qe)
+		}
+	})
+}

@@ -3,11 +3,13 @@
 import (
 	"math"
 	"math/big"
+
+	"github.com/iden3/go-iden3-crypto/ff"
 )
 
 type rootsT struct {
-	roots [][]*big.Int
-	w     []*big.Int
+	roots [][]*ff.Element
+	w     []*ff.Element
 }
 
 func newRootsT() rootsT {
@@ -19,15 +21,15 @@ for rem.Bit(0) == 0 { // rem.Bit==0 when even
 		s++
 		rem = new(big.Int).Rsh(rem, 1)
 	}
-	roots.w = make([]*big.Int, s+1)
-	roots.w[s] = fExp(big.NewInt(5), rem)
+	roots.w = make([]*ff.Element, s+1)
+	roots.w[s] = ff.NewElement().SetBigInt(fExp(big.NewInt(5), rem))
 
 	n := s - 1
 	for n >= 0 {
-		roots.w[n] = fMul(roots.w[n+1], roots.w[n+1])
+		roots.w[n] = ff.NewElement().Mul(roots.w[n+1], roots.w[n+1])
 		n--
 	}
-	roots.roots = make([][]*big.Int, 50) // TODO WIP
+	roots.roots = make([][]*ff.Element, 50) // TODO WIP
 
 	roots.setRoots(15)
 	return roots
@@ -35,25 +37,25 @@ }
 
 func (roots rootsT) setRoots(n int) {
 	for i := n; i >= 0 && nil == roots.roots[i]; i-- { // TODO tmp i<=len(r)
-		r := big.NewInt(1)
+		r := ff.NewElement().SetBigInt(big.NewInt(1))
 		nroots := 1 << i
-		var rootsi []*big.Int
+		var rootsi []*ff.Element
 		for j := 0; j < nroots; j++ {
 			rootsi = append(rootsi, r)
-			r = fMul(r, roots.w[i])
+			r = ff.NewElement().Mul(r, roots.w[i])
 		}
 		roots.roots[i] = rootsi
 	}
 }
 
-func fft(roots rootsT, pall []*big.Int, bits, offset, step int) []*big.Int {
+func fft(roots rootsT, pall []*ff.Element, bits, offset, step int) []*ff.Element {
 	n := 1 << bits
 	if n == 1 {
-		return []*big.Int{pall[offset]}
+		return []*ff.Element{pall[offset]}
 	} else if n == 2 {
-		return []*big.Int{
-			fAdd(pall[offset], pall[offset+step]), // TODO tmp
-			fSub(pall[offset], pall[offset+step]),
+		return []*ff.Element{
+			ff.NewElement().Add(pall[offset], pall[offset+step]), // TODO tmp
+			ff.NewElement().Sub(pall[offset], pall[offset+step]),
 		}
 	}
 
@@ -61,17 +63,16 @@ ndiv2 := n >> 1
 	p1 := fft(roots, pall, bits-1, offset, step*2)
 	p2 := fft(roots, pall, bits-1, offset+step, step*2)
 
-	// var out []*big.Int
-	out := make([]*big.Int, n)
+	out := make([]*ff.Element, n)
 	for i := 0; i < ndiv2; i++ {
 		// fmt.Println(i, len(roots.roots))
-		out[i] = fAdd(p1[i], fMul(roots.roots[bits][i], p2[i]))
-		out[i+ndiv2] = fSub(p1[i], fMul(roots.roots[bits][i], p2[i]))
+		out[i] = ff.NewElement().Add(p1[i], ff.NewElement().Mul(roots.roots[bits][i], p2[i]))
+		out[i+ndiv2] = ff.NewElement().Sub(p1[i], ff.NewElement().Mul(roots.roots[bits][i], p2[i]))
 	}
 	return out
 }
 
-func ifft(p []*big.Int) []*big.Int {
+func ifft(p []*ff.Element) []*ff.Element {
 	if len(p) <= 1 {
 		return p
 	}
@@ -82,20 +83,20 @@ m := 1 << int(bits)
 	ep := extend(p, m)
 	res := fft(roots, ep, int(bits), 0, 1)
 
-	twoinvm := fInv(fMul(big.NewInt(1), big.NewInt(int64(m))))
+	twoinvm := ff.NewElement().SetBigInt(fInv(fMul(big.NewInt(1), big.NewInt(int64(m)))))
 
-	var resn []*big.Int
+	var resn []*ff.Element
 	for i := 0; i < m; i++ {
-		resn = append(resn, fMul(res[(m-i)%m], twoinvm))
+		resn = append(resn, ff.NewElement().Mul(res[(m-i)%m], twoinvm))
 	}
 
 	return resn
 }
 
-func extend(p []*big.Int, e int) []*big.Int {
+func extend(p []*ff.Element, e int) []*ff.Element {
 	if e == len(p) {
 		return p
 	}
-	z := arrayOfZeroes(e - len(p))
+	z := arrayOfZeroesE(e - len(p))
 	return append(p, z...)
 }

@@ -6,6 +6,7 @@ "math/big"
 
 	bn256 "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare"
 	"github.com/iden3/go-circom-prover-verifier/types"
+	"github.com/iden3/go-iden3-crypto/utils"
 )
 
 // Proof is the data structure of the Groth16 zkSNARK proof
@@ -123,13 +124,18 @@ for j := range pk.PolsC[i] {
 			polCT[j] = fAdd(polCT[j], fMul(w[i], pk.PolsC[i][j]))
 		}
 	}
-	polAS := ifft(polAT)
-	polBS := ifft(polBT)
+	polATe := utils.BigIntArrayToElementArray(polAT)
+	polBTe := utils.BigIntArrayToElementArray(polBT)
+	polCTe := utils.BigIntArrayToElementArray(polCT)
+
+	polASe := ifft(polATe)
+	polBSe := ifft(polBTe)
+	polABSe := polynomialMulE(polASe, polBSe)
+
+	polCSe := ifft(polCTe)
 
-	polABS := polynomialMul(polAS, polBS)
-	polCS := ifft(polCT)
-	polABCS := polynomialSub(polABS, polCS)
+	polABCSe := polynomialSubE(polABSe, polCSe)
 
-	hS := polABCS[m:]
-	return hS
+	hSe := polABCSe[m:]
+	return ElementArrayToBigIntArray(hSe)
 }