Update to goff v0.2.0

2026-02-07 11:36:41 +01:00 · 2020-04-08 10:47:26 +02:00
21 changed files with 1487 additions and 461 deletions
--- a/.github/workflows/lint.yml
+++ b/.github/workflows/lint.yml
@@ -1,16 +0,0 @@
-on: [ push, pull_request ]
-name: Lint
-jobs:
-  lint:
-    runs-on: ubuntu-latest
-    steps:
-    - name: Install Go
-      uses: actions/setup-go@v1
-      with:
-        go-version: 1.14.x
-    - name: Checkout code
-      uses: actions/checkout@v2
-    - name: Lint
-      run: |
-        curl -sSfL https://raw.githubusercontent.com/golangci/golangci-lint/master/install.sh | sh -s -- -b $(go env GOPATH)/bin v1.24.0
-        $(go env GOPATH)/bin/golangci-lint run
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -1,28 +0,0 @@
-on: [ push, pull_request ]
-name: Test
-jobs:
-  test:
-    strategy:
-      matrix:
-        go-version: [ 1.13.x, 1.14.x ]
-        goarch: [ "amd64", "386" ]
-    runs-on: ubuntu-latest
-    steps:
-    - name: Install Go
-      uses: actions/setup-go@v1
-      with:
-        go-version: ${{ matrix.go-version }}
-      env:
-        GOARCH: ${{ matrix.goarch }}
-    - name: Checkout code
-      uses: actions/checkout@v2
-    - uses: actions/cache@v1
-      with:
-          path: ~/go/pkg/mod
-          key: ${{ runner.os }}-go-${{ hashFiles('**/go.sum') }}
-          restore-keys: |
-              ${{ runner.os }}-go-
-    - name: Test
-      env:
-        GOARCH: ${{ matrix.goarch }}
-      run: go test ./...
--- a/README.md
+++ b/README.md
@@ -1,4 +1,4 @@
-# go-iden3-crypto [![Go Report Card](https://goreportcard.com/badge/github.com/iden3/go-iden3-crypto)](https://goreportcard.com/report/github.com/iden3/go-iden3-crypto) [![Test Status](https://github.com/iden3/go-iden3-crypto/workflows/Test/badge.svg)](https://github.com/iden3/go-iden3-crypto/actions?query=workflow%3ATest) [![Lint Status](https://github.com/iden3/go-iden3-crypto/workflows/Lint/badge.svg)](https://github.com/iden3/go-iden3-crypto/actions?query=workflow%3ALint) [![GoDoc](https://godoc.org/github.com/iden3/go-iden3-crypto?status.svg)](https://godoc.org/github.com/iden3/go-iden3-crypto)
+# go-iden3-crypto [![Go Report Card](https://goreportcard.com/badge/github.com/iden3/go-iden3-crypto)](https://goreportcard.com/report/github.com/iden3/go-iden3-crypto) [![Build Status](https://travis-ci.org/iden3/go-iden3-crypto.svg?branch=master)](https://travis-ci.org/iden3/go-iden3-crypto) [![GoDoc](https://godoc.org/github.com/iden3/go-iden3-crypto?status.svg)](https://godoc.org/github.com/iden3/go-iden3-crypto)

 Go implementation of some cryptographic primitives (that fit inside the SNARK field) used in iden3

--- a/babyjub/babyjub.go
+++ b/babyjub/babyjub.go
@@ -154,7 +154,10 @@ func (p *Point) InSubGroup() bool {
 // PointCoordSign returns the sign of the curve point coordinate.  It returns
 // false if the sign is positive and false if the sign is negative.
 func PointCoordSign(c *big.Int) bool {
-	return c.Cmp(new(big.Int).Rsh(constants.Q, 1)) == 1
+	if c.Cmp(new(big.Int).Rsh(constants.Q, 1)) == 1 {
+		return true
+	}
+	return false
 }

 func PackPoint(ay *big.Int, sign bool) [32]byte {
--- a/babyjub/eddsa.go
+++ b/babyjub/eddsa.go
@@ -36,13 +36,6 @@ func NewRandPrivKey() PrivateKey {
 // Scalar converts a private key into the scalar value s following the EdDSA
 // standard, and using blake-512 hash.
 func (k *PrivateKey) Scalar() *PrivKeyScalar {
-	s := SkToBigInt(k)
-	return NewPrivKeyScalar(s)
-}
-
-// SkToBigInt converts a private key into the *big.Int value following the
-// EdDSA standard, and using blake-512 hash
-func SkToBigInt(k *PrivateKey) *big.Int {
 	sBuf := Blake512(k[:])
 	sBuf32 := [32]byte{}
 	copy(sBuf32[:], sBuf[:32])
@@ -50,7 +43,7 @@ func SkToBigInt(k *PrivateKey) *big.Int {
 	s := new(big.Int)
 	utils.SetBigIntFromLEBytes(s, sBuf32[:])
 	s.Rsh(s, 3)
-	return s
+	return NewPrivKeyScalar(s)
 }

 // Pub returns the public key corresponding to a private key.
--- a/babyjub/eddsa_test.go
+++ b/babyjub/eddsa_test.go
@@ -1,6 +1,7 @@
 package babyjub

 import (
+	"crypto/rand"
 	"encoding/hex"
 	"fmt"
 	"math/big"
@@ -9,13 +10,25 @@ import (
 	"github.com/iden3/go-iden3-crypto/constants"
 	"github.com/iden3/go-iden3-crypto/utils"
 	"github.com/stretchr/testify/assert"
-	"github.com/stretchr/testify/require"
 )

+func genInputs() (*PrivateKey, *big.Int) {
+	k := NewRandPrivKey()
+	fmt.Println("k", hex.EncodeToString(k[:]))
+
+	msgBuf := [32]byte{}
+	rand.Read(msgBuf[:])
+	msg := utils.SetBigIntFromLEBytes(new(big.Int), msgBuf[:])
+	msg.Mod(msg, constants.Q)
+	fmt.Println("msg", msg)
+
+	return &k, msg
+}
+
 func TestPublicKey(t *testing.T) {
 	var k PrivateKey
-	for i := 0; i < 32; i++ {
-		k[i] = byte(i)
+	for i := 0; i < 256; i++ {
+		hex.Decode(k[:], []byte{byte(i)})
 	}
 	pk := k.Public()
 	assert.True(t, pk.X.Cmp(constants.Q) == -1)
@@ -24,8 +37,7 @@ func TestPublicKey(t *testing.T) {

 func TestSignVerifyMimc7(t *testing.T) {
 	var k PrivateKey
-	_, err := hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
-	require.Nil(t, err)
+	hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
 	msgBuf, err := hex.DecodeString("00010203040506070809")
 	if err != nil {
 		panic(err)
@@ -69,8 +81,7 @@ func TestSignVerifyMimc7(t *testing.T) {

 func TestSignVerifyPoseidon(t *testing.T) {
 	var k PrivateKey
-	_, err := hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
-	require.Nil(t, err)
+	hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
 	msgBuf, err := hex.DecodeString("00010203040506070809")
 	if err != nil {
 		panic(err)
@@ -114,8 +125,7 @@ func TestSignVerifyPoseidon(t *testing.T) {

 func TestCompressDecompress(t *testing.T) {
 	var k PrivateKey
-	_, err := hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
-	require.Nil(t, err)
+	hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
 	pk := k.Public()
 	for i := 0; i < 64; i++ {
 		msgBuf, err := hex.DecodeString(fmt.Sprintf("000102030405060708%02d", i))
@@ -134,8 +144,7 @@ func TestCompressDecompress(t *testing.T) {

 func BenchmarkBabyjubEddsa(b *testing.B) {
 	var k PrivateKey
-	_, err := hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
-	require.Nil(b, err)
+	hex.Decode(k[:], []byte("0001020304050607080900010203040506070809000102030405060708090001"))
 	pk := k.Public()

 	const n = 256
--- a/babyjub/helpers.go
+++ b/babyjub/helpers.go
@@ -8,9 +8,6 @@ import (
 // the original blake from the SHA3 competition and not the new blake2 version.
 func Blake512(m []byte) []byte {
 	h := blake512.New()
-	_, err := h.Write(m[:])
-	if err != nil {
-		panic(err)
-	}
+	h.Write(m[:])
 	return h.Sum(nil)
 }
--- a/ff/arith.go
+++ b/ff/arith.go
@@ -12,14 +12,19 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.

-// Code generated by goff DO NOT EDIT
+// Code generated by goff (v0.2.0) DO NOT EDIT

+// Package ff contains field arithmetic operations
 package ff

 import (
 	"math/bits"
+
+	"golang.org/x/sys/cpu"
 )

+var supportAdx = cpu.X86.HasADX && cpu.X86.HasBMI2
+
 func madd(a, b, t, u, v uint64) (uint64, uint64, uint64) {
 	var carry uint64
 	hi, lo := bits.Mul64(a, b)
--- a/ff/element.go
+++ b/ff/element.go
@@ -12,29 +12,33 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.

-// field modulus q =
-//
-// 21888242871839275222246405745257275088548364400416034343698204186575808495617
-// Code generated by goff DO NOT EDIT
-// goff version:  - build:
-// Element are assumed to be in Montgomery form in all methods
+// Code generated by goff (v0.2.0) DO NOT EDIT

-// Package ff (generated by goff) contains field arithmetics operations
+// Package ff contains field arithmetic operations
 package ff

+// /!\ WARNING /!\
+// this code has not been audited and is provided as-is. In particular,
+// there is no security guarantees such as constant time implementation
+// or side-channel attack resistance
+// /!\ WARNING /!\
+
 import (
 	"crypto/rand"
 	"encoding/binary"
 	"io"
 	"math/big"
 	"math/bits"
+	"strconv"
 	"sync"
-
 	"unsafe"
 )

 // Element represents a field element stored on 4 words (uint64)
 // Element are assumed to be in Montgomery form in all methods
+// field modulus q =
+//
+// 21888242871839275222246405745257275088548364400416034343698204186575808495617
 type Element [4]uint64

 // ElementLimbs number of 64 bits words needed to represent Element
@@ -311,6 +315,7 @@ func (z *Element) SetRandom() *Element {
 	z[3] %= 3486998266802970665

 	// if z > q --> z -= q
+	// note: this is NOT constant time
 	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
 		var b uint64
 		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
@@ -322,6 +327,38 @@ func (z *Element) SetRandom() *Element {
 	return z
 }

+// One returns 1 (in montgommery form)
+func One() Element {
+	var one Element
+	one.SetOne()
+	return one
+}
+
+// FromInterface converts i1 from uint64, int, string, or Element, big.Int into Element
+// panic if provided type is not supported
+func FromInterface(i1 interface{}) Element {
+	var val Element
+
+	switch c1 := i1.(type) {
+	case uint64:
+		val.SetUint64(c1)
+	case int:
+		val.SetString(strconv.Itoa(c1))
+	case string:
+		val.SetString(c1)
+	case big.Int:
+		val.SetBigInt(&c1)
+	case Element:
+		val = c1
+	case *Element:
+		val.Set(c1)
+	default:
+		panic("invalid type")
+	}
+
+	return val
+}
+
 // Add z = x + y mod q
 func (z *Element) Add(x, y *Element) *Element {
 	var carry uint64
@@ -332,6 +369,7 @@ func (z *Element) Add(x, y *Element) *Element {
 	z[3], _ = bits.Add64(x[3], y[3], carry)

 	// if z > q --> z -= q
+	// note: this is NOT constant time
 	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
 		var b uint64
 		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
@@ -352,6 +390,7 @@ func (z *Element) AddAssign(x *Element) *Element {
 	z[3], _ = bits.Add64(z[3], x[3], carry)

 	// if z > q --> z -= q
+	// note: this is NOT constant time
 	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
 		var b uint64
 		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
@@ -372,6 +411,7 @@ func (z *Element) Double(x *Element) *Element {
 	z[3], _ = bits.Add64(x[3], x[3], carry)

 	// if z > q --> z -= q
+	// note: this is NOT constant time
 	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
 		var b uint64
 		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
@@ -416,18 +456,31 @@ func (z *Element) SubAssign(x *Element) *Element {
 	return z
 }

-// Exp z = x^e mod q
-func (z *Element) Exp(x Element, e uint64) *Element {
-	if e == 0 {
+// Exp z = x^exponent mod q
+// (not optimized)
+// exponent (non-montgomery form) is ordered from least significant word to most significant word
+func (z *Element) Exp(x Element, exponent ...uint64) *Element {
+	r := 0
+	msb := 0
+	for i := len(exponent) - 1; i >= 0; i-- {
+		if exponent[i] == 0 {
+			r++
+		} else {
+			msb = (i * 64) + bits.Len64(exponent[i])
+			break
+		}
+	}
+	exponent = exponent[:len(exponent)-r]
+	if len(exponent) == 0 {
 		return z.SetOne()
 	}

 	z.Set(&x)

-	l := bits.Len64(e) - 2
+	l := msb - 2
 	for i := l; i >= 0; i-- {
 		z.Square(z)
-		if e&(1<<uint(i)) != 0 {
+		if exponent[i/64]&(1<<uint(i%64)) != 0 {
 			z.MulAssign(&x)
 		}
 	}
@@ -478,6 +531,7 @@ func (z *Element) FromMont() *Element {
 	}

 	// if z > q --> z -= q
+	// note: this is NOT constant time
 	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
 		var b uint64
 		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
@@ -549,6 +603,19 @@ func (z *Element) SetBigInt(v *big.Int) *Element {
 	zero := big.NewInt(0)
 	q := elementModulusBigInt()

+	// fast path
+	c := v.Cmp(q)
+	if c == 0 {
+		return z
+	} else if c != 1 && v.Cmp(zero) != -1 {
+		// v should
+		vBits := v.Bits()
+		for i := 0; i < len(vBits); i++ {
+			z[i] = uint64(vBits[i])
+		}
+		return z.ToMont()
+	}
+
 	// copy input
 	vv := new(big.Int).Set(v)

@@ -591,202 +658,97 @@ func (z *Element) SetString(s string) *Element {
 	return z.SetBigInt(x)
 }

-// Mul z = x * y mod q
-func (z *Element) Mul(x, y *Element) *Element {
+// Legendre returns the Legendre symbol of z (either +1, -1, or 0.)
+func (z *Element) Legendre() int {
+	var l Element
+	// z^((q-1)/2)
+	l.Exp(*z,
+		11669102379873075200,
+		10671829228508198984,
+		15863968012492123182,
+		1743499133401485332,
+	)

-	var t [4]uint64
-	var c [3]uint64
-	{
-		// round 0
-		v := x[0]
-		c[1], c[0] = bits.Mul64(v, y[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd1(v, y[1], c[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd1(v, y[2], c[1])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd1(v, y[3], c[1])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
-	}
-	{
-		// round 1
-		v := x[1]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
-	}
-	{
-		// round 2
-		v := x[2]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
-	}
-	{
-		// round 3
-		v := x[3]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], z[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], z[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		z[3], z[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	if l.IsZero() {
+		return 0
 	}

-	// if z > q --> z -= q
-	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
-		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
-		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
-		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	// if l == 1
+	if (l[3] == 1011752739694698287) && (l[2] == 7381016538464732718) && (l[1] == 3962172157175319849) && (l[0] == 12436184717236109307) {
+		return 1
 	}
-	return z
+	return -1
 }

-// MulAssign z = z * x mod q
-func (z *Element) MulAssign(x *Element) *Element {
+// Sqrt z = √x mod q
+// if the square root doesn't exist (x is not a square mod q)
+// Sqrt leaves z unchanged and returns nil
+func (z *Element) Sqrt(x *Element) *Element {
+	// q ≡ 1 (mod 4)
+	// see modSqrtTonelliShanks in math/big/int.go
+	// using https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf

-	var t [4]uint64
-	var c [3]uint64
-	{
-		// round 0
-		v := z[0]
-		c[1], c[0] = bits.Mul64(v, x[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd1(v, x[1], c[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd1(v, x[2], c[1])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd1(v, x[3], c[1])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	var y, b, t, w Element
+	// w = x^((s-1)/2))
+	w.Exp(*x,
+		14829091926808964255,
+		867720185306366531,
+		688207751544974772,
+		6495040407,
+	)
+
+	// y = x^((s+1)/2)) = w * x
+	y.Mul(x, &w)
+
+	// b = x^s = w * w * x = y * x
+	b.Mul(&w, &y)
+
+	// g = nonResidue ^ s
+	var g = Element{
+		7164790868263648668,
+		11685701338293206998,
+		6216421865291908056,
+		1756667274303109607,
 	}
-	{
-		// round 1
-		v := z[1]
-		c[1], c[0] = madd1(v, x[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, x[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, x[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, x[3], c[1], t[3])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	r := uint64(28)
+
+	// compute legendre symbol
+	// t = x^((q-1)/2) = r-1 squaring of x^s
+	t = b
+	for i := uint64(0); i < r-1; i++ {
+		t.Square(&t)
 	}
-	{
-		// round 2
-		v := z[2]
-		c[1], c[0] = madd1(v, x[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, x[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, x[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, x[3], c[1], t[3])
-		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	if t.IsZero() {
+		return z.SetZero()
 	}
-	{
-		// round 3
-		v := z[3]
-		c[1], c[0] = madd1(v, x[0], t[0])
-		m := c[0] * 14042775128853446655
-		c[2] = madd0(m, 4891460686036598785, c[0])
-		c[1], c[0] = madd2(v, x[1], c[1], t[1])
-		c[2], z[0] = madd2(m, 2896914383306846353, c[2], c[0])
-		c[1], c[0] = madd2(v, x[2], c[1], t[2])
-		c[2], z[1] = madd2(m, 13281191951274694749, c[2], c[0])
-		c[1], c[0] = madd2(v, x[3], c[1], t[3])
-		z[3], z[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	if !((t[3] == 1011752739694698287) && (t[2] == 7381016538464732718) && (t[1] == 3962172157175319849) && (t[0] == 12436184717236109307)) {
+		// t != 1, we don't have a square root
+		return nil
+	}
+	for {
+		var m uint64
+		t = b
+
+		// for t != 1
+		for !((t[3] == 1011752739694698287) && (t[2] == 7381016538464732718) && (t[1] == 3962172157175319849) && (t[0] == 12436184717236109307)) {
+			t.Square(&t)
+			m++
 		}

-	// if z > q --> z -= q
-	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
-		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
-		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
-		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+		if m == 0 {
+			return z.Set(&y)
+		}
+		// t = g^(2^(r-m-1)) mod q
+		ge := int(r - m - 1)
+		t = g
+		for ge > 0 {
+			t.Square(&t)
+			ge--
+		}
+
+		g.Square(&t)
+		y.MulAssign(&t)
+		b.MulAssign(&g)
+		r = m
 	}
-	return z
-}
-
-// Square z = x * x mod q
-func (z *Element) Square(x *Element) *Element {
-
-	var p [4]uint64
-
-	var u, v uint64
-	{
-		// round 0
-		u, p[0] = bits.Mul64(x[0], x[0])
-		m := p[0] * 14042775128853446655
-		C := madd0(m, 4891460686036598785, p[0])
-		var t uint64
-		t, u, v = madd1sb(x[0], x[1], u)
-		C, p[0] = madd2(m, 2896914383306846353, v, C)
-		t, u, v = madd1s(x[0], x[2], t, u)
-		C, p[1] = madd2(m, 13281191951274694749, v, C)
-		_, u, v = madd1s(x[0], x[3], t, u)
-		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
-	}
-	{
-		// round 1
-		m := p[0] * 14042775128853446655
-		C := madd0(m, 4891460686036598785, p[0])
-		u, v = madd1(x[1], x[1], p[1])
-		C, p[0] = madd2(m, 2896914383306846353, v, C)
-		var t uint64
-		t, u, v = madd2sb(x[1], x[2], p[2], u)
-		C, p[1] = madd2(m, 13281191951274694749, v, C)
-		_, u, v = madd2s(x[1], x[3], p[3], t, u)
-		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
-	}
-	{
-		// round 2
-		m := p[0] * 14042775128853446655
-		C := madd0(m, 4891460686036598785, p[0])
-		C, p[0] = madd2(m, 2896914383306846353, p[1], C)
-		u, v = madd1(x[2], x[2], p[2])
-		C, p[1] = madd2(m, 13281191951274694749, v, C)
-		_, u, v = madd2sb(x[2], x[3], p[3], u)
-		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
-	}
-	{
-		// round 3
-		m := p[0] * 14042775128853446655
-		C := madd0(m, 4891460686036598785, p[0])
-		C, z[0] = madd2(m, 2896914383306846353, p[1], C)
-		C, z[1] = madd2(m, 13281191951274694749, p[2], C)
-		u, v = madd1(x[3], x[3], p[3])
-		z[3], z[2] = madd3(m, 3486998266802970665, v, C, u)
-	}
-
-	// if z > q --> z -= q
-	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
-		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
-		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
-		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
-	}
-	return z
 }
--- a/ff/element_mul.go
+++ b/ff/element_mul.go
@@ -0,0 +1,170 @@
+// +build !amd64
+
+// Copyright 2020 ConsenSys AG
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+// Package ff contains field arithmetic operations
+package ff
+
+// /!\ WARNING /!\
+// this code has not been audited and is provided as-is. In particular,
+// there is no security guarantees such as constant time implementation
+// or side-channel attack resistance
+// /!\ WARNING /!\
+
+import "math/bits"
+
+// Mul z = x * y mod q
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) Mul(x, y *Element) *Element {
+
+	var t [4]uint64
+	var c [3]uint64
+	{
+		// round 0
+		v := x[0]
+		c[1], c[0] = bits.Mul64(v, y[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd1(v, y[1], c[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd1(v, y[2], c[1])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd1(v, y[3], c[1])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 1
+		v := x[1]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 2
+		v := x[2]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 3
+		v := x[3]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], z[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], z[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		z[3], z[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
+		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
+		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
+		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	}
+	return z
+}
+
+// MulAssign z = z * x mod q
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) MulAssign(x *Element) *Element {
+
+	var t [4]uint64
+	var c [3]uint64
+	{
+		// round 0
+		v := z[0]
+		c[1], c[0] = bits.Mul64(v, x[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd1(v, x[1], c[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd1(v, x[2], c[1])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd1(v, x[3], c[1])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 1
+		v := z[1]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 2
+		v := z[2]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 3
+		v := z[3]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], z[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], z[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		z[3], z[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
+		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
+		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
+		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	}
+	return z
+}
--- a/ff/element_mul_amd64.go
+++ b/ff/element_mul_amd64.go
@@ -0,0 +1,39 @@
+// Copyright 2020 ConsenSys AG
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+// Package ff contains field arithmetic operations
+package ff
+
+// MulAssignElement z = z * x mod q (constant time)
+// calling this instead of z.MulAssign(x) is prefered for performance critical path
+//go:noescape
+func MulAssignElement(res, y *Element)
+
+// Mul z = x * y mod q (constant time)
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) Mul(x, y *Element) *Element {
+	res := *x
+	MulAssignElement(&res, y)
+	z.Set(&res)
+	return z
+}
+
+// MulAssign z = z * x mod q (constant time)
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) MulAssign(x *Element) *Element {
+	MulAssignElement(z, x)
+	return z
+}
--- a/ff/element_mul_amd64.s
+++ b/ff/element_mul_amd64.s
@@ -0,0 +1,695 @@
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+#include "textflag.h"
+
+// func MulAssignElement(res,y *Element)
+// montgomery multiplication of res by y
+// stores the result in res
+TEXT ·MulAssignElement(SB), NOSPLIT, $0-16
+    
+    // dereference our parameters
+    MOVQ res+0(FP), DI
+    MOVQ y+8(FP), R8
+    
+    // check if we support adx and mulx
+    CMPB ·supportAdx(SB), $1
+    JNE no_adx
+    
+    // the algorithm is described here
+    // https://hackmd.io/@zkteam/modular_multiplication
+    // however, to benefit from the ADCX and ADOX carry chains
+    // we split the inner loops in 2:
+    // for i=0 to N-1
+    // 		for j=0 to N-1
+    // 		    (A,t[j])  := t[j] + a[j]*b[i] + A
+    // 		m := t[0]*q'[0] mod W
+    // 		C,_ := t[0] + m*q[0]
+    // 		for j=1 to N-1
+    // 		    (C,t[j-1]) := t[j] + m*q[j] + C
+    // 		t[N-1] = C + A
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 0
+    
+    // clear up the carry flags
+    XORQ R9 , R9
+    
+    // R12 = y[0]
+    MOVQ 0(R8), R12
+    
+    // for j=0 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    
+    // DX = res[0]
+    MOVQ 0(DI), DX
+    MULXQ R12, CX ,  R9
+    
+    // DX = res[1]
+    MOVQ 8(DI), DX
+    MOVQ R9, BX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BX
+    
+    // DX = res[2]
+    MOVQ 16(DI), DX
+    MOVQ R9, BP
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BP
+    
+    // DX = res[3]
+    MOVQ 24(DI), DX
+    MOVQ R9, SI
+    MULXQ R12, AX,  R9
+    ADOXQ AX, SI
+    
+    // add the last carries to R9
+    MOVQ $0, DX
+    ADCXQ DX, R9
+    ADOXQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, DX
+    MULXQ CX,R11, DX
+    
+    // clear the carry flags
+    XORQ DX, DX
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, DX
+    MULXQ R11, AX, R10
+    ADCXQ CX ,AX
+    
+    // for j=1 to N-1
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    
+    MOVQ $0x2833e84879b97091, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BX, R10
+    ADOXQ AX, R10
+    MOVQ R10, CX
+    MOVQ DX, R10
+    
+    MOVQ $0xb85045b68181585d, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BP, R10
+    ADOXQ AX, R10
+    MOVQ R10, BX
+    MOVQ DX, R10
+    
+    MOVQ $0x30644e72e131a029, DX
+    MULXQ R11, AX, DX
+    ADCXQ  SI, R10
+    ADOXQ AX, R10
+    MOVQ R10, BP
+    MOVQ $0, AX
+    ADCXQ AX, DX
+    ADOXQ DX, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 1
+    
+    // clear up the carry flags
+    XORQ R9 , R9
+    
+    // R12 = y[1]
+    MOVQ 8(R8), R12
+    
+    // for j=0 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    
+    // DX = res[0]
+    MOVQ 0(DI), DX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, CX
+    
+    // DX = res[1]
+    MOVQ 8(DI), DX
+    ADCXQ R9, BX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BX
+    
+    // DX = res[2]
+    MOVQ 16(DI), DX
+    ADCXQ R9, BP
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BP
+    
+    // DX = res[3]
+    MOVQ 24(DI), DX
+    ADCXQ R9, SI
+    MULXQ R12, AX,  R9
+    ADOXQ AX, SI
+    
+    // add the last carries to R9
+    MOVQ $0, DX
+    ADCXQ DX, R9
+    ADOXQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, DX
+    MULXQ CX,R11, DX
+    
+    // clear the carry flags
+    XORQ DX, DX
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, DX
+    MULXQ R11, AX, R10
+    ADCXQ CX ,AX
+    
+    // for j=1 to N-1
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    
+    MOVQ $0x2833e84879b97091, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BX, R10
+    ADOXQ AX, R10
+    MOVQ R10, CX
+    MOVQ DX, R10
+    
+    MOVQ $0xb85045b68181585d, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BP, R10
+    ADOXQ AX, R10
+    MOVQ R10, BX
+    MOVQ DX, R10
+    
+    MOVQ $0x30644e72e131a029, DX
+    MULXQ R11, AX, DX
+    ADCXQ  SI, R10
+    ADOXQ AX, R10
+    MOVQ R10, BP
+    MOVQ $0, AX
+    ADCXQ AX, DX
+    ADOXQ DX, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 2
+    
+    // clear up the carry flags
+    XORQ R9 , R9
+    
+    // R12 = y[2]
+    MOVQ 16(R8), R12
+    
+    // for j=0 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    
+    // DX = res[0]
+    MOVQ 0(DI), DX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, CX
+    
+    // DX = res[1]
+    MOVQ 8(DI), DX
+    ADCXQ R9, BX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BX
+    
+    // DX = res[2]
+    MOVQ 16(DI), DX
+    ADCXQ R9, BP
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BP
+    
+    // DX = res[3]
+    MOVQ 24(DI), DX
+    ADCXQ R9, SI
+    MULXQ R12, AX,  R9
+    ADOXQ AX, SI
+    
+    // add the last carries to R9
+    MOVQ $0, DX
+    ADCXQ DX, R9
+    ADOXQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, DX
+    MULXQ CX,R11, DX
+    
+    // clear the carry flags
+    XORQ DX, DX
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, DX
+    MULXQ R11, AX, R10
+    ADCXQ CX ,AX
+    
+    // for j=1 to N-1
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    
+    MOVQ $0x2833e84879b97091, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BX, R10
+    ADOXQ AX, R10
+    MOVQ R10, CX
+    MOVQ DX, R10
+    
+    MOVQ $0xb85045b68181585d, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BP, R10
+    ADOXQ AX, R10
+    MOVQ R10, BX
+    MOVQ DX, R10
+    
+    MOVQ $0x30644e72e131a029, DX
+    MULXQ R11, AX, DX
+    ADCXQ  SI, R10
+    ADOXQ AX, R10
+    MOVQ R10, BP
+    MOVQ $0, AX
+    ADCXQ AX, DX
+    ADOXQ DX, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 3
+    
+    // clear up the carry flags
+    XORQ R9 , R9
+    
+    // R12 = y[3]
+    MOVQ 24(R8), R12
+    
+    // for j=0 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    
+    // DX = res[0]
+    MOVQ 0(DI), DX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, CX
+    
+    // DX = res[1]
+    MOVQ 8(DI), DX
+    ADCXQ R9, BX
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BX
+    
+    // DX = res[2]
+    MOVQ 16(DI), DX
+    ADCXQ R9, BP
+    MULXQ R12, AX,  R9
+    ADOXQ AX, BP
+    
+    // DX = res[3]
+    MOVQ 24(DI), DX
+    ADCXQ R9, SI
+    MULXQ R12, AX,  R9
+    ADOXQ AX, SI
+    
+    // add the last carries to R9
+    MOVQ $0, DX
+    ADCXQ DX, R9
+    ADOXQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, DX
+    MULXQ CX,R11, DX
+    
+    // clear the carry flags
+    XORQ DX, DX
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, DX
+    MULXQ R11, AX, R10
+    ADCXQ CX ,AX
+    
+    // for j=1 to N-1
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    
+    MOVQ $0x2833e84879b97091, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BX, R10
+    ADOXQ AX, R10
+    MOVQ R10, CX
+    MOVQ DX, R10
+    
+    MOVQ $0xb85045b68181585d, DX
+    MULXQ R11, AX, DX
+    ADCXQ  BP, R10
+    ADOXQ AX, R10
+    MOVQ R10, BX
+    MOVQ DX, R10
+    
+    MOVQ $0x30644e72e131a029, DX
+    MULXQ R11, AX, DX
+    ADCXQ  SI, R10
+    ADOXQ AX, R10
+    MOVQ R10, BP
+    MOVQ $0, AX
+    ADCXQ AX, DX
+    ADOXQ DX, R9
+    MOVQ R9, SI
+    
+    reduce:
+    // reduce, constant time version
+    // first we copy registers storing t in a separate set of registers
+    // as SUBQ modifies the 2nd operand
+    MOVQ CX, DX
+    MOVQ BX, R8
+    MOVQ BP, R9
+    MOVQ SI, R10
+    MOVQ $0x43e1f593f0000001, R11
+    SUBQ  R11, DX
+    MOVQ $0x2833e84879b97091, R11
+    SBBQ  R11, R8
+    MOVQ $0xb85045b68181585d, R11
+    SBBQ  R11, R9
+    MOVQ $0x30644e72e131a029, R11
+    SBBQ  R11, R10
+    JCS t_is_smaller // no borrow, we return t
+    
+    // borrow is set, we return u
+    MOVQ DX, (DI)
+    MOVQ R8, 8(DI)
+    MOVQ R9, 16(DI)
+    MOVQ R10, 24(DI)
+    RET
+    t_is_smaller:
+    MOVQ CX, 0(DI)
+    MOVQ BX, 8(DI)
+    MOVQ BP, 16(DI)
+    MOVQ SI, 24(DI)
+    RET
+    
+    no_adx:
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 0
+    
+    // (A,t[0]) := t[0] + x[0]*y[0]
+    MOVQ (DI), AX // x[0]
+    MOVQ 0(R8), R12
+    MULQ R12 // x[0] * y[0]
+    MOVQ DX, R9
+    MOVQ AX, CX
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, R11
+    IMULQ CX , R11
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, AX
+    MULQ R11
+    ADDQ CX ,AX
+    ADCQ $0, DX
+    MOVQ  DX, R10
+    
+    // for j=1 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    MOVQ 8(DI), AX
+    MULQ R12 // x[1] * y[0]
+    MOVQ R9, BX
+    ADDQ AX, BX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x2833e84879b97091, AX
+    MULQ R11
+    ADDQ  BX, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, CX
+    MOVQ DX, R10
+    MOVQ 16(DI), AX
+    MULQ R12 // x[2] * y[0]
+    MOVQ R9, BP
+    ADDQ AX, BP
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0xb85045b68181585d, AX
+    MULQ R11
+    ADDQ  BP, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BX
+    MOVQ DX, R10
+    MOVQ 24(DI), AX
+    MULQ R12 // x[3] * y[0]
+    MOVQ R9, SI
+    ADDQ AX, SI
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x30644e72e131a029, AX
+    MULQ R11
+    ADDQ  SI, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BP
+    MOVQ DX, R10
+    
+    ADDQ R10, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 1
+    
+    // (A,t[0]) := t[0] + x[0]*y[1]
+    MOVQ (DI), AX // x[0]
+    MOVQ 8(R8), R12
+    MULQ R12 // x[0] * y[1]
+    ADDQ AX, CX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, R11
+    IMULQ CX , R11
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, AX
+    MULQ R11
+    ADDQ CX ,AX
+    ADCQ $0, DX
+    MOVQ  DX, R10
+    
+    // for j=1 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    MOVQ 8(DI), AX
+    MULQ R12 // x[1] * y[1]
+    ADDQ R9, BX
+    ADCQ $0, DX
+    ADDQ AX, BX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x2833e84879b97091, AX
+    MULQ R11
+    ADDQ  BX, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, CX
+    MOVQ DX, R10
+    MOVQ 16(DI), AX
+    MULQ R12 // x[2] * y[1]
+    ADDQ R9, BP
+    ADCQ $0, DX
+    ADDQ AX, BP
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0xb85045b68181585d, AX
+    MULQ R11
+    ADDQ  BP, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BX
+    MOVQ DX, R10
+    MOVQ 24(DI), AX
+    MULQ R12 // x[3] * y[1]
+    ADDQ R9, SI
+    ADCQ $0, DX
+    ADDQ AX, SI
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x30644e72e131a029, AX
+    MULQ R11
+    ADDQ  SI, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BP
+    MOVQ DX, R10
+    
+    ADDQ R10, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 2
+    
+    // (A,t[0]) := t[0] + x[0]*y[2]
+    MOVQ (DI), AX // x[0]
+    MOVQ 16(R8), R12
+    MULQ R12 // x[0] * y[2]
+    ADDQ AX, CX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, R11
+    IMULQ CX , R11
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, AX
+    MULQ R11
+    ADDQ CX ,AX
+    ADCQ $0, DX
+    MOVQ  DX, R10
+    
+    // for j=1 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    MOVQ 8(DI), AX
+    MULQ R12 // x[1] * y[2]
+    ADDQ R9, BX
+    ADCQ $0, DX
+    ADDQ AX, BX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x2833e84879b97091, AX
+    MULQ R11
+    ADDQ  BX, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, CX
+    MOVQ DX, R10
+    MOVQ 16(DI), AX
+    MULQ R12 // x[2] * y[2]
+    ADDQ R9, BP
+    ADCQ $0, DX
+    ADDQ AX, BP
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0xb85045b68181585d, AX
+    MULQ R11
+    ADDQ  BP, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BX
+    MOVQ DX, R10
+    MOVQ 24(DI), AX
+    MULQ R12 // x[3] * y[2]
+    ADDQ R9, SI
+    ADCQ $0, DX
+    ADDQ AX, SI
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x30644e72e131a029, AX
+    MULQ R11
+    ADDQ  SI, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BP
+    MOVQ DX, R10
+    
+    ADDQ R10, R9
+    MOVQ R9, SI
+    
+    // ---------------------------------------------------------------------------------------------
+    // outter loop 3
+    
+    // (A,t[0]) := t[0] + x[0]*y[3]
+    MOVQ (DI), AX // x[0]
+    MOVQ 24(R8), R12
+    MULQ R12 // x[0] * y[3]
+    ADDQ AX, CX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    // m := t[0]*q'[0] mod W
+    MOVQ $0xc2e1f593efffffff, R11
+    IMULQ CX , R11
+    
+    // C,_ := t[0] + m*q[0]
+    MOVQ $0x43e1f593f0000001, AX
+    MULQ R11
+    ADDQ CX ,AX
+    ADCQ $0, DX
+    MOVQ  DX, R10
+    
+    // for j=1 to N-1
+    //    (A,t[j])  := t[j] + x[j]*y[i] + A
+    //    (C,t[j-1]) := t[j] + m*q[j] + C
+    MOVQ 8(DI), AX
+    MULQ R12 // x[1] * y[3]
+    ADDQ R9, BX
+    ADCQ $0, DX
+    ADDQ AX, BX
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x2833e84879b97091, AX
+    MULQ R11
+    ADDQ  BX, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, CX
+    MOVQ DX, R10
+    MOVQ 16(DI), AX
+    MULQ R12 // x[2] * y[3]
+    ADDQ R9, BP
+    ADCQ $0, DX
+    ADDQ AX, BP
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0xb85045b68181585d, AX
+    MULQ R11
+    ADDQ  BP, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BX
+    MOVQ DX, R10
+    MOVQ 24(DI), AX
+    MULQ R12 // x[3] * y[3]
+    ADDQ R9, SI
+    ADCQ $0, DX
+    ADDQ AX, SI
+    ADCQ $0, DX
+    MOVQ DX, R9
+    
+    MOVQ $0x30644e72e131a029, AX
+    MULQ R11
+    ADDQ  SI, R10
+    ADCQ $0, DX
+    ADDQ AX, R10
+    ADCQ $0, DX
+    
+    MOVQ R10, BP
+    MOVQ DX, R10
+    
+    ADDQ R10, R9
+    MOVQ R9, SI
+    
+    JMP reduce
--- a/ff/element_square.go
+++ b/ff/element_square.go
@@ -0,0 +1,93 @@
+// +build !amd64
+
+// Copyright 2020 ConsenSys AG
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+// Package ff contains field arithmetic operations
+package ff
+
+// /!\ WARNING /!\
+// this code has not been audited and is provided as-is. In particular,
+// there is no security guarantees such as constant time implementation
+// or side-channel attack resistance
+// /!\ WARNING /!\
+
+import "math/bits"
+
+// Square z = x * x mod q
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) Square(x *Element) *Element {
+
+	var p [4]uint64
+
+	var u, v uint64
+	{
+		// round 0
+		u, p[0] = bits.Mul64(x[0], x[0])
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		var t uint64
+		t, u, v = madd1sb(x[0], x[1], u)
+		C, p[0] = madd2(m, 2896914383306846353, v, C)
+		t, u, v = madd1s(x[0], x[2], t, u)
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd1s(x[0], x[3], t, u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 1
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		u, v = madd1(x[1], x[1], p[1])
+		C, p[0] = madd2(m, 2896914383306846353, v, C)
+		var t uint64
+		t, u, v = madd2sb(x[1], x[2], p[2], u)
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd2s(x[1], x[3], p[3], t, u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 2
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		C, p[0] = madd2(m, 2896914383306846353, p[1], C)
+		u, v = madd1(x[2], x[2], p[2])
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd2sb(x[2], x[3], p[3], u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 3
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		C, z[0] = madd2(m, 2896914383306846353, p[1], C)
+		C, z[1] = madd2(m, 13281191951274694749, p[2], C)
+		u, v = madd1(x[3], x[3], p[3])
+		z[3], z[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
+		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
+		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
+		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	}
+	return z
+
+}
--- a/ff/element_square_amd64.go
+++ b/ff/element_square_amd64.go
@@ -0,0 +1,34 @@
+// Copyright 2020 ConsenSys AG
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+// Package ff contains field arithmetic operations
+package ff
+
+// SquareElement z = x * x mod q
+// calling this instead of z.Square(x) is prefered for performance critical path
+// go - noescape
+// func SquareElement(res,x *Element)
+
+// Square z = x * x mod q
+// see https://hackmd.io/@zkteam/modular_multiplication
+func (z *Element) Square(x *Element) *Element {
+	if z != x {
+		z.Set(x)
+	}
+	MulAssignElement(z, x)
+	// SquareElement(z, x)
+	return z
+}
--- a/ff/element_test.go
+++ b/ff/element_test.go
@@ -1,9 +1,26 @@
-// Code generated by goff DO NOT EDIT
+// Copyright 2020 ConsenSys AG
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+// Code generated by goff (v0.2.0) DO NOT EDIT
+
+// Package ff contains field arithmetic operations
 package ff

 import (
 	"crypto/rand"
 	"math/big"
+	"math/bits"
 	mrand "math/rand"
 	"testing"
 )
@@ -21,7 +38,14 @@ func TestELEMENTCorrectnessAgainstBigInt(t *testing.T) {

 	modulusMinusOne.Sub(modulus, &one)

-	for i := 0; i < 1000; i++ {
+	var n int
+	if testing.Short() {
+		n = 10
+	} else {
+		n = 500
+	}
+
+	for i := 0; i < n; i++ {

 		// sample 2 random big int
 		b1, _ := rand.Int(rand.Reader, modulus)
@@ -57,7 +81,7 @@ func TestELEMENTCorrectnessAgainstBigInt(t *testing.T) {

 		rbExp := new(big.Int).SetUint64(rExp)

-		var bMul, bAdd, bSub, bDiv, bNeg, bLsh, bInv, bExp, bSquare big.Int
+		var bMul, bAdd, bSub, bDiv, bNeg, bLsh, bInv, bExp, bExp2, bSquare big.Int

 		// e1 = mont(b1), e2 = mont(b2)
 		var e1, e2, eMul, eAdd, eSub, eDiv, eNeg, eLsh, eInv, eExp, eSquare, eMulAssign, eSubAssign, eAddAssign Element
@@ -106,12 +130,40 @@ func TestELEMENTCorrectnessAgainstBigInt(t *testing.T) {
 		cmpEandB(&eNeg, &bNeg, "Neg")
 		cmpEandB(&eInv, &bInv, "Inv")
 		cmpEandB(&eExp, &bExp, "Exp")
+
 		cmpEandB(&eLsh, &bLsh, "Lsh")
+
+		// legendre symbol
+		if e1.Legendre() != big.Jacobi(b1, modulus) {
+			t.Fatal("legendre symbol computation failed")
+		}
+		if e2.Legendre() != big.Jacobi(b2, modulus) {
+			t.Fatal("legendre symbol computation failed")
+		}
+
+		// these are slow, killing circle ci
+		if n <= 5 {
+			// sqrt
+			var eSqrt, eExp2 Element
+			var bSqrt big.Int
+			bSqrt.ModSqrt(b1, modulus)
+			eSqrt.Sqrt(&e1)
+			cmpEandB(&eSqrt, &bSqrt, "Sqrt")
+
+			bits := b2.Bits()
+			exponent := make([]uint64, len(bits))
+			for k := 0; k < len(bits); k++ {
+				exponent[k] = uint64(bits[k])
+			}
+			eExp2.Exp(e1, exponent...)
+			bExp2.Exp(b1, b2, modulus)
+			cmpEandB(&eExp2, &bExp2, "Exp multi words")
+		}
 	}
 }

 func TestELEMENTIsRandom(t *testing.T) {
-	for i := 0; i < 1000; i++ {
+	for i := 0; i < 50; i++ {
 		var x, y Element
 		x.SetRandom()
 		y.SetRandom()
@@ -125,7 +177,6 @@ func TestELEMENTIsRandom(t *testing.T) {
 // benchmarks
 // most benchmarks are rudimentary and should sample a large number of random inputs
 // or be run multiple times to ensure it didn't measure the fastest path of the function
-// TODO: clean up and push benchmarking branch

 var benchResElement Element

@@ -219,6 +270,15 @@ func BenchmarkSquareELEMENT(b *testing.B) {
 	}
 }

+func BenchmarkSqrtELEMENT(b *testing.B) {
+	var a Element
+	a.SetRandom()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		benchResElement.Sqrt(&a)
+	}
+}
+
 func BenchmarkMulAssignELEMENT(b *testing.B) {
 	x := Element{
 		1997599621687373223,
@@ -232,3 +292,183 @@ func BenchmarkMulAssignELEMENT(b *testing.B) {
 		benchResElement.MulAssign(&x)
 	}
 }
+
+func BenchmarkMulAssignASMELEMENT(b *testing.B) {
+	x := Element{
+		1997599621687373223,
+		6052339484930628067,
+		10108755138030829701,
+		150537098327114917,
+	}
+	benchResElement.SetOne()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		MulAssignElement(&benchResElement, &x)
+	}
+}
+
+func TestELEMENTAsm(t *testing.T) {
+	// ensure ASM implementations matches the ones using math/bits
+	modulus, _ := new(big.Int).SetString("21888242871839275222246405745257275088548364400416034343698204186575808495617", 10)
+	for i := 0; i < 500; i++ {
+		// sample 2 random big int
+		b1, _ := rand.Int(rand.Reader, modulus)
+		b2, _ := rand.Int(rand.Reader, modulus)
+
+		// e1 = mont(b1), e2 = mont(b2)
+		var e1, e2, eTestMul, eMulAssign, eSquare, eTestSquare Element
+		e1.SetBigInt(b1)
+		e2.SetBigInt(b2)
+
+		eTestMul = e1
+		eTestMul.testMulAssign(&e2)
+		eMulAssign = e1
+		eMulAssign.MulAssign(&e2)
+
+		if !eTestMul.Equal(&eMulAssign) {
+			t.Fatal("inconsisntencies between MulAssign and testMulAssign --> check if MulAssign is calling ASM implementaiton on amd64")
+		}
+
+		// square
+		eSquare.Square(&e1)
+		eTestSquare.testSquare(&e1)
+
+		if !eTestSquare.Equal(&eSquare) {
+			t.Fatal("inconsisntencies between Square and testSquare --> check if Square is calling ASM implementaiton on amd64")
+		}
+	}
+}
+
+// this is here for consistency purposes, to ensure MulAssign on AMD64 using asm implementation gives consistent results
+func (z *Element) testMulAssign(x *Element) *Element {
+
+	var t [4]uint64
+	var c [3]uint64
+	{
+		// round 0
+		v := z[0]
+		c[1], c[0] = bits.Mul64(v, x[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd1(v, x[1], c[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd1(v, x[2], c[1])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd1(v, x[3], c[1])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 1
+		v := z[1]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 2
+		v := z[2]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+	{
+		// round 3
+		v := z[3]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 14042775128853446655
+		c[2] = madd0(m, 4891460686036598785, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], z[0] = madd2(m, 2896914383306846353, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], z[1] = madd2(m, 13281191951274694749, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		z[3], z[2] = madd3(m, 3486998266802970665, c[0], c[2], c[1])
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
+		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
+		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
+		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	}
+	return z
+}
+
+// this is here for consistency purposes, to ensure Square on AMD64 using asm implementation gives consistent results
+func (z *Element) testSquare(x *Element) *Element {
+
+	var p [4]uint64
+
+	var u, v uint64
+	{
+		// round 0
+		u, p[0] = bits.Mul64(x[0], x[0])
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		var t uint64
+		t, u, v = madd1sb(x[0], x[1], u)
+		C, p[0] = madd2(m, 2896914383306846353, v, C)
+		t, u, v = madd1s(x[0], x[2], t, u)
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd1s(x[0], x[3], t, u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 1
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		u, v = madd1(x[1], x[1], p[1])
+		C, p[0] = madd2(m, 2896914383306846353, v, C)
+		var t uint64
+		t, u, v = madd2sb(x[1], x[2], p[2], u)
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd2s(x[1], x[3], p[3], t, u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 2
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		C, p[0] = madd2(m, 2896914383306846353, p[1], C)
+		u, v = madd1(x[2], x[2], p[2])
+		C, p[1] = madd2(m, 13281191951274694749, v, C)
+		_, u, v = madd2sb(x[2], x[3], p[3], u)
+		p[3], p[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+	{
+		// round 3
+		m := p[0] * 14042775128853446655
+		C := madd0(m, 4891460686036598785, p[0])
+		C, z[0] = madd2(m, 2896914383306846353, p[1], C)
+		C, z[1] = madd2(m, 13281191951274694749, p[2], C)
+		u, v = madd1(x[3], x[3], p[3])
+		z[3], z[2] = madd3(m, 3486998266802970665, v, C, u)
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 3486998266802970665 || (z[3] == 3486998266802970665 && (z[2] < 13281191951274694749 || (z[2] == 13281191951274694749 && (z[1] < 2896914383306846353 || (z[1] == 2896914383306846353 && (z[0] < 4891460686036598785))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 4891460686036598785, 0)
+		z[1], b = bits.Sub64(z[1], 2896914383306846353, b)
+		z[2], b = bits.Sub64(z[2], 13281191951274694749, b)
+		z[3], _ = bits.Sub64(z[3], 3486998266802970665, b)
+	}
+	return z
+
+}
--- a/go.mod
+++ b/go.mod
@@ -4,7 +4,8 @@ go 1.12

 require (
 	github.com/dchest/blake512 v1.0.0
-	github.com/ethereum/go-ethereum v1.9.12
-	github.com/stretchr/testify v1.4.0
-	golang.org/x/crypto v0.0.0-20200311171314-f7b00557c8c4
+	github.com/ethereum/go-ethereum v1.8.27
+	github.com/stretchr/testify v1.3.0
+	golang.org/x/crypto v0.0.0-20190621222207-cc06ce4a13d4
+	golang.org/x/sys v0.0.0-20190412213103-97732733099d
 )
--- a/go.sum
+++ b/go.sum
@@ -1,172 +1,20 @@
-github.com/Azure/azure-pipeline-go v0.2.1/go.mod h1:UGSo8XybXnIGZ3epmeBw7Jdz+HiUVpqIlpz/HKHylF4=
-github.com/Azure/azure-pipeline-go v0.2.2/go.mod h1:4rQ/NZncSvGqNkkOsNpOU1tgoNuIlp9AfUH5G1tvCHc=
-github.com/Azure/azure-storage-blob-go v0.7.0/go.mod h1:f9YQKtsG1nMisotuTPpO0tjNuEjKRYAcJU8/ydDI++4=
-github.com/Azure/go-autorest/autorest v0.9.0/go.mod h1:xyHB1BMZT0cuDHU7I0+g046+BFDTQ8rEZB0s4Yfa6bI=
-github.com/Azure/go-autorest/autorest/adal v0.5.0/go.mod h1:8Z9fGy2MpX0PvDjB1pEgQTmVqjGhiHBW7RJJEciWzS0=
-github.com/Azure/go-autorest/autorest/adal v0.8.0/go.mod h1:Z6vX6WXXuyieHAXwMj0S6HY6e6wcHn37qQMBQlvY3lc=
-github.com/Azure/go-autorest/autorest/date v0.1.0/go.mod h1:plvfp3oPSKwf2DNjlBjWF/7vwR+cUD/ELuzDCXwHUVA=
-github.com/Azure/go-autorest/autorest/date v0.2.0/go.mod h1:vcORJHLJEh643/Ioh9+vPmf1Ij9AEBM5FuBIXLmIy0g=
-github.com/Azure/go-autorest/autorest/mocks v0.1.0/go.mod h1:OTyCOPRA2IgIlWxVYxBee2F5Gr4kF2zd2J5cFRaIDN0=
-github.com/Azure/go-autorest/autorest/mocks v0.2.0/go.mod h1:OTyCOPRA2IgIlWxVYxBee2F5Gr4kF2zd2J5cFRaIDN0=
-github.com/Azure/go-autorest/autorest/mocks v0.3.0/go.mod h1:a8FDP3DYzQ4RYfVAxAN3SVSiiO77gL2j2ronKKP0syM=
-github.com/Azure/go-autorest/logger v0.1.0/go.mod h1:oExouG+K6PryycPJfVSxi/koC6LSNgds39diKLz7Vrc=
-github.com/Azure/go-autorest/tracing v0.5.0/go.mod h1:r/s2XiOKccPW3HrqB+W0TQzfbtp2fGCgRFtBroKn4Dk=
-github.com/BurntSushi/toml v0.3.1/go.mod h1:xHWCNGjB5oqiDr8zfno3MHue2Ht5sIBksp03qcyfWMU=
-github.com/OneOfOne/xxhash v1.2.2/go.mod h1:HSdplMjZKSmBqAxg5vPj2TmRDmfkzw+cTzAElWljhcU=
-github.com/OneOfOne/xxhash v1.2.5/go.mod h1:eZbhyaAYD41SGSSsnmcpxVoRiQ/MPUTjUdIIOT9Um7Q=
-github.com/StackExchange/wmi v0.0.0-20180116203802-5d049714c4a6 h1:fLjPD/aNc3UIOA6tDi6QXUemppXK3P9BI7mr2hd6gx8=
-github.com/StackExchange/wmi v0.0.0-20180116203802-5d049714c4a6/go.mod h1:3eOhrUMpNV+6aFIbp5/iudMxNCF27Vw2OZgy4xEx0Fg=
-github.com/VictoriaMetrics/fastcache v1.5.3 h1:2odJnXLbFZcoV9KYtQ+7TH1UOq3dn3AssMgieaezkR4=
-github.com/VictoriaMetrics/fastcache v1.5.3/go.mod h1:+jv9Ckb+za/P1ZRg/sulP5Ni1v49daAVERr0H3CuscE=
-github.com/alecthomas/template v0.0.0-20160405071501-a0175ee3bccc/go.mod h1:LOuyumcjzFXgccqObfd/Ljyb9UuFJ6TxHnclSeseNhc=
-github.com/alecthomas/units v0.0.0-20151022065526-2efee857e7cf/go.mod h1:ybxpYRFXyAe+OPACYpWeL0wqObRcbAqCMya13uyzqw0=
-github.com/allegro/bigcache v1.2.1-0.20190218064605-e24eb225f156/go.mod h1:Cb/ax3seSYIx7SuZdm2G2xzfwmv3TPSk2ucNfQESPXM=
-github.com/aristanetworks/goarista v0.0.0-20170210015632-ea17b1a17847 h1:rtI0fD4oG/8eVokGVPYJEW1F88p1ZNgXiEIs9thEE4A=
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-golang.org/x/sync v0.0.0-20180314180146-1d60e4601c6f/go.mod h1:RxMgew5VJxzue5/jJTE5uejpjVlOe/izrB70Jof72aM=
-golang.org/x/sync v0.0.0-20181108010431-42b317875d0f/go.mod h1:RxMgew5VJxzue5/jJTE5uejpjVlOe/izrB70Jof72aM=
-golang.org/x/sys v0.0.0-20180909124046-d0be0721c37e/go.mod h1:STP8DvDyc/dI5b8T5hshtkjS+E42TnysNCUPdjciGhY=
-golang.org/x/sys v0.0.0-20181107165924-66b7b1311ac8/go.mod h1:STP8DvDyc/dI5b8T5hshtkjS+E42TnysNCUPdjciGhY=
 golang.org/x/sys v0.0.0-20190215142949-d0b11bdaac8a/go.mod h1:STP8DvDyc/dI5b8T5hshtkjS+E42TnysNCUPdjciGhY=
 golang.org/x/sys v0.0.0-20190412213103-97732733099d h1:+R4KGOnez64A81RvjARKc4UT5/tI9ujCIVX+P5KiHuI=
 golang.org/x/sys v0.0.0-20190412213103-97732733099d/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
-golang.org/x/sys v0.0.0-20200302150141-5c8b2ff67527 h1:uYVVQ9WP/Ds2ROhcaGPeIdVq0RIXVLwsHlnvJ+cT1So=
-golang.org/x/sys v0.0.0-20200302150141-5c8b2ff67527/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs=
 golang.org/x/text v0.3.0/go.mod h1:NqM8EUOU14njkJ3fqMW+pc6Ldnwhi/IjpwHt7yyuwOQ=
-golang.org/x/text v0.3.2/go.mod h1:bEr9sfX3Q8Zfm5fL9x+3itogRgK3+ptLWKqgva+5dAk=
-golang.org/x/time v0.0.0-20190308202827-9d24e82272b4/go.mod h1:tRJNPiyCQ0inRvYxbN9jk5I+vvW/OXSQhTDSoE431IQ=
-golang.org/x/tools v0.0.0-20180917221912-90fa682c2a6e/go.mod h1:n7NCudcB/nEzxVGmLbDWY5pfWTLqBcC2KZ6jyYvM4mQ=
-gopkg.in/alecthomas/kingpin.v2 v2.2.6/go.mod h1:FMv+mEhP44yOT+4EoQTLFTRgOQ1FBLkstjWtayDeSgw=
-gopkg.in/check.v1 v0.0.0-20161208181325-20d25e280405/go.mod h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0=
-gopkg.in/check.v1 v1.0.0-20180628173108-788fd7840127/go.mod h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0=
-gopkg.in/fsnotify.v1 v1.4.7/go.mod h1:Tz8NjZHkW78fSQdbUxIjBTcgA1z1m8ZHf0WmKUhAMys=
-gopkg.in/natefinch/npipe.v2 v2.0.0-20160621034901-c1b8fa8bdcce/go.mod h1:5AcXVHNjg+BDxry382+8OKon8SEWiKktQR07RKPsv1c=
-gopkg.in/olebedev/go-duktape.v3 v3.0.0-20190213234257-ec84240a7772/go.mod h1:uAJfkITjFhyEEuUfm7bsmCZRbW5WRq8s9EY8HZ6hCns=
-gopkg.in/tomb.v1 v1.0.0-20141024135613-dd632973f1e7/go.mod h1:dt/ZhP58zS4L8KSrWDmTeBkI65Dw0HsyUHuEVlX15mw=
-gopkg.in/urfave/cli.v1 v1.20.0/go.mod h1:vuBzUtMdQeixQj8LVd+/98pzhxNGQoyuPBlsXHOQNO0=
-gopkg.in/yaml.v2 v2.2.1/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI=
-gopkg.in/yaml.v2 v2.2.2 h1:ZCJp+EgiOT7lHqUV2J862kp8Qj64Jo6az82+3Td9dZw=
-gopkg.in/yaml.v2 v2.2.2/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI=
-gotest.tools v2.2.0+incompatible/go.mod h1:DsYFclhRJ6vuDpmuTbkuFWG+y2sxOXAzmJt81HFBacw=
--- a/mimc7/mimc7_test.go
+++ b/mimc7/mimc7_test.go
@@ -87,6 +87,6 @@ func BenchmarkMIMC7(b *testing.B) {
 	bigArray4 := []*big.Int{b12, b45, b78, b41}

 	for i := 0; i < b.N; i++ {
-		Hash(bigArray4, nil) //nolint:errcheck
+		Hash(bigArray4, nil)
 	}
 }
--- a/poseidon/poseidon_test.go
+++ b/poseidon/poseidon_test.go
@@ -104,7 +104,7 @@ func BenchmarkPoseidon(b *testing.B) {
 	bigArray4 := []*big.Int{b12, b45, b78, b41}

 	for i := 0; i < b.N; i++ {
-		Hash(bigArray4) //nolint:errcheck
+		Hash(bigArray4)
 	}
 }

@@ -117,6 +117,6 @@ func BenchmarkPoseidonLarge(b *testing.B) {
 	bigArray4 := []*big.Int{b12, b45, b78, b41}

 	for i := 0; i < b.N; i++ {
-		Hash(bigArray4) //nolint:errcheck
+		Hash(bigArray4)
 	}
 }
--- a/utils/showarchbits_test.go
+++ b/utils/showarchbits_test.go
@@ -1,11 +0,0 @@
-package utils
-
-import (
-	"fmt"
-	"math/bits"
-	"testing"
-)
-
-func TestShowArchBits(t *testing.T) {
-	fmt.Printf("Architecture is %v bits\n", bits.UintSize)
-}
--- a/utils/utils.go
+++ b/utils/utils.go
@@ -66,7 +66,9 @@ func HexEncode(bs []byte) string {

 // HexDecode decodes a hex string into an array of bytes.
 func HexDecode(h string) ([]byte, error) {
-	h = strings.TrimPrefix(h, "0x")
+	if strings.HasPrefix(h, "0x") {
+		h = h[2:]
+	}
 	return hex.DecodeString(h)
 }

@@ -90,7 +92,10 @@ func HexDecodeInto(dst []byte, h []byte) error {

 // CheckBigIntInField checks if given *big.Int fits in a Field Q element
 func CheckBigIntInField(a *big.Int) bool {
-	return a.Cmp(constants.Q) == -1
+	if a.Cmp(constants.Q) != -1 {
+		return false
+	}
+	return true
 }

 // CheckBigIntArrayInField checks if given *big.Int fits in a Field Q element
@@ -103,7 +108,6 @@ func CheckBigIntArrayInField(arr []*big.Int) bool {
 	return true
 }

-// BigIntArrayToElementArray converts an array of *big.Int into an array of *ff.Element
 func BigIntArrayToElementArray(bi []*big.Int) []*ff.Element {
 	var o []*ff.Element
 	for i := range bi {
@@ -111,15 +115,3 @@ func BigIntArrayToElementArray(bi []*big.Int) []*ff.Element {
 	}
 	return o
 }
-
-// ElementArrayToBigIntArray converts an array of *ff.Element into an array of *big.Int
-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
-}