go-iden3-crypto/ffg/element.go

// Copyright 2020 ConsenSys Software Inc.
//
// 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 consensys/gnark-crypto DO NOT EDIT

package ffg

// /!\ 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"
	"errors"
	"io"
	"math/big"
	"math/bits"
	"reflect"
	"strconv"
	"sync"
)

// Element represents a field element stored on 1 words (uint64)
// Element are assumed to be in Montgomery form in all methods
// field modulus q =
//
// 18446744069414584321
type Element [1]uint64

// Limbs number of 64 bits words needed to represent Element
const Limbs = 1

// Bits number bits needed to represent Element
const Bits = 64

// Bytes number bytes needed to represent Element
const Bytes = Limbs * 8

// field modulus stored as big.Int
var _modulus big.Int

// Modulus returns q as a big.Int
// q =
//
// 18446744069414584321
func Modulus() *big.Int {
	return new(big.Int).Set(&_modulus)
}

// q (modulus)
var qElement = Element{
	18446744069414584321,
}

// rSquare
var rSquare = Element{
	18446744065119617025,
}

var bigIntPool = sync.Pool{
	New: func() interface{} {
		return new(big.Int)
	},
}

func init() {
	_modulus.SetString("18446744069414584321", 10)
}

// NewElement returns a new Element
func NewElement() *Element {
	return &Element{}
}

// NewElementFromUint64 returns a new Element from a uint64 value
//
// it is equivalent to
// 		var v NewElement
// 		v.SetUint64(...)
func NewElementFromUint64(v uint64) *Element {
	z := Element{v}
	z.Mul(&z, &rSquare)
	return &z
}

// SetUint64 z = v, sets z LSB to v (non-Montgomery form) and convert z to Montgomery form
func (z *Element) SetUint64(v uint64) *Element {
	*z = Element{v}
	return z.Mul(z, &rSquare) // z.ToMont()
}

// Set z = x
func (z *Element) Set(x *Element) *Element {
	z[0] = x[0]
	return z
}

// SetInterface converts provided interface into Element
// returns an error if provided type is not supported
// supported types: Element, *Element, uint64, int, string (interpreted as base10 integer),
// *big.Int, big.Int, []byte
func (z *Element) SetInterface(i1 interface{}) (*Element, error) {
	switch c1 := i1.(type) {
	case Element:
		return z.Set(&c1), nil
	case *Element:
		return z.Set(c1), nil
	case uint64:
		return z.SetUint64(c1), nil
	case int:
		return z.SetString(strconv.Itoa(c1)), nil
	case string:
		return z.SetString(c1), nil
	case *big.Int:
		return z.SetBigInt(c1), nil
	case big.Int:
		return z.SetBigInt(&c1), nil
	case []byte:
		return z.SetBytes(c1), nil
	default:
		return nil, errors.New("can't set ffg.Element from type " + reflect.TypeOf(i1).String())
	}
}

// SetZero z = 0
func (z *Element) SetZero() *Element {
	z[0] = 0
	return z
}

// SetOne z = 1 (in Montgomery form)
func (z *Element) SetOne() *Element {
	z[0] = 4294967295
	return z
}

// Div z = x*y^-1 mod q
func (z *Element) Div(x, y *Element) *Element {
	var yInv Element
	yInv.Inverse(y)
	z.Mul(x, &yInv)
	return z
}

// Bit returns the i'th bit, with lsb == bit 0.
// It is the responsability of the caller to convert from Montgomery to Regular form if needed
func (z *Element) Bit(i uint64) uint64 {
	j := i / 64
	if j >= 1 {
		return 0
	}
	return uint64(z[j] >> (i % 64) & 1)
}

// Equal returns z == x
func (z *Element) Equal(x *Element) bool {
	return (z[0] == x[0])
}

// IsZero returns z == 0
func (z *Element) IsZero() bool {
	return (z[0]) == 0
}

// IsUint64 returns true if z[0] >= 0 and all other words are 0
func (z *Element) IsUint64() bool {
	return true
}

// Cmp compares (lexicographic order) z and x and returns:
//
//   -1 if z <  x
//    0 if z == x
//   +1 if z >  x
//
func (z *Element) Cmp(x *Element) int {
	_z := *z
	_x := *x
	_z.FromMont()
	_x.FromMont()
	if _z[0] > _x[0] {
		return 1
	} else if _z[0] < _x[0] {
		return -1
	}
	return 0
}

// LexicographicallyLargest returns true if this element is strictly lexicographically
// larger than its negation, false otherwise
func (z *Element) LexicographicallyLargest() bool {
	// adapted from github.com/zkcrypto/bls12_381
	// we check if the element is larger than (q-1) / 2
	// if z - (((q -1) / 2) + 1) have no underflow, then z > (q-1) / 2

	_z := *z
	_z.FromMont()

	var b uint64
	_, b = bits.Sub64(_z[0], 9223372034707292161, 0)

	return b == 0
}

// SetRandom sets z to a random element < q
func (z *Element) SetRandom() (*Element, error) {
	var bytes [8]byte
	if _, err := io.ReadFull(rand.Reader, bytes[:]); err != nil {
		return nil, err
	}
	z[0] = binary.BigEndian.Uint64(bytes[0:8])
	z[0] %= 18446744069414584321

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		// var b uint64
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}

	return z, nil
}

// One returns 1 (in montgommery form)
func One() Element {
	var one Element
	one.SetOne()
	return one
}

// Halve sets z to z / 2 (mod p)
func (z *Element) Halve() {
	var twoInv Element
	twoInv.SetOne().Double(&twoInv).Inverse(&twoInv)
	z.Mul(z, &twoInv)

}

// API with assembly impl

// Mul z = x * y mod q
// see https://hackmd.io/@zkteam/modular_multiplication
func (z *Element) Mul(x, y *Element) *Element {
	mul(z, x, y)
	return z
}

// Square z = x * x mod q
// see https://hackmd.io/@zkteam/modular_multiplication
func (z *Element) Square(x *Element) *Element {
	mul(z, x, x)
	return z
}

// FromMont converts z in place (i.e. mutates) from Montgomery to regular representation
// sets and returns z = z * 1
func (z *Element) FromMont() *Element {
	fromMont(z)
	return z
}

// Add z = x + y mod q
func (z *Element) Add(x, y *Element) *Element {
	add(z, x, y)
	return z
}

// Double z = x + x mod q, aka Lsh 1
func (z *Element) Double(x *Element) *Element {
	double(z, x)
	return z
}

// Sub  z = x - y mod q
func (z *Element) Sub(x, y *Element) *Element {
	sub(z, x, y)
	return z
}

// Neg z = q - x
func (z *Element) Neg(x *Element) *Element {
	neg(z, x)
	return z
}

// Generic (no ADX instructions, no AMD64) versions of multiplication and squaring algorithms

func _mulGeneric(z, x, y *Element) {

	var t [2]uint64
	var D uint64
	var m, C uint64
	// -----------------------------------
	// First loop

	C, t[0] = bits.Mul64(y[0], x[0])

	t[1], D = bits.Add64(t[1], C, 0)

	// m = t[0]n'[0] mod W
	m = t[0] * 18446744069414584319

	// -----------------------------------
	// Second loop
	C = madd0(m, 18446744069414584321, t[0])

	t[0], C = bits.Add64(t[1], C, 0)
	t[1], _ = bits.Add64(0, D, C)

	if t[1] != 0 {
		// we need to reduce, we have a result on 2 words
		// var b uint64
		z[0], _ = bits.Sub64(t[0], 18446744069414584321, 0)

		return

	}

	// copy t into z
	z[0] = t[0]

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}
}

func _fromMontGeneric(z *Element) {
	// the following lines implement z = z * 1
	// with a modified CIOS montgomery multiplication
	{
		// m = z[0]n'[0] mod W
		m := z[0] * 18446744069414584319
		C := madd0(m, 18446744069414584321, z[0])
		z[0] = C
	}

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}
}

func _addGeneric(z, x, y *Element) {
	var carry uint64

	z[0], carry = bits.Add64(x[0], y[0], 0)
	// if we overflowed the last addition, z >= q
	// if z >= q, z = z - q
	if carry != 0 {
		// we overflowed, so z >= q
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
		return
	}

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}
}

func _doubleGeneric(z, x *Element) {
	var carry uint64

	z[0], carry = bits.Add64(x[0], x[0], 0)
	// if we overflowed the last addition, z >= q
	// if z >= q, z = z - q
	if carry != 0 {
		// we overflowed, so z >= q
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
		return
	}

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}
}

func _subGeneric(z, x, y *Element) {
	var b uint64
	z[0], b = bits.Sub64(x[0], y[0], 0)
	if b != 0 {
		// var c uint64
		z[0], _ = bits.Add64(z[0], 18446744069414584321, 0)
	}
}

func _negGeneric(z, x *Element) {
	if x.IsZero() {
		z.SetZero()
		return
	}
	z[0], _ = bits.Sub64(18446744069414584321, x[0], 0)
}

func _reduceGeneric(z *Element) {

	// if z > q --> z -= q
	// note: this is NOT constant time
	if !(z[0] < 18446744069414584321) {
		z[0], _ = bits.Sub64(z[0], 18446744069414584321, 0)
	}
}

func mulByConstant(z *Element, c uint8) {
	switch c {
	case 0:
		z.SetZero()
		return
	case 1:
		return
	case 2:
		z.Double(z)
		return
	case 3:
		_z := *z
		z.Double(z).Add(z, &_z)
	case 5:
		_z := *z
		z.Double(z).Double(z).Add(z, &_z)
	default:
		var y Element
		y.SetUint64(uint64(c))
		z.Mul(z, &y)
	}
}

// BatchInvert returns a new slice with every element inverted.
// Uses Montgomery batch inversion trick
func BatchInvert(a []Element) []Element {
	res := make([]Element, len(a))
	if len(a) == 0 {
		return res
	}

	zeroes := make([]bool, len(a))
	accumulator := One()

	for i := 0; i < len(a); i++ {
		if a[i].IsZero() {
			zeroes[i] = true
			continue
		}
		res[i] = accumulator
		accumulator.Mul(&accumulator, &a[i])
	}

	accumulator.Inverse(&accumulator)

	for i := len(a) - 1; i >= 0; i-- {
		if zeroes[i] {
			continue
		}
		res[i].Mul(&res[i], &accumulator)
		accumulator.Mul(&accumulator, &a[i])
	}

	return res
}

func _butterflyGeneric(a, b *Element) {
	t := *a
	a.Add(a, b)
	b.Sub(&t, b)
}

// BitLen returns the minimum number of bits needed to represent z
// returns 0 if z == 0
func (z *Element) BitLen() int {
	return bits.Len64(z[0])
}

// Exp z = x^exponent mod q
func (z *Element) Exp(x Element, exponent *big.Int) *Element {
	var bZero big.Int
	if exponent.Cmp(&bZero) == 0 {
		return z.SetOne()
	}

	z.Set(&x)

	for i := exponent.BitLen() - 2; i >= 0; i-- {
		z.Square(z)
		if exponent.Bit(i) == 1 {
			z.Mul(z, &x)
		}
	}

	return z
}

// ToMont converts z to Montgomery form
// sets and returns z = z * r^2
func (z *Element) ToMont() *Element {
	return z.Mul(z, &rSquare)
}

// ToRegular returns z in regular form (doesn't mutate z)
func (z Element) ToRegular() Element {
	return *z.FromMont()
}

// String returns the string form of an Element in Montgomery form
func (z *Element) String() string {
	zz := *z
	zz.FromMont()
	if zz.IsUint64() {
		return strconv.FormatUint(zz[0], 10)
	} else {
		var zzNeg Element
		zzNeg.Neg(z)
		zzNeg.FromMont()
		if zzNeg.IsUint64() {
			return "-" + strconv.FormatUint(zzNeg[0], 10)
		}
	}
	vv := bigIntPool.Get().(*big.Int)
	defer bigIntPool.Put(vv)
	return zz.ToBigInt(vv).String()
}

// ToBigInt returns z as a big.Int in Montgomery form
func (z *Element) ToBigInt(res *big.Int) *big.Int {
	var b [Limbs * 8]byte
	binary.BigEndian.PutUint64(b[0:8], z[0])

	return res.SetBytes(b[:])
}

// ToBigIntRegular returns z as a big.Int in regular form
func (z Element) ToBigIntRegular(res *big.Int) *big.Int {
	z.FromMont()
	return z.ToBigInt(res)
}

// ToUint64Regular returns z as a uint64 in regular form
func (z Element) ToUint64Regular() uint64 {
	z.FromMont()
	return z[0]
}

// Bytes returns the regular (non montgomery) value
// of z as a big-endian byte array.
func (z *Element) Bytes() (res [Limbs * 8]byte) {
	_z := z.ToRegular()
	binary.BigEndian.PutUint64(res[0:8], _z[0])

	return
}

// Marshal returns the regular (non montgomery) value
// of z as a big-endian byte slice.
func (z *Element) Marshal() []byte {
	b := z.Bytes()
	return b[:]
}

// SetBytes interprets e as the bytes of a big-endian unsigned integer,
// sets z to that value (in Montgomery form), and returns z.
func (z *Element) SetBytes(e []byte) *Element {
	// get a big int from our pool
	vv := bigIntPool.Get().(*big.Int)
	vv.SetBytes(e)

	// set big int
	z.SetBigInt(vv)

	// put temporary object back in pool
	bigIntPool.Put(vv)

	return z
}

// SetBigInt sets z to v (regular form) and returns z in Montgomery form
func (z *Element) SetBigInt(v *big.Int) *Element {
	z.SetZero()

	var zero big.Int

	// fast path
	c := v.Cmp(&_modulus)
	if c == 0 {
		// v == 0
		return z
	} else if c != 1 && v.Cmp(&zero) != -1 {
		// 0 < v < q
		return z.setBigInt(v)
	}

	// get temporary big int from the pool
	vv := bigIntPool.Get().(*big.Int)

	// copy input + modular reduction
	vv.Set(v)
	vv.Mod(v, &_modulus)

	// set big int byte value
	z.setBigInt(vv)

	// release object into pool
	bigIntPool.Put(vv)
	return z
}

// setBigInt assumes 0 <= v < q
func (z *Element) setBigInt(v *big.Int) *Element {
	vBits := v.Bits()

	if bits.UintSize == 64 {
		for i := 0; i < len(vBits); i++ {
			z[i] = uint64(vBits[i])
		}
	} else {
		for i := 0; i < len(vBits); i++ {
			if i%2 == 0 {
				z[i/2] = uint64(vBits[i])
			} else {
				z[i/2] |= uint64(vBits[i]) << 32
			}
		}
	}

	return z.ToMont()
}

// SetString creates a big.Int with s (in base 10) and calls SetBigInt on z
func (z *Element) SetString(s string) *Element {
	// get temporary big int from the pool
	vv := bigIntPool.Get().(*big.Int)

	if _, ok := vv.SetString(s, 10); !ok {
		panic("Element.SetString failed -> can't parse number in base10 into a big.Int")
	}
	z.SetBigInt(vv)

	// release object into pool
	bigIntPool.Put(vv)

	return z
}

var (
	_bLegendreExponentElement *big.Int
	_bSqrtExponentElement     *big.Int
)

func init() {
	_bLegendreExponentElement, _ = new(big.Int).SetString("7fffffff80000000", 16)
	const sqrtExponentElement = "7fffffff"
	_bSqrtExponentElement, _ = new(big.Int).SetString(sqrtExponentElement, 16)
}

// 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, _bLegendreExponentElement)

	if l.IsZero() {
		return 0
	}

	// if l == 1
	if l[0] == 4294967295 {
		return 1
	}
	return -1
}

// 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 y, b, t, w Element
	// w = x^((s-1)/2))
	w.Exp(*x, _bSqrtExponentElement)

	// 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{
		15733474329512464024,
	}
	r := uint64(32)

	// 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)
	}
	if t.IsZero() {
		return z.SetZero()
	}
	if !(t[0] == 4294967295) {
		// t != 1, we don't have a square root
		return nil
	}
	for {
		var m uint64
		t = b

		// for t != 1
		for !(t[0] == 4294967295) {
			t.Square(&t)
			m++
		}

		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.Mul(&y, &t)
		b.Mul(&b, &g)
		r = m
	}
}

// Inverse z = x^-1 mod q
// note: allocates a big.Int (math/big)
func (z *Element) Inverse(x *Element) *Element {
	var _xNonMont big.Int
	x.ToBigIntRegular(&_xNonMont)
	_xNonMont.ModInverse(&_xNonMont, Modulus())
	z.SetBigInt(&_xNonMont)
	return z
}