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element.go
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element.go
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// 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 fp
// /!\ 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 6 words (uint64)
// Element are assumed to be in Montgomery form in all methods
// field modulus q =
//
// 258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177
type Element [6]uint64
// Limbs number of 64 bits words needed to represent Element
const Limbs = 6
// Bits number bits needed to represent Element
const Bits = 377
// 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 =
//
// 258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177
func Modulus() *big.Int {
return new(big.Int).Set(&_modulus)
}
// q (modulus)
var qElement = Element{
9586122913090633729,
1660523435060625408,
2230234197602682880,
1883307231910630287,
14284016967150029115,
121098312706494698,
}
// rSquare
var rSquare = Element{
13224372171368877346,
227991066186625457,
2496666625421784173,
13825906835078366124,
9475172226622360569,
30958721782860680,
}
var bigIntPool = sync.Pool{
New: func() interface{} {
return new(big.Int)
},
}
func init() {
_modulus.SetString("258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177", 10)
}
// 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]
z[1] = x[1]
z[2] = x[2]
z[3] = x[3]
z[4] = x[4]
z[5] = x[5]
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 fp.Element from type " + reflect.TypeOf(i1).String())
}
}
// SetZero z = 0
func (z *Element) SetZero() *Element {
z[0] = 0
z[1] = 0
z[2] = 0
z[3] = 0
z[4] = 0
z[5] = 0
return z
}
// SetOne z = 1 (in Montgomery form)
func (z *Element) SetOne() *Element {
z[0] = 202099033278250856
z[1] = 5854854902718660529
z[2] = 11492539364873682930
z[3] = 8885205928937022213
z[4] = 5545221690922665192
z[5] = 39800542322357402
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 >= 6 {
return 0
}
return uint64(z[j] >> (i % 64) & 1)
}
// Equal returns z == x
func (z *Element) Equal(x *Element) bool {
return (z[5] == x[5]) && (z[4] == x[4]) && (z[3] == x[3]) && (z[2] == x[2]) && (z[1] == x[1]) && (z[0] == x[0])
}
// IsZero returns z == 0
func (z *Element) IsZero() bool {
return (z[5] | z[4] | z[3] | z[2] | z[1] | z[0]) == 0
}
// IsUint64 returns true if z[0] >= 0 and all other words are 0
func (z *Element) IsUint64() bool {
return (z[5] | z[4] | z[3] | z[2] | z[1]) == 0
}
// 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[5] > _x[5] {
return 1
} else if _z[5] < _x[5] {
return -1
}
if _z[4] > _x[4] {
return 1
} else if _z[4] < _x[4] {
return -1
}
if _z[3] > _x[3] {
return 1
} else if _z[3] < _x[3] {
return -1
}
if _z[2] > _x[2] {
return 1
} else if _z[2] < _x[2] {
return -1
}
if _z[1] > _x[1] {
return 1
} else if _z[1] < _x[1] {
return -1
}
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], 4793061456545316865, 0)
_, b = bits.Sub64(_z[1], 830261717530312704, b)
_, b = bits.Sub64(_z[2], 10338489135656117248, b)
_, b = bits.Sub64(_z[3], 10165025652810090951, b)
_, b = bits.Sub64(_z[4], 7142008483575014557, b)
_, b = bits.Sub64(_z[5], 60549156353247349, b)
return b == 0
}
// SetRandom sets z to a random element < q
func (z *Element) SetRandom() (*Element, error) {
var bytes [48]byte
if _, err := io.ReadFull(rand.Reader, bytes[:]); err != nil {
return nil, err
}
z[0] = binary.BigEndian.Uint64(bytes[0:8])
z[1] = binary.BigEndian.Uint64(bytes[8:16])
z[2] = binary.BigEndian.Uint64(bytes[16:24])
z[3] = binary.BigEndian.Uint64(bytes[24:32])
z[4] = binary.BigEndian.Uint64(bytes[32:40])
z[5] = binary.BigEndian.Uint64(bytes[40:48])
z[5] %= 121098312706494698
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
return z, nil
}
// One returns 1 (in montgommery form)
func One() Element {
var one Element
one.SetOne()
return one
}
// 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 [6]uint64
var c [3]uint64
{
// round 0
v := x[0]
c[1], c[0] = bits.Mul64(v, y[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd1(v, y[1], c[1])
c[2], t[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd1(v, y[2], c[1])
c[2], t[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd1(v, y[3], c[1])
c[2], t[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd1(v, y[4], c[1])
c[2], t[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd1(v, y[5], c[1])
t[5], t[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
{
// round 1
v := x[1]
c[1], c[0] = madd1(v, y[0], t[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd2(v, y[1], c[1], t[1])
c[2], t[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd2(v, y[2], c[1], t[2])
c[2], t[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd2(v, y[3], c[1], t[3])
c[2], t[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd2(v, y[4], c[1], t[4])
c[2], t[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd2(v, y[5], c[1], t[5])
t[5], t[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
{
// round 2
v := x[2]
c[1], c[0] = madd1(v, y[0], t[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd2(v, y[1], c[1], t[1])
c[2], t[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd2(v, y[2], c[1], t[2])
c[2], t[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd2(v, y[3], c[1], t[3])
c[2], t[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd2(v, y[4], c[1], t[4])
c[2], t[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd2(v, y[5], c[1], t[5])
t[5], t[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
{
// round 3
v := x[3]
c[1], c[0] = madd1(v, y[0], t[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd2(v, y[1], c[1], t[1])
c[2], t[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd2(v, y[2], c[1], t[2])
c[2], t[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd2(v, y[3], c[1], t[3])
c[2], t[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd2(v, y[4], c[1], t[4])
c[2], t[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd2(v, y[5], c[1], t[5])
t[5], t[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
{
// round 4
v := x[4]
c[1], c[0] = madd1(v, y[0], t[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd2(v, y[1], c[1], t[1])
c[2], t[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd2(v, y[2], c[1], t[2])
c[2], t[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd2(v, y[3], c[1], t[3])
c[2], t[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd2(v, y[4], c[1], t[4])
c[2], t[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd2(v, y[5], c[1], t[5])
t[5], t[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
{
// round 5
v := x[5]
c[1], c[0] = madd1(v, y[0], t[0])
m := c[0] * 9586122913090633727
c[2] = madd0(m, 9586122913090633729, c[0])
c[1], c[0] = madd2(v, y[1], c[1], t[1])
c[2], z[0] = madd2(m, 1660523435060625408, c[2], c[0])
c[1], c[0] = madd2(v, y[2], c[1], t[2])
c[2], z[1] = madd2(m, 2230234197602682880, c[2], c[0])
c[1], c[0] = madd2(v, y[3], c[1], t[3])
c[2], z[2] = madd2(m, 1883307231910630287, c[2], c[0])
c[1], c[0] = madd2(v, y[4], c[1], t[4])
c[2], z[3] = madd2(m, 14284016967150029115, c[2], c[0])
c[1], c[0] = madd2(v, y[5], c[1], t[5])
z[5], z[4] = madd3(m, 121098312706494698, c[0], c[2], c[1])
}
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
}
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] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
{
// m = z[0]n'[0] mod W
m := z[0] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
{
// m = z[0]n'[0] mod W
m := z[0] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
{
// m = z[0]n'[0] mod W
m := z[0] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
{
// m = z[0]n'[0] mod W
m := z[0] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
{
// m = z[0]n'[0] mod W
m := z[0] * 9586122913090633727
C := madd0(m, 9586122913090633729, z[0])
C, z[0] = madd2(m, 1660523435060625408, z[1], C)
C, z[1] = madd2(m, 2230234197602682880, z[2], C)
C, z[2] = madd2(m, 1883307231910630287, z[3], C)
C, z[3] = madd2(m, 14284016967150029115, z[4], C)
C, z[4] = madd2(m, 121098312706494698, z[5], C)
z[5] = C
}
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
}
func _addGeneric(z, x, y *Element) {
var carry uint64
z[0], carry = bits.Add64(x[0], y[0], 0)
z[1], carry = bits.Add64(x[1], y[1], carry)
z[2], carry = bits.Add64(x[2], y[2], carry)
z[3], carry = bits.Add64(x[3], y[3], carry)
z[4], carry = bits.Add64(x[4], y[4], carry)
z[5], _ = bits.Add64(x[5], y[5], carry)
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
}
func _doubleGeneric(z, x *Element) {
var carry uint64
z[0], carry = bits.Add64(x[0], x[0], 0)
z[1], carry = bits.Add64(x[1], x[1], carry)
z[2], carry = bits.Add64(x[2], x[2], carry)
z[3], carry = bits.Add64(x[3], x[3], carry)
z[4], carry = bits.Add64(x[4], x[4], carry)
z[5], _ = bits.Add64(x[5], x[5], carry)
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
}
func _subGeneric(z, x, y *Element) {
var b uint64
z[0], b = bits.Sub64(x[0], y[0], 0)
z[1], b = bits.Sub64(x[1], y[1], b)
z[2], b = bits.Sub64(x[2], y[2], b)
z[3], b = bits.Sub64(x[3], y[3], b)
z[4], b = bits.Sub64(x[4], y[4], b)
z[5], b = bits.Sub64(x[5], y[5], b)
if b != 0 {
var c uint64
z[0], c = bits.Add64(z[0], 9586122913090633729, 0)
z[1], c = bits.Add64(z[1], 1660523435060625408, c)
z[2], c = bits.Add64(z[2], 2230234197602682880, c)
z[3], c = bits.Add64(z[3], 1883307231910630287, c)
z[4], c = bits.Add64(z[4], 14284016967150029115, c)
z[5], _ = bits.Add64(z[5], 121098312706494698, c)
}
}
func _negGeneric(z, x *Element) {
if x.IsZero() {
z.SetZero()
return
}
var borrow uint64
z[0], borrow = bits.Sub64(9586122913090633729, x[0], 0)
z[1], borrow = bits.Sub64(1660523435060625408, x[1], borrow)
z[2], borrow = bits.Sub64(2230234197602682880, x[2], borrow)
z[3], borrow = bits.Sub64(1883307231910630287, x[3], borrow)
z[4], borrow = bits.Sub64(14284016967150029115, x[4], borrow)
z[5], _ = bits.Sub64(121098312706494698, x[5], borrow)
}
func _reduceGeneric(z *Element) {
// if z > q --> z -= q
// note: this is NOT constant time
if !(z[5] < 121098312706494698 || (z[5] == 121098312706494698 && (z[4] < 14284016967150029115 || (z[4] == 14284016967150029115 && (z[3] < 1883307231910630287 || (z[3] == 1883307231910630287 && (z[2] < 2230234197602682880 || (z[2] == 2230234197602682880 && (z[1] < 1660523435060625408 || (z[1] == 1660523435060625408 && (z[0] < 9586122913090633729))))))))))) {
var b uint64
z[0], b = bits.Sub64(z[0], 9586122913090633729, 0)
z[1], b = bits.Sub64(z[1], 1660523435060625408, b)
z[2], b = bits.Sub64(z[2], 2230234197602682880, b)
z[3], b = bits.Sub64(z[3], 1883307231910630287, b)
z[4], b = bits.Sub64(z[4], 14284016967150029115, b)
z[5], _ = bits.Sub64(z[5], 121098312706494698, b)
}
}
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 {
if z[5] != 0 {
return 320 + bits.Len64(z[5])
}
if z[4] != 0 {
return 256 + bits.Len64(z[4])
}
if z[3] != 0 {
return 192 + bits.Len64(z[3])
}
if z[2] != 0 {
return 128 + bits.Len64(z[2])
}
if z[1] != 0 {
return 64 + bits.Len64(z[1])
}
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[40:48], z[0])
binary.BigEndian.PutUint64(b[32:40], z[1])
binary.BigEndian.PutUint64(b[24:32], z[2])
binary.BigEndian.PutUint64(b[16:24], z[3])
binary.BigEndian.PutUint64(b[8:16], z[4])
binary.BigEndian.PutUint64(b[0:8], z[5])
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)
}
// 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[40:48], _z[0])
binary.BigEndian.PutUint64(res[32:40], _z[1])
binary.BigEndian.PutUint64(res[24:32], _z[2])
binary.BigEndian.PutUint64(res[16:24], _z[3])
binary.BigEndian.PutUint64(res[8:16], _z[4])
binary.BigEndian.PutUint64(res[0:8], _z[5])
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("d71d230be28875631d82e03650a49d8d116cf9807a89c78f79b117dd04a4000b85aea2180000004284600000000000", 16)
const sqrtExponentElement = "35c748c2f8a21d58c760b80d94292763445b3e601ea271e3de6c45f741290002e16ba88600000010a11"
_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[5] == 39800542322357402) && (l[4] == 5545221690922665192) && (l[3] == 8885205928937022213) && (l[2] == 11492539364873682930) && (l[1] == 5854854902718660529) && (l[0] == 202099033278250856) {
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{
7563926049028936178,
2688164645460651601,
12112688591437172399,
3177973240564633687,
14764383749841851163,
52487407124055189,
}
r := uint64(46)
// 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[5] == 39800542322357402) && (t[4] == 5545221690922665192) && (t[3] == 8885205928937022213) && (t[2] == 11492539364873682930) && (t[1] == 5854854902718660529) && (t[0] == 202099033278250856)) {
// t != 1, we don't have a square root
return nil
}
for {
var m uint64
t = b
// for t != 1
for !((t[5] == 39800542322357402) && (t[4] == 5545221690922665192) && (t[3] == 8885205928937022213) && (t[2] == 11492539364873682930) && (t[1] == 5854854902718660529) && (t[0] == 202099033278250856)) {
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
// Algorithm 16 in "Efficient Software-Implementation of Finite Fields with Applications to Cryptography"
// if x == 0, sets and returns z = x
func (z *Element) Inverse(x *Element) *Element {
if x.IsZero() {
z.SetZero()
return z