fp

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Published: Mar 25, 2022 License: Apache-2.0 Imports: 11 Imported by: 1

Documentation

Overview

Package fp contains field arithmetic operations for modulus = 0x3eeb04...000001.

The API is similar to math/big (big.Int), but the operations are significantly faster (up to 20x for the modular multiplication on amd64, see also https://hackmd.io/@gnark/modular_multiplication)

The modulus is hardcoded in all the operations.

Field elements are represented as an array, and assumed to be in Montgomery form in all methods:

type Element [6]uint64

Example API signature

// Mul z = x * y mod q
func (z *Element) Mul(x, y *Element) *Element

and can be used like so:

var a, b Element
a.SetUint64(2)
b.SetString("984896738")
a.Mul(a, b)
a.Sub(a, a)
 .Add(a, b)
 .Inv(a)
b.Exp(b, new(big.Int).SetUint64(42))

Modulus

0x3eeb0416684d19053cb5d240ed107a284059eb647102326980dc360d0a49d7fce97f76a822c00009948a20000000001 // base 16
605248206075306171733248481581800960739847691770924913753520744034740935903401304776283802348837311170974282940417 // base 10

Index

Constants

View Source
const Bits = 378

Bits number bits needed to represent Element

View Source
const Bytes = Limbs * 8

Bytes number bytes needed to represent Element

View Source
const Limbs = 6

Limbs number of 64 bits words needed to represent Element

Variables

This section is empty.

Functions

func Butterfly

func Butterfly(a, b *Element)

func Modulus

func Modulus() *big.Int

Modulus returns q as a big.Int q =

605248206075306171733248481581800960739847691770924913753520744034740935903401304776283802348837311170974282940417

func MulBy13

func MulBy13(x *Element)

func MulBy3

func MulBy3(x *Element)

func MulBy5

func MulBy5(x *Element)

Types

type Element

type Element [6]uint64

Element represents a field element stored on 6 words (uint64) Element are assumed to be in Montgomery form in all methods field modulus q =

605248206075306171733248481581800960739847691770924913753520744034740935903401304776283802348837311170974282940417

func BatchInvert

func BatchInvert(a []Element) []Element

BatchInvert returns a new slice with every element inverted. Uses Montgomery batch inversion trick

func NewElement

func NewElement(v uint64) Element

NewElement returns a new Element from a uint64 value

it is equivalent to

var v NewElement
v.SetUint64(...)

func One

func One() Element

One returns 1 (in montgommery form)

func (*Element) Add

func (z *Element) Add(x, y *Element) *Element

Add z = x + y mod q

func (*Element) Bit

func (z *Element) Bit(i uint64) uint64

Bit returns the i'th bit, with lsb == bit 0. It is the responsibility of the caller to convert from Montgomery to Regular form if needed

func (*Element) BitLen

func (z *Element) BitLen() int

BitLen returns the minimum number of bits needed to represent z returns 0 if z == 0

func (*Element) Bytes

func (z *Element) Bytes() (res [Limbs * 8]byte)

Bytes returns the regular (non montgomery) value of z as a big-endian byte array.

func (*Element) Cmp

func (z *Element) Cmp(x *Element) int

Cmp compares (lexicographic order) z and x and returns:

-1 if z <  x
 0 if z == x
+1 if z >  x

func (*Element) Div

func (z *Element) Div(x, y *Element) *Element

Div z = x*y^-1 mod q

func (*Element) Double

func (z *Element) Double(x *Element) *Element

Double z = x + x mod q, aka Lsh 1

func (*Element) Equal

func (z *Element) Equal(x *Element) bool

Equal returns z == x; constant-time

func (*Element) Exp

func (z *Element) Exp(x Element, exponent *big.Int) *Element

Exp z = x^exponent mod q

func (*Element) FitsOnOneWord

func (z *Element) FitsOnOneWord() bool

FitsOnOneWord reports whether z words (except the least significant word) are 0

func (*Element) FromMont

func (z *Element) FromMont() *Element

FromMont converts z in place (i.e. mutates) from Montgomery to regular representation sets and returns z = z * 1

func (*Element) Halve

func (z *Element) Halve()

Halve sets z to z / 2 (mod p)

func (*Element) Inverse

func (z *Element) Inverse(x *Element) *Element

Inverse z = x⁻¹ mod q Implements "Optimized Binary GCD for Modular Inversion" https://github.com/pornin/bingcd/blob/main/doc/bingcd.pdf

func (*Element) IsOne

func (z *Element) IsOne() bool

IsOne returns z == 1

func (*Element) IsUint64

func (z *Element) IsUint64() bool

IsUint64 reports whether z can be represented as an uint64.

func (*Element) IsZero

func (z *Element) IsZero() bool

IsZero returns z == 0

func (*Element) Legendre

func (z *Element) Legendre() int

Legendre returns the Legendre symbol of z (either +1, -1, or 0.)

func (*Element) LexicographicallyLargest

func (z *Element) LexicographicallyLargest() bool

LexicographicallyLargest returns true if this element is strictly lexicographically larger than its negation, false otherwise

func (*Element) Marshal

func (z *Element) Marshal() []byte

Marshal returns the regular (non montgomery) value of z as a big-endian byte slice.

func (*Element) MarshalJSON

func (z *Element) MarshalJSON() ([]byte, error)

MarshalJSON returns json encoding of z (z.Text(10)) If z == nil, returns null

func (*Element) Mul

func (z *Element) Mul(x, y *Element) *Element

Mul z = x * y mod q see https://hackmd.io/@gnark/modular_multiplication

func (*Element) Neg

func (z *Element) Neg(x *Element) *Element

Neg z = q - x

func (*Element) NotEqual

func (z *Element) NotEqual(x *Element) uint64

NotEqual returns 0 if and only if z == x; constant-time

func (*Element) Select

func (z *Element) Select(c int, x0 *Element, x1 *Element) *Element

Select is a constant-time conditional move. If c=0, z = x0. Else z = x1

func (*Element) Set

func (z *Element) Set(x *Element) *Element

Set z = x

func (*Element) SetBigInt

func (z *Element) SetBigInt(v *big.Int) *Element

SetBigInt sets z to v (regular form) and returns z in Montgomery form

func (*Element) SetBytes

func (z *Element) SetBytes(e []byte) *Element

SetBytes interprets e as the bytes of a big-endian unsigned integer, sets z to that value (in Montgomery form), and returns z.

func (*Element) SetInt64

func (z *Element) SetInt64(v int64) *Element

SetInt64 sets z to v and returns z

func (*Element) SetInterface

func (z *Element) SetInterface(i1 interface{}) (*Element, error)

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 (*Element) SetOne

func (z *Element) SetOne() *Element

SetOne z = 1 (in Montgomery form)

func (*Element) SetRandom

func (z *Element) SetRandom() (*Element, error)

SetRandom sets z to a random element < q

func (*Element) SetString

func (z *Element) SetString(number string) *Element

SetString creates a big.Int with number and calls SetBigInt on z

The number prefix determines the actual base: A prefix of ”0b” or ”0B” selects base 2, ”0”, ”0o” or ”0O” selects base 8, and ”0x” or ”0X” selects base 16. Otherwise, the selected base is 10 and no prefix is accepted.

For base 16, lower and upper case letters are considered the same: The letters 'a' to 'f' and 'A' to 'F' represent digit values 10 to 15.

An underscore character ”_” may appear between a base prefix and an adjacent digit, and between successive digits; such underscores do not change the value of the number. Incorrect placement of underscores is reported as a panic if there are no other errors.

func (*Element) SetUint64

func (z *Element) SetUint64(v uint64) *Element

SetUint64 sets z to v and returns z

func (*Element) SetZero

func (z *Element) SetZero() *Element

SetZero z = 0

func (*Element) Sqrt

func (z *Element) Sqrt(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 (*Element) Square

func (z *Element) Square(x *Element) *Element

Square z = x * x mod q see https://hackmd.io/@gnark/modular_multiplication

func (*Element) String

func (z *Element) String() string

String returns the decimal representation of z as generated by z.Text(10).

func (*Element) Sub

func (z *Element) Sub(x, y *Element) *Element

Sub z = x - y mod q

func (*Element) Text

func (z *Element) Text(base int) string

Text returns the string representation of z in the given base. Base must be between 2 and 36, inclusive. The result uses the lower-case letters 'a' to 'z' for digit values 10 to 35. No prefix (such as "0x") is added to the string. If z is a nil pointer it returns "<nil>". If base == 10 and -z fits in a uint64 prefix "-" is added to the string.

func (*Element) ToBigInt

func (z *Element) ToBigInt(res *big.Int) *big.Int

ToBigInt returns z as a big.Int in Montgomery form

func (Element) ToBigIntRegular

func (z Element) ToBigIntRegular(res *big.Int) *big.Int

ToBigIntRegular returns z as a big.Int in regular form

func (*Element) ToMont

func (z *Element) ToMont() *Element

ToMont converts z to Montgomery form sets and returns z = z * r²

func (Element) ToRegular

func (z Element) ToRegular() Element

ToRegular returns z in regular form (doesn't mutate z)

func (*Element) Uint64

func (z *Element) Uint64() uint64

Uint64 returns the uint64 representation of x. If x cannot be represented in a uint64, the result is undefined.

func (*Element) UnmarshalJSON

func (z *Element) UnmarshalJSON(data []byte) error

UnmarshalJSON accepts numbers and strings as input See Element.SetString for valid prefixes (0x, 0b, ...)

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