fr

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Published: Nov 11, 2022 License: Apache-2.0 Imports: 11 Imported by: 0

Documentation

Overview

Package fr contains field arithmetic operations for modulus = 0x4c23a0...300001.

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 [5]uint64

Usage

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 q =

q[base10] = 39705142709513438335025689890408969744933502416914749335064285505637884093126342347073617133569
q[base16] = 0x4c23a02b586d650d3f7498be97c5eafdec1d01aa27a1ae0421ee5da52bde5026fe802ff40300001

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.

Index

Constants

View Source
const (
	Limbs = 5         // number of 64 bits words needed to represent a Element
	Bits  = 315       // number of bits needed to represent a Element
	Bytes = Limbs * 8 // number of bytes needed to represent a Element
)

Variables

This section is empty.

Functions

func Butterfly

func Butterfly(a, b *Element)

Butterfly sets

a = a + b (mod q)
b = a - b (mod q)

func Modulus

func Modulus() *big.Int

Modulus returns q as a big.Int

q[base10] = 39705142709513438335025689890408969744933502416914749335064285505637884093126342347073617133569
q[base16] = 0x4c23a02b586d650d3f7498be97c5eafdec1d01aa27a1ae0421ee5da52bde5026fe802ff40300001

func MulBy13

func MulBy13(x *Element)

func MulBy3

func MulBy3(x *Element)

func MulBy5

func MulBy5(x *Element)

Types

type Element

type Element [5]uint64

Element represents a field element stored on 5 words (uint64)

Element are assumed to be in Montgomery form in all methods.

Modulus q =

q[base10] = 39705142709513438335025689890408969744933502416914749335064285505637884093126342347073617133569
q[base16] = 0x4c23a02b586d650d3f7498be97c5eafdec1d01aa27a1ae0421ee5da52bde5026fe802ff40300001

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.

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 Element
v.SetUint64(...)

func One

func One() Element

One returns 1

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 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, k *big.Int) *Element

Exp z = xᵏ (mod q)

func (*Element) FitsOnOneWord

func (z *Element) FitsOnOneWord() bool

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

It is the responsibility of the caller to convert from Montgomery to Regular form if needed.

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 q)

func (*Element) Inverse

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

Inverse z = x⁻¹ (mod q)

if x == 0, sets and returns z = x

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 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)

x and y must be strictly inferior to q

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 and returns z

func (*Element) SetBigInt

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

SetBigInt sets z to v and returns z

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, 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 (see SetString for valid formats)
*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 uniform random value in [0, q).

This might error only if reading from crypto/rand.Reader errors, in which case, value of z is undefined.

func (*Element) SetString

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

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.

If the number is invalid this method leaves z unchanged and returns nil, error.

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)

x must be strictly inferior to q

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 uint16 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, ...)

Directories

Path Synopsis
Package fft provides in-place discrete Fourier transform.
Package fft provides in-place discrete Fourier transform.
Package fri provides the FRI (multiplicative) commitment scheme.
Package fri provides the FRI (multiplicative) commitment scheme.
Package kzg provides a KZG commitment scheme.
Package kzg provides a KZG commitment scheme.
Package mimc provides MiMC hash function using Miyaguchi–Preneel construction.
Package mimc provides MiMC hash function using Miyaguchi–Preneel construction.
Package permutation provides an API to build permutation proofs.
Package permutation provides an API to build permutation proofs.
Package plookup provides an API to build plookup proofs.
Package plookup provides an API to build plookup proofs.
Package polynomial provides polynomial methods and commitment schemes.
Package polynomial provides polynomial methods and commitment schemes.

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