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
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] = 605248206075306171733248481581800960739847691770924913753520744034740935903401304776283802348837311170974282940417 q[base16] = 0x3eeb0416684d19053cb5d240ed107a284059eb647102326980dc360d0a49d7fce97f76a822c00009948a20000000001
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
- func Butterfly(a, b *Element)
- func Modulus() *big.Int
- func MulBy13(x *Element)
- func MulBy3(x *Element)
- func MulBy5(x *Element)
- type Element
- func (z *Element) Add(x, y *Element) *Element
- func (z *Element) Bit(i uint64) uint64
- func (z *Element) BitLen() int
- func (z *Element) Bytes() (res [Limbs * 8]byte)
- func (z *Element) Cmp(x *Element) int
- func (z *Element) Div(x, y *Element) *Element
- func (z *Element) Double(x *Element) *Element
- func (z *Element) Equal(x *Element) bool
- func (z *Element) Exp(x Element, k *big.Int) *Element
- func (z *Element) FitsOnOneWord() bool
- func (z *Element) FromMont() *Element
- func (z *Element) Halve()
- func (z *Element) Inverse(x *Element) *Element
- func (z *Element) IsOne() bool
- func (z *Element) IsUint64() bool
- func (z *Element) IsZero() bool
- func (z *Element) Legendre() int
- func (z *Element) LexicographicallyLargest() bool
- func (z *Element) Marshal() []byte
- func (z *Element) MarshalJSON() ([]byte, error)
- func (z *Element) Mul(x, y *Element) *Element
- func (z *Element) Neg(x *Element) *Element
- func (z *Element) NotEqual(x *Element) uint64
- func (z *Element) Select(c int, x0 *Element, x1 *Element) *Element
- func (z *Element) Set(x *Element) *Element
- func (z *Element) SetBigInt(v *big.Int) *Element
- func (z *Element) SetBytes(e []byte) *Element
- func (z *Element) SetInt64(v int64) *Element
- func (z *Element) SetInterface(i1 interface{}) (*Element, error)
- func (z *Element) SetOne() *Element
- func (z *Element) SetRandom() (*Element, error)
- func (z *Element) SetString(number string) (*Element, error)
- func (z *Element) SetUint64(v uint64) *Element
- func (z *Element) SetZero() *Element
- func (z *Element) Sqrt(x *Element) *Element
- func (z *Element) Square(x *Element) *Element
- func (z *Element) String() string
- func (z *Element) Sub(x, y *Element) *Element
- func (z *Element) Text(base int) string
- func (z *Element) ToBigInt(res *big.Int) *big.Int
- func (z Element) ToBigIntRegular(res *big.Int) *big.Int
- func (z *Element) ToMont() *Element
- func (z Element) ToRegular() Element
- func (z *Element) Uint64() uint64
- func (z *Element) UnmarshalJSON(data []byte) error
Constants ¶
const ( Limbs = 6 // number of 64 bits words needed to represent a Element Bits = 378 // 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 ¶
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.
Modulus q =
q[base10] = 605248206075306171733248481581800960739847691770924913753520744034740935903401304776283802348837311170974282940417 q[base16] = 0x3eeb0416684d19053cb5d240ed107a284059eb647102326980dc360d0a49d7fce97f76a822c00009948a20000000001
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 ¶
BatchInvert returns a new slice with every element inverted. Uses Montgomery batch inversion trick
func NewElement ¶
NewElement returns a new Element from a uint64 value
it is equivalent to
var v Element v.SetUint64(...)
func (*Element) Bit ¶
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 ¶
BitLen returns the minimum number of bits needed to represent z returns 0 if z == 0
func (*Element) Cmp ¶
Cmp compares (lexicographic order) z and x and returns:
-1 if z < x 0 if z == x +1 if z > x
func (*Element) FitsOnOneWord ¶
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 ¶
FromMont converts z in place (i.e. mutates) from Montgomery to regular representation sets and returns z = z * 1
func (*Element) LexicographicallyLargest ¶
LexicographicallyLargest returns true if this element is strictly lexicographically larger than its negation, false otherwise
func (*Element) MarshalJSON ¶
MarshalJSON returns json encoding of z (z.Text(10)) If z == nil, returns null
func (*Element) SetBytes ¶
SetBytes interprets e as the bytes of a big-endian unsigned integer, sets z to that value, and returns z.
func (*Element) SetInterface ¶
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) SetRandom ¶
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 ¶
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) Sqrt ¶
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) Text ¶
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) ToBigIntRegular ¶
ToBigIntRegular returns z as a big.Int in regular form
func (*Element) Uint64 ¶
Uint64 returns the uint64 representation of x. If x cannot be represented in a uint64, the result is undefined.
func (*Element) UnmarshalJSON ¶
UnmarshalJSON accepts numbers and strings as input See Element.SetString for valid prefixes (0x, 0b, ...)
Source Files