ff

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 Go to latest Published: Jan 1, 2024 License: BSD-3-Clause Imports: 8 Imported by: 2

Documentation

Overview

Package ff provides finite fields of characteristic P381.

Package ff provides finite fields and groups useful for the BLS12-381 curve.

Fp

Fp are elements of the prime field GF(p), where 

p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab

The binary representation takes FpSize = 48 bytes encoded in big-endian form.

Fp2

Fp2 are elements of the finite field GF(p^2) = Fp[u]/(u^2+1) represented as 

(a[1]u + a[0]) in Fp2, where a[0],a[1] in Fp

The binary representation takes Fp2Size = 96 bytes encoded as a[1] || a[0] all in big-endian form.

Fp4

Fp4 is GF(p^4)=Fp2[t]/(t^2-(u+1)). We use the representation a[1]v+a[0]. There is no fixed external form.

Fp6

Fp6 are elements of the finite field GF(p^6) = Fp2[v]/(v^3-u-1) represented as 

(a[2]v^2 + a[1]v + a[0]) in Fp6, where a[0],a[1],a[2] in Fp2

The binary representation takes Fp6Size = 288 bytes encoded as a[2] || a[1] || a[0] all in big-endian form.

Fp12

Fp12 are elements of the finite field GF(p^12) = Fp6[w]/(w^2-v) represented as 

(a[1]w + a[0]) in Fp12, where a[0],a[1] in Fp6

The binary representation takes Fp12Size = 576 bytes encoded as a[1] || a[0] all in big-endian form.

We can also represent this field via Fp4[w]/(w^3-t). This is the struct Fp12alt, used to accelerate the pairing calculation.

Scalar

Scalar are elements of the prime field GF(r), where 

r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001

The binary representation takes ScalarSize = 32 bytes encoded in big-endian form.

Groups

Cyclo6 are elements of the 6th cyclotomic group contained in Fp12. For efficient arithmetic see Granger-Scott "Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions" (https://eprint.iacr.org/2009/565).

URoot are elements of the r-roots of unity group contained in Fp12.

Index

Constants

View Source 
const Fp12Size = 2 * Fp6Size

Fp12Size is the length in bytes of an Fp12 element.

View Source 
const Fp2Size = 2 * FpSize

Fp2Size is the length in bytes of an Fp2 element.

View Source 
const Fp4Size = 4 * FpSize

Fp4Size is the size of an Fp4 element

View Source 
const Fp6Size = 3 * Fp2Size

Fp6Size is the length in bytes of an Fp6 element.

View Source 
const FpSize = 48

FpSize is the length in bytes of an Fp element.

View Source 
const ScalarSize = 32

ScalarSize is the length in bytes of a Scalar.

View Source 
const URootSize = Fp12Size

URootSize is the length in bytes of a root of unit.

Variables

This section is empty.

Functions

func EasyExponentiation 

func EasyExponentiation(g *Cyclo6, f *Fp12)

EasyExponentiation calculates g = f^(p^6-1)(p^2+1), where g becomes an element of the 6-th cyclotomic group.

func FpOrder 

func FpOrder() []byte

FpOrder is the order of the base field for towering returned as a big-endian slice. 

FpOrder = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab.

func HardExponentiation 

func HardExponentiation(u *URoot, g *Cyclo6)

HardExponentiation calculates u = g^(Cy_6(p)/r), where u is a root of unity.

func ScalarOrder 

func ScalarOrder() []byte

ScalarOrder is the order of the scalar field of the pairing groups, order is returned as a big-endian slice. 

ScalarOrder = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001

Types

type Cyclo6 

type Cyclo6 Fp12

Cyclo6 represents an element of the 6th cyclotomic group.

func (*Cyclo6) Frob 

func (z *Cyclo6) Frob(x *Cyclo6)

func (*Cyclo6) Inv 

func (z *Cyclo6) Inv(x *Cyclo6)

func (Cyclo6) IsEqual 

func (z Cyclo6) IsEqual(x *Cyclo6) int

func (Cyclo6) IsIdentity 

func (z Cyclo6) IsIdentity() int

func (*Cyclo6) Mul 

func (z *Cyclo6) Mul(x, y *Cyclo6)

func (*Cyclo6) PowToX 

func (z *Cyclo6) PowToX(x *Cyclo6)

PowToX computes z = x^paramX, where paramX is the parameter of the BLS curve.

func (*Cyclo6) Sqr 

func (z *Cyclo6) Sqr(x *Cyclo6)

func (Cyclo6) String 

func (z Cyclo6) String() string

type Fp 

type Fp struct {
	// contains filtered or unexported fields
}

Fp represents prime field elements as positive integers less than FpOrder.

func (*Fp) Add 

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

func (*Fp) CMov 

func (z *Fp) CMov(x, y *Fp, b int)

CMov sets z=x if b == 0 and z=y if b == 1. Its behavior is undefined if b takes any other value.

func (*Fp) ExpVarTime 

func (z *Fp) ExpVarTime(x *Fp, n []byte)

ExpVarTime calculates z=x^n, where n is the exponent in big-endian order.

func (*Fp) Inv 

func (z *Fp) Inv(x *Fp)

func (Fp) IsEqual 

func (z Fp) IsEqual(x *Fp) int

IsEqual returns 1 if z == x and 0 otherwise.

func (Fp) IsNegative 

func (z Fp) IsNegative() int

IsNegative returns 0 if the least absolute residue for z is in [0,(p-1)/2], and 1 otherwise. Equivalently, this function returns 1 if z is lexicographically larger than -z.

func (Fp) IsZero 

func (z Fp) IsZero() int

IsZero returns 1 if z == 0 and 0 otherwise.

func (*Fp) MarshalBinary 

func (z *Fp) MarshalBinary() ([]byte, error)

MarshalBinary returns a slice of FpSize bytes that contains the minimal residue of z such that 0 <= z < FpOrder (in big-endian order).

func (*Fp) Mul 

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

func (*Fp) Neg 

func (z *Fp) Neg()

func (*Fp) Random 

func (z *Fp) Random(r io.Reader) error

func (*Fp) SetBytes added in v1.2.0 

func (z *Fp) SetBytes(data []byte)

SetBytes assigns to z the number modulo FpOrder stored in the slice (in big-endian order).

func (*Fp) SetOne 

func (z *Fp) SetOne()

func (*Fp) SetString 

func (z *Fp) SetString(s string) error

SetString reconstructs a Fp from a numeric string from 0 to FpOrder-1.

func (*Fp) SetUint64 

func (z *Fp) SetUint64(n uint64)

func (Fp) Sgn0 

func (z Fp) Sgn0() int

func (*Fp) Sqr 

func (z *Fp) Sqr(x *Fp)

func (*Fp) Sqrt 

func (z *Fp) Sqrt(x *Fp) int

Sqrt returns 1 and sets z=sqrt(x) only if x is a quadratic-residue; otherwise, returns 0 and z is unmodified.

func (Fp) String 

func (z Fp) String() string

func (*Fp) Sub 

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

func (*Fp) UnmarshalBinary 

func (z *Fp) UnmarshalBinary(b []byte) error

UnmarshalBinary reconstructs a Fp from a slice that must have at least FpSize bytes and contain a number (in big-endian order) from 0 to FpOrder-1.

type Fp12 

type Fp12 [2]Fp6

Fp12 represents an element of the field Fp12 = Fp6[w]/(w^2-v)., where v in Fp6.

func (*Fp12) Add 

func (z *Fp12) Add(x, y *Fp12)

func (*Fp12) CMov 

func (z *Fp12) CMov(x, y *Fp12, b int)

func (*Fp12) Cjg 

func (z *Fp12) Cjg()

func (*Fp12) Exp 

func (z *Fp12) Exp(x *Fp12, n []byte)

Exp calculates z=x^n, where n is the exponent in big-endian order.

func (*Fp12) Frob 

func (z *Fp12) Frob(x *Fp12)

func (*Fp12) FromFp12Alt 

func (z *Fp12) FromFp12Alt(x *Fp12Cubic)

func (*Fp12) Inv 

func (z *Fp12) Inv(x *Fp12)

func (Fp12) IsEqual 

func (z Fp12) IsEqual(x *Fp12) int

func (Fp12) IsZero 

func (z Fp12) IsZero() int

func (Fp12) MarshalBinary 

func (z Fp12) MarshalBinary() (b []byte, e error)

func (*Fp12) Mul 

func (z *Fp12) Mul(x, y *Fp12)

func (*Fp12) MulBeta 

func (z *Fp12) MulBeta()

func (*Fp12) Neg 

func (z *Fp12) Neg()

func (*Fp12) SetOne 

func (z *Fp12) SetOne()

func (*Fp12) Sqr 

func (z *Fp12) Sqr(x *Fp12)

func (Fp12) String 

func (z Fp12) String() string

func (*Fp12) Sub 

func (z *Fp12) Sub(x, y *Fp12)

func (*Fp12) UnmarshalBinary 

func (z *Fp12) UnmarshalBinary(b []byte) error

type Fp12Cubic 

type Fp12Cubic [3]Fp4

Fp12Cubic represents elements of Fp4[w]/w^3-t

func (*Fp12Cubic) Add 

func (z *Fp12Cubic) Add(x *Fp12Cubic, y *Fp12Cubic)

func (*Fp12Cubic) FromFp12 

func (z *Fp12Cubic) FromFp12(x *Fp12)

func (Fp12Cubic) IsEqual 

func (z Fp12Cubic) IsEqual(x *Fp12Cubic) int

func (*Fp12Cubic) Mul 

func (z *Fp12Cubic) Mul(x *Fp12Cubic, y *Fp12Cubic)

func (*Fp12Cubic) MulLine 

func (z *Fp12Cubic) MulLine(x *Fp12Cubic, y *LineValue)

func (*Fp12Cubic) SetOne 

func (z *Fp12Cubic) SetOne()

func (*Fp12Cubic) Sqr 

func (z *Fp12Cubic) Sqr(x *Fp12Cubic)

func (Fp12Cubic) String 

func (z Fp12Cubic) String() string

type Fp2 

type Fp2 [2]Fp

func (*Fp2) Add 

func (z *Fp2) Add(x, y *Fp2)

func (*Fp2) CMov 

func (z *Fp2) CMov(x, y *Fp2, b int)

func (*Fp2) Cjg 

func (z *Fp2) Cjg()

func (*Fp2) ExpVarTime added in v1.2.0 

func (z *Fp2) ExpVarTime(x *Fp2, n []byte)

ExpVarTime calculates z=x^n, where n is the exponent in big-endian order.

func (*Fp2) Frob 

func (z *Fp2) Frob(x *Fp2)

func (*Fp2) Inv 

func (z *Fp2) Inv(x *Fp2)

func (Fp2) IsEqual 

func (z Fp2) IsEqual(x *Fp2) int

func (Fp2) IsNegative 

func (z Fp2) IsNegative() int

IsNegative returns 1 if z is lexicographically larger than -z; otherwise returns 0.

func (Fp2) IsZero 

func (z Fp2) IsZero() int

func (Fp2) MarshalBinary 

func (z Fp2) MarshalBinary() (b []byte, e error)

func (*Fp2) Mul 

func (z *Fp2) Mul(x, y *Fp2)

func (*Fp2) MulBeta 

func (z *Fp2) MulBeta()

func (*Fp2) Neg 

func (z *Fp2) Neg()

func (*Fp2) SetOne 

func (z *Fp2) SetOne()

func (*Fp2) SetString added in v1.2.0 

func (z *Fp2) SetString(s0, s1 string) (err error)

SetString reconstructs a Fp2 element as s0+s1*i, where s0 and s1 are numeric strings from 0 to FpOrder-1.

func (Fp2) Sgn0 added in v1.2.0 

func (z Fp2) Sgn0() int

func (*Fp2) Sqr 

func (z *Fp2) Sqr(x *Fp2)

func (*Fp2) Sqrt 

func (z *Fp2) Sqrt(x *Fp2) int

Sqrt returns 1 and sets z=sqrt(x) only if x is a quadratic-residue; otherwise, returns 0 and z is unmodified.

func (Fp2) String 

func (z Fp2) String() string

func (*Fp2) Sub 

func (z *Fp2) Sub(x, y *Fp2)

func (*Fp2) UnmarshalBinary 

func (z *Fp2) UnmarshalBinary(b []byte) error

type Fp4 

type Fp4 [2]Fp2

Fp4 is obtained by adjoining t, the square root of u+1 to Fp2

func (*Fp4) Add 

func (z *Fp4) Add(x *Fp4, y *Fp4)

func (*Fp4) Inv 

func (z *Fp4) Inv(x *Fp4)

func (*Fp4) IsEqual 

func (z *Fp4) IsEqual(x *Fp4) int

func (*Fp4) IsZero 

func (z *Fp4) IsZero() int

func (*Fp4) Mul 

func (z *Fp4) Mul(x *Fp4, y *Fp4)

func (*Fp4) Neg 

func (z *Fp4) Neg()

func (*Fp4) SetOne 

func (z *Fp4) SetOne()

func (*Fp4) Sqr 

func (z *Fp4) Sqr(x *Fp4)

func (Fp4) String 

func (z Fp4) String() string

func (*Fp4) Sub 

func (z *Fp4) Sub(x *Fp4, y *Fp4)

type Fp6 

type Fp6 [3]Fp2

func (*Fp6) Add 

func (z *Fp6) Add(x, y *Fp6)

func (*Fp6) CMov 

func (z *Fp6) CMov(x, y *Fp6, b int)

func (*Fp6) Frob 

func (z *Fp6) Frob(x *Fp6)

func (*Fp6) Inv 

func (z *Fp6) Inv(x *Fp6)

func (Fp6) IsEqual 

func (z Fp6) IsEqual(x *Fp6) int

func (Fp6) IsZero 

func (z Fp6) IsZero() int

func (Fp6) MarshalBinary 

func (z Fp6) MarshalBinary() (b []byte, e error)

func (*Fp6) Mul 

func (z *Fp6) Mul(x, y *Fp6)

func (*Fp6) MulBeta 

func (z *Fp6) MulBeta()

func (*Fp6) Neg 

func (z *Fp6) Neg()

func (*Fp6) SetOne 

func (z *Fp6) SetOne()

func (*Fp6) Sqr 

func (z *Fp6) Sqr(x *Fp6)

func (Fp6) String 

func (z Fp6) String() string

func (*Fp6) Sub 

func (z *Fp6) Sub(x, y *Fp6)

func (*Fp6) UnmarshalBinary 

func (z *Fp6) UnmarshalBinary(b []byte) error

type LineValue 

type LineValue [3]Fp2

LineValue a represents a[0]+a[1]*w^2+a[2]*w^3, with all values in Fp2. This lets us shave off a number of Fp2 multiplications.

func (*LineValue) IsZero 

func (z *LineValue) IsZero() int

func (*LineValue) SetOne 

func (z *LineValue) SetOne()

type Scalar 

type Scalar struct {
	// contains filtered or unexported fields
}

Scalar represents positive integers less than ScalarOrder.

func (*Scalar) Add 

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

func (*Scalar) Inv 

func (z *Scalar) Inv(x *Scalar)

func (Scalar) IsEqual 

func (z Scalar) IsEqual(x *Scalar) int

func (Scalar) IsZero 

func (z Scalar) IsZero() int

func (*Scalar) MarshalBinary 

func (z *Scalar) MarshalBinary() ([]byte, error)

MarshalBinary returns a slice of ScalarSize bytes that contains the minimal residue of z such that 0 <= z < ScalarOrder (in big-endian order).

func (*Scalar) Mul 

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

func (*Scalar) Neg 

func (z *Scalar) Neg()

func (*Scalar) Random 

func (z *Scalar) Random(r io.Reader) error

func (*Scalar) Set 

func (z *Scalar) Set(x *Scalar)

func (*Scalar) SetBytes 

func (z *Scalar) SetBytes(data []byte)

SetBytes assigns to z the number modulo ScalarOrder stored in the slice (in big-endian order).

func (*Scalar) SetOne 

func (z *Scalar) SetOne()

func (*Scalar) SetString 

func (z *Scalar) SetString(s string) error

SetString reconstructs a Fp from a numeric string from 0 to ScalarOrder-1.

func (*Scalar) SetUint64 

func (z *Scalar) SetUint64(n uint64)

func (*Scalar) Sqr 

func (z *Scalar) Sqr(x *Scalar)

func (Scalar) String 

func (z Scalar) String() string

func (*Scalar) Sub 

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

func (*Scalar) UnmarshalBinary 

func (z *Scalar) UnmarshalBinary(data []byte) error

UnmarshalBinary reconstructs a Scalar from a slice that must have at least ScalarSize bytes and contain a number (in big-endian order) from 0 to ScalarOrder-1.

type URoot 

type URoot Cyclo6

URoot represents an n-th root of unit, that is an element x in Cyclo6 such that x^n=1, where n = ScalarOrder().

func (*URoot) Exp 

func (z *URoot) Exp(x *URoot, n []byte)

func (*URoot) Inv 

func (z *URoot) Inv(x *URoot)

func (URoot) IsEqual 

func (z URoot) IsEqual(x *URoot) int

func (URoot) IsIdentity 

func (z URoot) IsIdentity() int

func (URoot) MarshalBinary 

func (z URoot) MarshalBinary() ([]byte, error)

func (*URoot) Mul 

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

func (*URoot) SetIdentity 

func (z *URoot) SetIdentity()

func (*URoot) Sqr 

func (z *URoot) Sqr(x *URoot)

func (URoot) String 

func (z URoot) String() string

func (*URoot) UnmarshalBinary 

func (z *URoot) UnmarshalBinary(b []byte) error

Source Files

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