org.bouncycastle.crypto.bls

Class Fp6Element

java.lang.Object

org.bouncycastle.crypto.bls.Fp6Element

public final class Fp6Element
extends java.lang.Object

Immutable element of Fp^6 = Fp^2[v] / (v^3 - (1 + I)), the cubic extension of Fp2Element used as the inner level of the BLS12-381 pairing field tower.

An element is represented as c0 + c1*v + c2*v^2 where each c_i is an Fp^2 element. Multiplication uses the relation v^3 = NON_RESIDUE where NON_RESIDUE = 1 + I, the standard BLS12-381 cubic non-residue choice.

Field Summary

Fields

Modifier and Type	Field and Description
static Fp6Element	ONE
static Fp6Element	ZERO

Method Summary

Modifier and Type	Method and Description
Fp6Element	add(Fp6Element other)
Fp2Element	c0()
Fp2Element	c1()
Fp2Element	c2()
boolean	equals(java.lang.Object other)
Fp6Element	frobenius() Frobenius: raise to the p power.
Fp6Element	frobeniusSquared() Frobenius²: raise to the p^2 power.
static Fp6Element	fromFp2(Fp2Element c0)
int	hashCode()
Fp6Element	inverse() Inverse via the standard cubic-extension formula: t0 = c0^2 - xi*c1*c2 t1 = xi*c2^2 - c0*c1 t2 = c1^2 - c0*c2 norm = c0*t0 + xi*c2*t1 + xi*c1*t2 inv = (t0 + t1*v + t2*v^2) / norm
boolean	isZero()
Fp6Element	modPow(java.math.BigInteger exponent) Modular exponentiation by a non-negative integer.
Fp6Element	mul(Fp6Element other) Schoolbook multiplication using v^3 = NON_RESIDUE: (a0 + a1*v + a2*v^2) * (b0 + b1*v + b2*v^2) = (a0*b0 + (a1*b2 + a2*b1)*xi) + (a0*b1 + a1*b0 + a2*b2*xi) * v + (a0*b2 + a1*b1 + a2*b0) * v^2 where xi = NON_RESIDUE.
Fp6Element	mulByV() Multiplies an element of Fp^6 by v, the polynomial generator.
Fp6Element	mulFp2(Fp2Element s) Multiplies by an Fp^2 scalar.
Fp6Element	neg()
static Fp6Element	of(Fp2Element c0, Fp2Element c1, Fp2Element c2)
Fp6Element	square() Squaring via the Chung-Hasan SQR3 algorithm: 6 Fp^2 multiplications versus 9 for the schoolbook product.
Fp6Element	sub(Fp6Element other)
java.lang.String	toString()

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

ZERO

public static final Fp6Element ZERO

ONE

public static final Fp6Element ONE

Method Detail

of

public static Fp6Element of(Fp2Element c0,
                            Fp2Element c1,
                            Fp2Element c2)

fromFp2

public static Fp6Element fromFp2(Fp2Element c0)

c0

public Fp2Element c0()

c1

public Fp2Element c1()

c2

public Fp2Element c2()

isZero

public boolean isZero()

add

public Fp6Element add(Fp6Element other)

sub

public Fp6Element sub(Fp6Element other)

neg

public Fp6Element neg()

mul

public Fp6Element mul(Fp6Element other)

Schoolbook multiplication using v^3 = NON_RESIDUE:

   (a0 + a1*v + a2*v^2) * (b0 + b1*v + b2*v^2)
   = (a0*b0 + (a1*b2 + a2*b1)*xi)
   + (a0*b1 + a1*b0 + a2*b2*xi) * v
   + (a0*b2 + a1*b1 + a2*b0) * v^2
 

where xi = NON_RESIDUE.

square

public Fp6Element square()

Squaring via the Chung-Hasan SQR3 algorithm: 6 Fp^2 multiplications versus 9 for the schoolbook product.

mulByV

public Fp6Element mulByV()

Multiplies an element of Fp^6 by v, the polynomial generator. Useful for the Fp^12 multiplication formula.

mulFp2

public Fp6Element mulFp2(Fp2Element s)

Multiplies by an Fp^2 scalar.

inverse

public Fp6Element inverse()

Inverse via the standard cubic-extension formula:

   t0 = c0^2 - xi*c1*c2
   t1 = xi*c2^2 - c0*c1
   t2 = c1^2 - c0*c2
   norm = c0*t0 + xi*c2*t1 + xi*c1*t2
   inv = (t0 + t1*v + t2*v^2) / norm
 

frobeniusSquared

public Fp6Element frobeniusSquared()

Frobenius²: raise to the p^2 power. Identity on Fp² components (Frobenius² in Fp² is identity), with the v-basis components scaled by precomputed Fp² constants.

frobenius

public Fp6Element frobenius()

Frobenius: raise to the p power. The Fp² components are conjugated (Fp² Frobenius is conjugation since p ≡ 3 mod 4), then the v-basis components are scaled by precomputed coefficients.

modPow

public Fp6Element modPow(java.math.BigInteger exponent)

Modular exponentiation by a non-negative integer.

equals

public boolean equals(java.lang.Object other)

Overrides:

equals in class java.lang.Object

hashCode

public int hashCode()

Overrides:

hashCode in class java.lang.Object

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object