Package org.bouncycastle.crypto.bls

Class Fp12Element

java.lang.Object

org.bouncycastle.crypto.bls.Fp12Element

public final class Fp12Element
extends java.lang.Object

Immutable element of Fp^12 = Fp^6[w] / (w^2 - v), the outer level of the BLS12-381 pairing field tower.

An element is represented as c0 + c1*w where each c_i is an Fp^6 element. The relation w^2 = v (the polynomial generator of Fp^6) gives a clean Karatsuba-style multiplication.

Field Summary

Fields

Modifier and Type	Field	Description
static Fp12Element	ONE
static Fp12Element	ZERO

Method Summary

Modifier and Type	Method	Description
Fp12Element	add(Fp12Element other)
Fp6Element	c0()
Fp6Element	c1()
Fp12Element	conjugate()	Conjugate: (c0 + c1*w) → (c0 - c1*w).
boolean	equals(java.lang.Object other)
Fp12Element	frobenius()	Frobenius: raise to the p power.
Fp12Element	frobeniusSquared()	Frobenius²: raise to the p^2 power.
static Fp12Element	fromFp6(Fp6Element c0)
int	hashCode()
Fp12Element	inverse()	Modular inverse:
boolean	isZero()
Fp12Element	modPow(java.math.BigInteger exponent)	Modular exponentiation by an integer exponent.
Fp12Element	mul(Fp12Element other)	Karatsuba multiplication using w^2 = v:
Fp12Element	neg()
static Fp12Element	of(Fp6Element c0, Fp6Element c1)
Fp12Element	square()	Squaring via complex-style: (a0 + a1*w)^2 = (a0+a1)(a0+a1*v) - a0*a1 - a0*a1*v + 2*a0*a1*w.
Fp12Element	sub(Fp12Element 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 Fp12Element ZERO

ONE

public static final Fp12Element ONE

Method Detail

of

public static Fp12Element of(Fp6Element c0,
                             Fp6Element c1)

fromFp6

public static Fp12Element fromFp6(Fp6Element c0)

c0

public Fp6Element c0()

c1

public Fp6Element c1()

isZero

public boolean isZero()

add

public Fp12Element add(Fp12Element other)

sub

public Fp12Element sub(Fp12Element other)

neg

public Fp12Element neg()

mul

public Fp12Element mul(Fp12Element other)

Karatsuba multiplication using w^2 = v:

   (a0 + a1*w)(b0 + b1*w) = (a0*b0 + a1*b1*v) + ((a0+a1)(b0+b1) - a0*b0 - a1*b1) * w
 

square

public Fp12Element square()

Squaring via complex-style: (a0 + a1*w)^2 = (a0+a1)(a0+a1*v) - a0*a1 - a0*a1*v + 2*a0*a1*w. Three Fp^6 multiplications versus four for the Karatsuba product.

frobeniusSquared

public Fp12Element frobeniusSquared()

Frobenius²: raise to the p^2 power. Combines Fp6Element.frobeniusSquared() on each Fp&sup6; component with a multiplication by the precomputed Fp² coefficient for the w basis on the c1 component.

frobenius

public Fp12Element frobenius()

Frobenius: raise to the p power. Combines Fp6Element.frobenius() on each Fp&sup6; component with a multiplication by the precomputed Fp² coefficient for the w basis on the c1 component.

conjugate

public Fp12Element conjugate()

Conjugate: (c0 + c1*w) → (c0 - c1*w). For BLS12-381 this equals the Fp^12 Frobenius applied 6 times (i.e. raising to the p^6 power), since the cyclotomic structure makes p^6-Frobenius act as conjugation.

inverse

public Fp12Element inverse()

Modular inverse:

   (c0 + c1*w)^-1 = (c0 - c1*w) / (c0^2 - c1^2 * v)
 

modPow

public Fp12Element modPow(java.math.BigInteger exponent)

Modular exponentiation by an integer exponent. Uses right-to-left square-and-multiply on the bit representation of exponent; a negative exponent is handled by inverting and recursing on the absolute value.

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