Class Fp12Element

java.lang.Object
org.bouncycastle.crypto.bls.Fp12Element
public final class Fp12Element extends 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 Details

  • Method Details

    • 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(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(Object other)
      Overrides:
      equals in class Object

    • hashCode

      public int hashCode()
      Overrides:
      hashCode in class Object

    • toString

      public String toString()
      Overrides:
      toString in class Object