Class Fp6Element

java.lang.Object
org.bouncycastle.crypto.bls.Fp6Element
public final class Fp6Element extends 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 Details

  • Method Details

    • 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(BigInteger exponent)
      Modular exponentiation by a non-negative integer.

    • 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