Type F internals

Using carefully crafted polynomials, k = 12 pairings can be constructed. Only 160 bits are needed to represent elements of one group, and 320 bits for the other.

Also, embedding degree k = 12 allows higher security short signatures. (k = 6 curves cannot be used to scale security from 160-bits to say 256-bits because finite field attacks are subexponential.)

f_param struct fields:

  The curve is defined over Fq
  The order of the curve.
  E: y^2= x^3 + b
  A quadratic nonresidue in Fq: used in quadratic extension.
alpha0, alpha1:
  x^6 + alpha0 + alpha1 sqrt(beta) is irreducible: used in sextic extension.

Discovered by Barreto and Naehrig, "Pairing-friendly elliptic curves of prime order".