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:
q: The curve is defined over Fq r: The order of the curve. b: E: y^2= x^3 + b beta: 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".