# Elliptic curves with pairings at 128-bit security in libpbc?

I am using Ben Lynn's libpbc to implement a BLS threshold signature scheme and I am aiming for 128-bit security (i.e., a forgery attack should take around $2^{128}$ tries).

I was wondering what curves in libpbc would provide this level of security?

There are some indications about discrete log security here but I'm not sure how to interpret them. (I don't know too much about elliptic curve cryptography.)

For example, a "type A" curve with a "base field size" of 512 bits and embedding degree 2 is said to have 1024-bit discrete log security in the link above. Does that mean a successful attack on that curve takes around $2^{1024}$ operations? Or does it mean an attack on that curve is equivalent to an attack on a group $G(q) \subset \mathbb{Z}_p^*$ where $|q| = 1024$ bits?

Any tips would be appreciated!

Later edit: A more precise estimate of the security parameter of libpbc's curves would be nice!