I am looking into the application of pairings in CPABE in particular. I've notice that the scheme uses a supersingular curve as the basis of the pairing. Looking through Ben Lynn's thesis for the Pairing Based Cryptography Library, there is only a single supersingular elliptic curve implemented, which is a Type A curve. However, he also mentions that type E curves provide the possibility of a symmetric pairing. I am unclear as to how this works, and how to generate such a curve that is a symmetric pairing to be used in schemes like CPABE, as the thesis only explains arbitrary Type E curves.
This case for symmetric pairing was mentioned in http://eprint.iacr.org/2001/003.
To sum the results given in that paper, you can get symmetric pairing in a few cases: