Attribute-based encryption for circuits

In an attribute-based encryption (ABE) scheme, a ciphertext is associated with an l-bit public index pind and a message m, and a secret key is associated with a Boolean predicate P. The secret key allows to decrypt the ciphertext and learn m iff P(pind) = 1. Moreover, the scheme should be secure against collusions of users, namely, given secret keys for polynomially many predicates, an adversary learns nothing about the message if none of the secret keys can individually decrypt the ciphertext.