Attribute-Based Encryption (ABE) is a cryptographic primitive which supports fine-grained access control on encrypted data, making it an appealing building block for many applications. Multi-Authority Attribute-Based Encryption (MA-ABE) is a generalization of ABE where the central authority is distributed across several independent parties. We provide the first MA-ABE scheme from prime-order pairings where no trusted setup is needed and where the attribute universe of each authority is unbounded. Our constructions rely on a common modular blueprint that uses an Identity-Based Functional Encryption scheme for inner products (ID-IPFE) as an underlying primitive. Our presentation leads to simple proofs of security and brings new insight into the algebraic design choices that seem common to existing schemes. In particular, the well-known MA-ABE construction by Lewko and Waters (EUROCRYPT 2011) can be seen as a specific instantiation of our modular construction. Our schemes enjoy all of their advantageous features, and the improvements mentioned. Furthermore, different instantiations of the core ID-IPFE primitive lead to various security/efficiency trade-offs: we propose an adaptively secure construction proven in the generic group model and a selectively secure one that relies on SXDH. As in previous work, we rely on a hash function (to generate matching randomness for the same user across different authorities while preserving collusion resistance) that is modeled as a random oracle.