Attribute-based broadcast encryption with respect to ABE What is the exact difference between attribute-based encryption (ABE) and attribute-based broadcast encryption (ABBE)?
Is ABBE equivalent to broadcast encryption on top of ABE?
At first, the abbreviation “ABBE” is used by Zhou et al. in [2] as a name for his ABE scheme. Zhou builds on top of a CP-ABE scheme which in addition supports “constant” ciphertexts (named CCP-ABE). From this perspective ABBE is a specific ABE scheme based on CP-ABE supporting constant ciphertexts.

However, in a more general way, I see ABE in the context of Broadcast communication as:

A merge ofthe notion of ABE and Broadcast Encryption. Broadcast related use-cases have special requirements which you have to take into consideration not only when applying ABE.

There are broadcast-enabled ABE schemes which build on top of KP-ABE[1,3] or CP-ABE[1,2,2a]. More or less broadcast-enabled ABE schemes face the same challanges and thus focus on similar properties/aspects. 

Some often recurring aspects in ABE-based broadcast scenarios are: 


Size of the ciphertext: a constant message size is often advantage. In addition the message size itself is of interest.
Access Structure: The ciphertext size depends on the embedded access structure. The access structure is important not only when thinking about how to enforce revocation. 
General communication and storage overheads are more of importance in Broadcast scenarios - compare [2a]. 


[1] Attrapadung et al., 2009: Conjunctive Broadcast and Attribute-Based Encryption.

[2] Zhou et al., 2010:  On Efficient Ciphertext-Policy Attribute Based Encryption and Broadcast Encryption, Source: http://eprint.iacr.org/2010/395 

[2a] Zhou et al.2015: Efficient Privacy-Preserving Ciphertext-Policy Attribute Based Encryption and Broadcast Encryption

[3] Sun et al., 2013: A Key-Policy Attribute-based Broadcast Encryption.