Attribute-based encryption () is an extension of traditional public key encryption in which the encryption and decryption phases are based on user’s attributes. More precisely, we focus on ciphertext-policy (-) where the secret-key is associated to a set of attributes and the ciphertext is generated with an access policy. It then becomes feasible to decrypt a ciphertext only if one’s attributes satisfy the used access policy. - scheme with constant-size ciphertext supporting fine-grained access control has been investigated at AsiaCrypt’15 and then at TCC’16. The former makes use of the conversion technique between and spatial encryption, and the later studies the pair encodings framework.

In this paper, we give a new approach to construct such kind of - scheme. More precisely, we propose private - schemes with constant-size ciphertext, supporting CNF (Conjunctive Normal Form) access policy, with the simple restriction that each attribute can only appear times in the access formula. Our two constructions are based on the scheme at Crypto’05. The first scheme is basic selective secure (in the standard model) while our second one reaches the selective security (in the random oracle model).