Ciphertext-policy attribute-based encryption (CP-ABE) is a very promising cryptographic primitive that allows a data owner to encrypt messages and manage access policies themselves. Most of the existing CP-ABE schemes suffer from efficiency drawbacks due to long ciphertexts, which impacts their adoption in applications where data are shared and stored. In this work, we aim to address this gap by proposing a CP-ABE which features constant-size ciphertext and supports access policies of an AND-gate and a threshold, which make ciphertext policies more expressive and applicable to many practical applications. Prior CP-ABE schemes with short ciphertexts such as that of Herranz et al. (in: Public key cryptography—PKC, Springer, 2010) only allow access policies to be a single AND-gate or a single threshold only. Combinations between these short CP-ABE constructions will result in systems insecure against collusion attacks, which makes the effort to enable access policies with an AND-gate and a threshold gate at the same time becomes very challenging. We present such a scheme that solves this drawback. Our scheme is efficient, expressive and secure. In our construction, the encryptor chooses two subsets of a certain universe of attributes S1, S2 with a threshold value t1 that only users who have at least t1 attributes in S1 and all attributes in S2 can decrypt the ciphertext. The scheme is proven secure against selective chosen plaintext attacks in the standard model by reduction to the augmented multi-sequence of exponents decisional Diffie–Hellman (aMSE-DDH) problem.