In this paper, we propose a novel constant-size threshold attribute-based signcryption scheme for securely sharing data through public clouds. Our proposal has several advantages. First, it provides flexible cryptographic access control, while preserving users' privacy as the identifying information for satisfying the access control policy are not revealed. Second, the proposed scheme guarantees both data origin authentication and anonymity thanks to the novel use of attribute based signcryption mechanism, while ensuring the unlinkability between the different access sessions. Third, the proposed signcryption scheme has efficient computation cost and constant communication overhead whatever the number of involved attributes. Finally, our scheme satisfies strong security properties in the random oracle model, namely Indistinguishability against the Adaptive Chosen Ciphertext Attacks (IND-CCA2), Existential Unforgeability against Chosen Message Attacks (EUFCMA) and privacy preservation of the attributes involved in the signcryption process, based on the assumption that the augmented Multi-Sequence of Exponents Decisional Diffie-Hellman (aMSE-DDH) problem and the Computational Diffie Hellman Assumption (CDH) are hard