CP-ABE (Ciphertext-Policy Attribute-Based Encryption) with hidden access control policy enables data owners to share their encrypted data using cloud storage with authorized users while keeping the access control policies blinded. However, a mechanism to prevent users from achieving successive access to a data owner’s certain number of data objects, which present a conflict of interest or whose combination thereof is sensitive, has yet to be studied. In this paper, we analyze the underlying relations among these particular data objects, introduce the concept of the sensitive data set constraint, and propose a CP-ABE access control scheme with hidden attributes for the sensitive data set constraint. This scheme incorporates extensible, partially hidden constraint policy. In our scheme, due to the separation of duty principle, the duties of enforcing the access control policy and the constraint policy are divided into two independent entities to enhance security. The hidden constraint policy provides flexibility in that the data owner can partially change the sensitive data set constraint structure after the system has been set up.