Distributed broadcast encryption (DBE) is a specific kind of broadcast encryption (BE) where users independently generate their own public and private keys, and a sender can efficiently create a ciphertext for a subset of users by using the public keys of the subset users. Previously proposed DBE schemes have been proven in the adaptive chosen-plaintext attack (CPA) security model and have the disadvantage of requiring linear number of pairing operations when verifying the public key of a user. In this paper, we propose an efficient DBE scheme in bilinear groups and prove adaptive chosen-ciphertext attack (CCA) security for the first time. To do this, we first propose a semi-static CCA secure DBE scheme and prove the security under the $q$-Type assumption. Then, by modifying the generic transformation of Gentry and Waters that converts a semi-static CPA secure DBE scheme into an adaptive CPA secure DBE scheme to be applied to CCA secure DBE schemes, we propose an adaptive CCA secure DBE scheme and prove its adaptive CCA security. Our proposed DBE scheme is efficient because it requires constant size ciphertexts, constant size private keys, and linear size public keys, and the public key verification requires only a constant number of pairing operations and efficient group membership checks.