Certificate-based cryptography which is attracted great interest can solve the certificate management issue of the traditional public-key cryptography system, at the same time, which can also avoid the key escrow in the identity-based cryptography, thus, it has attracted attention of cryptography researchers. The traditional security models assume that any adversary cannot obtain the leakage information on the internal secret states, such as secret keys, however, some leakage can be leaked through various leakage attacks in the actual environment. In addition, many cryptographic schemes with broadcast communication function were created, because broadcast communication has higher efficiency of message transmission. To further provide leakage resilience and broadcast communication for certificate-based broadcast key encapsulation mechanism (CB-BKEM), a concrete construction of CB-BKEM is proposed, and the leakage-resilient chosen-ciphertext attacks security is proved based on decisional Diffie-Hellman assumption. To further improve the practicability of CB-BKEM, continuous leakage-resilient CB-BKEM is researched, and the continuous leakage resilience of CB-BKEM can be obtained by performing key update. The performance analysis shows that the proposed construction has higher computational efficiency while maintaining the provable security, the leakage resilience and the broadcast communication.