Identity-based encryption (IBE) is a generalization of public-key encryption (PKE) by allowing encryptions to be made to user identities. In this work, we seek to obtain IBE schemes that achieve key-dependent-message (KDM) security with respect to messages that depend on the master secret key. Previous KDM-secure schemes only achieved KDM security in simpler settings, in which messages may only depend on user secret keys. An important motivation behind studying master-KDM security is the application of this notion in obtaining generic constructions of KDM-CCA secure PKE, a primitive notoriously difficult to realize. We give the first IBE that achieves master-KDM security from standard assumptions in pairing groups. Our construction is modular and combines techniques from KDM-secure PKE based from hash-proof systems, together with IBE that admits a tight security proof in the multi-challenge setting, which happens to be unexpectedly relevant in the context of KDM security. In fact, to the best of our knowledge, this is the first setting where techniques developed in the context of realizing tightly secure cryptosystems have led to a new feasibility result. As a byproduct, our KDM-secure IBE, and thus the resulting KDM-CCA-secure PKE both enjoy a tight security reduction, independent of the number of challenge ciphertexts, which was not achieved before.