We present a security analysis of conference key agreement (CKA) in the most adversarial model of device independence (DI). Our protocol can be implemented by any experimental setup that is capable of performing Bell tests [specifically, the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality], and security can in principle be obtained for any violation of the MABK inequality that detects genuine multipartite entanglement among the N parties involved in the protocol. As our main tool, we derive a direct physical connection between the N-partite MABK inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality, showing that certain violations of the MABK inequality correspond to a violation of the CHSH inequality between one of the parties and the other N-1. We compare the asymptotic key rate for device-independent conference key agreement (DICKA) to the case where the parties use N-1 device-independent quantum key distribution protocols in order to generate a common key. We show that for some regime of noise the DICKA protocol leads to better rates.