The lack of authentication protection for bootstrapping messages broadcast by base-stations makes impossible for devices to differentiate between a legitimate and a fake base-station. This vulnerability has been widely acknowledged, but not yet fixed and thus enables law-enforcement agencies, motivated adversaries and nation-states to carry out attacks against targeted users. Although 5G cellular protocols have been enhanced to prevent some of these attacks, the root vulnerability for fake base-stations still exists. In this paper, we propose an efficient broadcast authentication protocol based on a hierarchical identity-based signature scheme, Schnorr-HIBS, which addresses the root cause of the fake base-station problem with minimal computation and communication overhead. We implement and evaluate our proposed protocol using off-the-shelf software-defined radios and open-source libraries. We also provide a comprehensive quantitative and qualitative comparison between our scheme and other candidate solutions for 5G base-station authentication proposed by 3GPP. Our proposed protocol achieves at least a 6x speedup in terms of end-to-end cryptographic delay and a communication cost reduction of 31% over other 3GPP proposals.