The Precision Time Protocol (PTP) aims to provide highly accurate and synchronised clocks. Its defining standard, IEEE 1588, has a security section ("Annex K") which relies on symmetric-key secrecy. In this paper we present a detailed threat analysis of the PTP standard, in which we highlight the security properties that should be addressed by any security extension. During this analysis we identify a sequence of new attacks and non-cryptographic network-based defenses that mitigate them. We then suggest to replace Annex K's symmetric cryptography by an efficient elliptic-curve Public-Key signatures. We implemented all our attacks to demonstrate their effectiveness, and also implemented and evaluated both the network and cryptographic defenses. Our results show that the proposed schemes are extremely practical, and much more secure than previous suggestions.