This paper concerns the problem of maintaining information freshness under passive eavesdropping attacks. The classical three-node wiretap channel model is considered, where a source aims to send its latest status wirelessly to its intended destination while protecting it from being overheard by an eavesdropper. Considering that conventional rate-oriented secrecy metrics, including secrecy capacity and secrecy outage probability, are no longer adequate to measure the timeliness of status update systems, we propose two new age-of-information-based metrics to evaluate the secrecy performance of the considered system. We further devised, analyzed, and optimized two new transmission policies for the source to enhance the secrecy performance. Simulation results are provided to validate our analysis and optimization.