Edge networks connected to the Internet need effective monitoring techniques to drive routing decisions and detect violations of Service Level Agreements (SLAs). However, existing measurement tools, like ping, traceroute, and trajectory sampling, are vulnerable to attacks that can make a path look better than it really is. In this paper, we design and analyze path-quality monitoring protocols that reliably raise an alarm when the packet-loss rate and delay exceed a threshold, even when an adversary tries to bias monitoring results by selectively delaying, dropping, modifying, injecting, or preferentially treating packets. Despite the strong threat model we consider in this paper, our protocols are efficient enough to run at line rate on high-speed routers. We present a secure sketching protocol for identifying when packet loss and delay degrade beyond a threshold. This protocol is extremely lightweight, requiring only 250-600 bytes of storage and periodic transmission of a comparably sized IP packet to monitor billions of packets. We also present secure sampling protocols that provide faster feedback and accurate round-trip delay estimates, at the expense of somewhat higher storage and communication costs. We prove that all our protocols satisfy a precise definition of secure path-quality monitoring and derive analytic expressions for the trade-off between statistical accuracy and system overhead. We also compare how our protocols perform in the client-server setting, when paths are asymmetric, and when packet marking is not permitted.