Attackers often take advantage of vulnerabilities in benign software, and the authors of benign software must search their code for bugs in hopes of finding vulnerabilities before they are exploited. But there has been little research on the converse question of whether defenders can turn the tables by finding vulnerabilities in malware. We provide a first affirmative answer to that question. We introduce a new technique, stitched dynamic symbolic execution, that makes it possible to use exploration techniques based on symbolic execution in the presence of functionalities that are common in malware and otherwise hard to analyze, such as decryption and checksums. The technique is based on decomposing the constraints induced by a program, solving only a subset, and then re-stitching the constraint solution into a complete input. We implement the approach in a system for x86 binaries, and apply it to 4 prevalent families of bots and other malware. We find 6 bugs that could be exploited by a network attacker to terminate or subvert the malware. These bugs have persisted across malware revisions for months, and even years. We discuss the possible applications and ethical considerations of this new capability.