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Timing channel attacks are emerging as real-world threats to computer security. In cryptographic systems, an effective countermeasure against timing attacks is the constant-time programming discipline. However, strictly enforcing the discipline manually is both time-consuming and error-prone. While various tools exist for analyzing/verifying constant-time programs, they sacrifice at least one feature among precision, soundness and efficiency. In this paper, we build CtChecker, a sound static analysis for constant-time programming. Under the hood, CtChecker uses a static information flow analysis to identify violations of constant-time discipline. Despite the common wisdom that sound, static information flow analysis lacks precision for real-world applications, we show that by enabling field-sensitivity, context-sensitivity and partial flow-sensitivity, CtChecker reports fewer false positives compared with existing sound tools. Evaluation on real-world cryptographic systems shows that CtChecker analyzes 24K lines of source code in under one minute. Moreover, CtChecker reveals that some repaired code generated by program rewriters supposedly remove timing channels are still not constant-time.