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Cache side-channel attacks leverage secret-dependent footprints in CPU cache to steal confidential information, such as encryption keys. Due to the lack of a proper abstraction for reasoning about cache side channels, existing static program analysis tools that can quantify or mitigate cache side channels are built on very different kinds of abstractions. As a consequence, it is hard to bridge advances in quantification and mitigation research. Moreover, existing abstractions lead to imprecise results. In this paper, we present a novel abstraction, called differential set, for analyzing cache side channels at compile time. A distinguishing feature of differential sets is that it allows compositional and precise reasoning about cache side channels. Moreover, it is the first abstraction that carries sufficient information for both side channel quantification and mitigation. Based on this new abstraction, we develop a static analysis tool DSA that automatically quantifies and mitigates cache side channel leakage at the same time. Experimental evaluation on a set of commonly used benchmarks shows that DSA can produce more precise leakage bound as well as mitigated code with fewer memory footprints, when compared with state-of-the-art tools that only quantify or mitigate cache side channel leakage.