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Security is a core responsibility for Function-as-a-Service (FaaS) providers. The prevailing approach has each function execute in its own container to isolate concurrent executions of different functions. However, successive invocations of the same function commonly reuse the runtime state of a previous invocation in order to avoid container cold-start delays when invoking a function. Although efficient, this container reuse has security implications for functions that are invoked on behalf of differently privileged users or administrative domains: bugs in a function's implementation, third-party library, or the language runtime may leak private data from one invocation of the function to subsequent invocations of the same function. Groundhog isolates sequential invocations of a function by efficiently reverting to a clean state, free from any private data, after each invocation. The system exploits two properties of typical FaaS platforms: each container executes at most one function at a time and legitimate functions do not retain state across invocations. This enables Groundhog to efficiently snapshot and restore function state between invocations in a manner that is independent of the programming language/runtime and does not require any changes to existing functions, libraries, language runtimes, or OS kernels. We describe the design of Groundhog and its implementation in OpenWhisk, a popular production-grade open-source FaaS framework. On three existing benchmark suites, Groundhog isolates sequential invocations with modest overhead on end-to-end latency (median: 1.5%, 95p: 7%) and throughput (median: 2.5%, 95p: 49.6%), relative to an insecure baseline that reuses the container and runtime state.