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This paper introduces a new cryptographic primitive called a private resource allocator (PRA) that can be used to allocate resources (e.g., network bandwidth, CPUs) to a set of clients without revealing to the clients whether any other clients received resources. We give several constructions of PRAs that provide guarantees ranging from information-theoretic to differential privacy. PRAs are useful in preventing a new class of attacks that we call allocation-based side-channel attacks. These attacks can be used, for example, to break the privacy guarantees of anonymous messaging systems that were designed specifically to defend against side-channel and traffic analysis attacks. Our implementation of PRAs in Alpenhorn, which is a recent anonymous messaging system, shows that PRAs increase the network resources required to start a conversation by up to 16× (can be made as low as 4×in some cases), but add no overhead once the conversation has been established.