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Contact tracing is an essential tool in containing infectious diseases such as COVID-19. Many countries and research groups have launched or announced mobile apps to facilitate contact tracing by recording contacts between users with some privacy considerations. Most of the focus has been on using random tokens, which are exchanged during encounters and stored locally on users' phones. Prior systems allow users to search over released tokens in order to learn if they have recently been in the proximity of a user that has since been diagnosed with the disease. However, prior approaches do not provide end-to-end privacy in the collection and querying of tokens. In particular, these approaches are vulnerable to either linkage attacks by users using token metadata, linkage attacks by the server, or false reporting by users. In this work, we introduce Epione, a lightweight system for contact tracing with strong privacy protections. Epione alerts users directly if any of their contacts have been diagnosed with the disease, while protecting the privacy of users' contacts from both central services and other users, and provides protection against false reporting. As a key building block, we present a new cryptographic tool for secure two-party private set intersection cardinality (PSI-CA), which allows two parties, each holding a set of items, to learn the intersection size of two private sets without revealing intersection items. We specifically tailor it to the case of large-scale contact tracing where clients have small input sets and the server's database of tokens is much larger.