Cloud computing is often utilized for file storage. Clients of cloud storage services want to ensure the privacy of their data, and both clients and servers want to use as little storage as possible. Cross-user deduplication is one method to reduce the amount of storage a server uses. Deduplication and privacy are naturally conflicting goals, especially for nearly-identical (``fuzzy'') deduplication, as some information about the data must be used to perform deduplication. Prior solutions thus utilize multiple servers, or only function for exact deduplication. In this paper, we present a single-server protocol for cross-user nearly-identical deduplication based on secure locality-sensitive hashing (SLSH). We formally define our ideal security, and rigorously prove our protocol secure against fully malicious, colluding adversaries with a proof by simulation. We show experimentally that the individual parts of the protocol are computationally feasible, and further discuss practical issues of security and efficiency.