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Fenske, Ellis, Mani, Akshaya, Johnson, Aaron, Sherr, Micah.  2017.  Distributed Measurement with Private Set-Union Cardinality. Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security. :2295–2312.

This paper introduces a cryptographic protocol for efficiently aggregating a count of unique items across a set of data parties privately - that is, without exposing any information other than the count. Our protocol allows for more secure and useful statistics gathering in privacy-preserving distributed systems such as anonymity networks; for example, it allows operators of anonymity networks such as Tor to securely answer the questions: how many unique users are using the distributed service? and how many hidden services are being accessed?. We formally prove the correctness and security of our protocol in the Universal Composability framework against an active adversary that compromises all but one of the aggregation parties. We also show that the protocol provides security against adaptive corruption of the data parties, which prevents them from being victims of targeted compromise. To ensure safe measurements, we also show how the output can satisfy differential privacy. We present a proof-of-concept implementation of the private set-union cardinality protocol (PSC) and use it to demonstrate that PSC operates with low computational overhead and reasonable bandwidth. In particular, for reasonable deployment sizes, the protocol run at timescales smaller than the typical measurement period would be and thus is suitable for distributed measurement.

Jansen, Rob, Johnson, Aaron.  2016.  Safely Measuring Tor. Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. :1553–1567.

Tor is a popular network for anonymous communication. The usage and operation of Tor is not well-understood, however, because its privacy goals make common measurement approaches ineffective or risky. We present PrivCount, a system for measuring the Tor network designed with user privacy as a primary goal. PrivCount securely aggregates measurements across Tor relays and over time to produce differentially private outputs. PrivCount improves on prior approaches by enabling flexible exploration of many diverse kinds of Tor measurements while maintaining accuracy and privacy for each. We use PrivCount to perform a measurement study of Tor of sufficient breadth and depth to inform accurate models of Tor users and traffic. Our results indicate that Tor has 710,000 users connected but only 550,000 active at a given time, that Web traffic now constitutes 91% of data bytes on Tor, and that the strictness of relays' connection policies significantly affects the type of application data they forward.