Anonymity of Memory-Limited Chaum Mixes Under Timing Analysis: An Information Theoretic Perspective

Abstract

Anonymous communication, where users communicate without revealing the identities of communicating parties or the paths of data flow is critical in data networks. On the Internet, Chaum mixes, intermediate nodes, or proxy servers, which use layered encryption and packet shuffling methods to hide source identities, are used to provide anonymity to network users. In this paper, an information theoretic framework is developed to study the maximum anonymity achievable by packet shuffling when the mixes are memory limited-in other words, they can store a finite number of packets. Using the Shannon entropy of the a posteriori distribution of packet sources from an eavesdropper's perspective as the measure of anonymity, the maximum achievable anonymity of a single mix with buffer size b (packets) serving two independent Poisson sources with equal arrival rates is shown to be log [2 cos (π/b+3)]. For a general multiuser b+3 system, the maximum anonymity as buffer size b → ∞ is shown to approach the entropy of the source arrival probabilities at a convergence rate no lesser than 1/b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . When the arrival probabilities of the general multiuser system can be expressed as a rational fraction k/2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> for some fixed n, this convergence rate is shown to be achievable. The anonymity analysis is extended to a general network of mixes connecting the sources to a common destination, where the source anonymity achievable on the destination link is shown to be lower bounded by a weighted sum of the anonymity achievable by each individual mix.