A Practical Complexity-Theoretic Analysis of Mix Systems

The Minimal-Hitting-Set attack (HS-attack) [10] is a well-known passive intersection attack against Mix-based anonymity systems, applicable in cases where communication behaviour is non-uniform and unknown. The attack allows an observer to identify uniquely the fixed set of communication partners of a particular user by observing the messages of all senders and receivers using a Mix. Whilst the attack makes use of a provably minimal number of observations, it also requires solving an NP-complete problem. No prior research, to our knowledge, analyses the average complexity of this attack as opposed to its worst case. We choose to explore the HS-attack, as opposed to statistical attacks, to provide a baseline metric and a practical attack for unambiguously identifying anonymous users. We show that the average complexity of the HS-attack can vary between a worst-case exponential complexity and a linear-time complexity according to the Mix parameters. We provide a closed formula for this relationship, giving a precise measure of the resistance of Mixes against the HS-attack in practice, and allowing adjustment of their parameters to reach a desired level of strength.