The Minimal-Hitting-Set attack (HS-attack) is a well-known, provably optimal exact attack against the anonymity provided by Chaumian Mixes (Threshold-Mixes). This attack allows an attacker to identify the fixed set of communication partners of a given user by observing all messages sent and received by a Chaum Mix. In contrast to this, the Statistical Disclosure attack (SDA) provides a guess of that user’s contacts, based on statistical analyses of the observed message exchanges.
We contribute the first closed formula that shows the influence of traffic distributions on the least number of observations of the Mix to complete the HS-attack. This measures when the Mix fails to hide a user’s partners, such that the user cannot plausibly deny the identified contacts. It reveals that the HS-attack requires asymptotically less observations to identify a user’s partners than the SDA, which guesses them with a given bias. This number of observations is
for the HS-attack and
for the SDA, where
the probability that the attacked user contacts his least frequent partner.