Entropy Bounds for Traffic Confirmation

Consider an open MIX-based anonymity system with N participants and a batch size of b. Assume a global passive adversary who targets a given participant Alice with a set RA of m communicating partners. Let H(RA | Bt) denote the entropy of RA as calculated by the adversary given t message sets (MIX batches) where Alice is a sender in each message set. Our main result is to express the rate at which the anonymity of Alice (as measured by RA) degrades over time as a function of the main parameters N , b and m. Assuming m, b < √ N , we prove that there is a threshold t∗ = O(m ln N) such that when t = t∗ + 5cm, for any integer c > 0, then Pr(H(RA | Bt) = 0) > 1− e−c (1) where the probabilities are computed over the random communication model. Thus once the attacker has collected a threshold t∗ of messages sets, each additional 5m message sets collected decreases H(RA | Bt) towards zero geometrically. We provide formulas for computing the implied constant in t∗ = O(m ln N) and also to improve the constant 5 in t = t∗ + 5cm. When N > O(b m lnm), or in general for large N , the threshold can be improved to t∗ = O(m(ln b + lnm)), which is independent of N . Further, if b = O(m) then the threshold simplifies to t∗ = O(m ln m) and this bound is asymptotically optimal since the coupon collector problem indicates that t∗ = Ω(m ln m). ∗The author can be contacted via email at [email protected].