With 512 PlanetLab nodes each advertising 50 KB/s as malicious Tor middle routers, the theoretical catch probability that at least one bridge circuit traverses a controlled node reaches P(512, 50, 30) ≈ 99% after only 30 circuits. In real-world validation, the 21st circuit created by a bridge client traversed one of the 512 controlled PlanetLab nodes, matching theory. The result generalizes: the 30-circuit exposure threshold applies to any adversary whose nodes' aggregated bandwidth reaches the equivalent of 512 × 50 KB/s = ~25.6 MB/s.

Tor's bandwidth-weighted path selection creates a structural amplification: 60% of middle routers selected across 430 circuits had bandwidth above 1 MB/s, yet only 10% of all Tor routers exceed 1 MB/s. This skew means that an adversary advertising a single high-bandwidth middle node achieves selection probability far exceeding its proportional count in the network, making high-bandwidth Sybil nodes highly cost-effective for bridge discovery.

§V-B, Fig. 8, Fig. 9 detection

The paper identifies three countermeasure classes against bridge discovery: (i) CAPTCHA on email/HTTPS distribution (limited by automated solving services); (ii) uniform random middle-node selection, which defeats bandwidth-Sybil attacks but degrades Tor throughput by routing through low-bandwidth nodes; (iii) DHT-based P2P architecture where no central server holds all bridge IPs, making systematic enumeration infeasible—though DHT systems introduce Sybil and eclipse-attack vulnerabilities of their own.

§VI defense

Large-scale email and HTTPS enumeration of Tor bridges using 500+ PlanetLab nodes and 2,000 Yahoo accounts discovered 2,365 distinct bridges over approximately one month. The bridge https server rate-limits distribution to 3 bridges per 24-bit IP prefix per day, and the email server to 1 reply per account per day; these controls are circumvented by sourcing requests from hundreds of distinct prefixes. Bridge distribution follows a weighted coupon collector model proportional to bridge bandwidth, not uniform probability.

§III-B, §III-C, §IV-A, §V-A detection

A single malicious Tor middle router advertising 10 MB/s bandwidth discovered 2,369 distinct bridges in 14 days. The catch probability is determined solely by the aggregated bandwidth M = k·b of malicious middle routers regardless of how that bandwidth is distributed across nodes: three routers at 10 MB/s each achieve strictly greater catch probability than 512 nodes at 50 KB/s each. This means a well-resourced single node is equivalent to or surpasses hundreds of low-bandwidth Sybil nodes.

§IV-B, §V-B, Theorem 3 detection