Degree of anonymity

As an example consider the network shown above, in here ${\displaystyle A,B,C}$ and ${\displaystyle D}$ are users (senders), ${\displaystyle Q,R,S}$, and ${\displaystyle T}$ are servers (receivers), the boxes are mixes, and ${\displaystyle \{A,B\}\in T}$, ${\displaystyle \{A,B,C\}\in S}$ and ${\displaystyle \{A,B,C,D\}\in Q,R}$ where ${\displaystyle \in }$ denotes the anonymity set. Now as there are pool mixes let the cap on the number of incoming messages to wait before sending be ${\displaystyle 2}$; as such if ${\displaystyle A,B}$, or ${\displaystyle C}$ is communicating with ${\displaystyle R}$ and ${\displaystyle S}$ receives a message then ${\displaystyle S}$ knows that it must have come from ${\displaystyle E}$ (as the links between the mixes can only have ${\displaystyle 1}$ message at a time). This is in no way reflected in ${\displaystyle S}$'s anonymity set, but should be taken into account in the analysis of the network.

These papers also served to give concise definitions of an attacker:

Internal/External

an internal attacker controls nodes in the network, whereas an external can only compromise communication channels between nodes.

Passive/Active

an active attacker can add, remove, and modify any messages, whereas a passive attacker can only listen to the messages.

Local/Global

a local attacker has access to only part of the network, whereas a global can access the entire network.

In Crowds there is a global probability of forwarding (${\displaystyle p_{f}}$), which is the probability a node will forward the message internally instead of routing it to the final destination. Let there be ${\displaystyle C}$ corrupt nodes and ${\displaystyle N}$ total nodes. In Crowds the attacker is internal, passive, and local. Trivially ${\displaystyle H_{M}\gets \lg(N-C)}$, and overall the entropy is ${\displaystyle H(x)\gets {\frac {N-p_{f}\cdot (N-C-1)}{N}}\cdot \lg \left[{\frac {N}{N-p_{f}\cdot (N-C-1)}}\right]+p_{f}\cdot {\frac {N-C-1}{N}}\cdot \lg \left[N/p_{f}\right]}$, ${\displaystyle d}$ is this value divided by ${\displaystyle H_{M}}$.

In onion routing let's assume the attacker can exclude a subset of the nodes from the network, then the entropy would easily be ${\displaystyle H(X)\gets \lg(S)}$, where ${\displaystyle S}$ is the size of the subset of non-excluded nodes. Under an attack model where a node can both globally listen to message passing and is a node on the path this decreases to ${\displaystyle H(X)\gets \lg(L)}$, where ${\displaystyle L}$ is the length of the onion route (this could be larger or smaller than ${\displaystyle S}$), as there is no attempt in onion routing to remove the correlation between the incoming and outgoing messages.

In 2004, Diaz, Sassaman, and DeWitte presented an analysis of two anonymous remailers using the Serjantov and Danezis metric, showing one of them to provide zero anonymity under certain realistic conditions.