Crowds provides perfect anonymity against a corrupt receiver (i.e. formula_23 see Degree of anonymity) as all members appear equally likely to have been the initiator. As we showed against collaborating corrupt nodes Crowds provides probable innocence as long as formula_24 (see the paper for the derivation of this), and provides a degree of anonymity formula_25. Against the predecessor attack Crowds succumbs in formula_26; this attack works by a corrupt node retaining the previous hop in the path, as this will be the sender more than any other node over the rounds of rebuilding the network it will become apparent who the initiator is. Reiter and Rubin mention this and recommend "long" (and if possible infinite) time between path reformations (caused when a node in the path leaves the network). Crowds is unable to protect against a global eavesdropper as it cannot use encryption on the links, this is because each node in Crowds is able to communicate with every other node (a fully connected graph), because of this setting up symmetric keys requires formula_27 pairwise keys; this is too large of a number to be feasible. Against a local eavesdropper again Crowds provides no protection as the eavesdropper will see a message coming out of a node that did not enter, and this positively identifies the node as the sender.