  The popularity of mobile devices and location-based services (LBS) has
created great concern regarding the location privacy of their users.
Anonymization is a common technique that is often used to protect the location
privacy of LBS users. Here, we present an information-theoretic approach to
define the notion of perfect location privacy. We show how LBS's should use the
anonymization method to ensure that their users can achieve perfect location
privacy. First, we assume that a user's current location is independent from
her past locations. Using this i.i.d model, we show that if the pseudonym of
the user is changed before $O(n^{\frac{2}{r-1}})$ observations are made by the
adversary for that user, then the user has perfect location privacy. Here, n is
the number of the users in the network and r is the number of all possible
locations that users can go to. Next, we model users' movements using Markov
chains to better model real-world movement patterns. We show that perfect
location privacy is achievable for a user if the user's pseudonym is changed
before $O(n^{\frac{2}{|E|-r}})$ observations are collected by the adversary for
the user, where |E| is the number of edges in the user's Markov chain model.
