  In many systems privacy of users depends on the number of participants
applying collectively some method to protect their security. Indeed, there are
numerous already classic results about revealing aggregated data from a set of
users. The conclusion is usually as follows: if you have enough friends to
"aggregate" the private data, you can safely reveal your private information.
  Apart from data aggregation, it has been noticed that in a wider context
privacy can be often reduced to being hidden in a crowd. Generally, the
problems is how to create such crowd. This task may be not easy in some
distributed systems, wherein gathering enough "individuals" is hard for
practical reasons.
  Such example are social networks (or similar systems), where users have only
a limited number of semi trusted contacts and their aim is to reveal some
aggregated data in a privacy preserving manner. This may be particularly
problematic in the presence of a strong adversary that can additionally corrupt
some users.
  We show two methods that allow to significantly amplify privacy with only
limited number of local operations and very moderate communication overhead.
Except theoretical analysis we show experimental results on topologies of
real-life social networks to demonstrate that our methods can significantly
amplify privacy of chosen aggregation protocols even facing a massive attack of
a powerful adversary.
  We believe however that our results can have much wider applications for
improving security of systems based on locally trusted relations.
