  We show that monogamy of Bell's inequality violations, which is strictly
weaker condition than no-signaling is enough to prove security of quantum key
distribution. We derive our results for a whole class of monogamy constraints
and generalize our results to any theory that communicating parties may have
access to. Some of these theories do not respect no-signaling principle yet
still allow for secure communication. This proves that no-signaling is only a
sufficient condition for the possibility of secure communication, but not the
necessary one. We also present some new qualitative results concerning the
security of existing quantum key distribution protocols.
