  The discovery of quantum key distribution by Bennett and Brassard (BB84)
bases on the fundamental quantum feature: incompatibility of measurements of
quantum non-commuting observables. In 1991 Ekert showed that cryptographic key
can be generated at a distance with help of entangled (correlated) quantum
particles. Recently Barrett, Hardy and Kent showed that the non-locality used
by Ekert is itself a good resource of cryptographic key even beyond quantum
mechanics. Their result paved the way to new generation of quantum
cryptographic protocols - secure even if the devices are built by the very
eavesdropper. However, there is a question, which is fundamental from both
practical and philosophical point of view: does Nature offer security on
operational level based on the original concept behind quantum cryptography -
that information gain about one observable must cause disturbance to another,
incompatible one?
  Here we resolve this problem by using another striking feature of quantum
world - contextuality. It is a strong version of incompatibility manifested in
the famous Kochen-Specker paradox. The crucial concept is the use of a new
class of families of bipartite probability distributions . We show that if two
persons share systems which locally exhibit the Kochen-Specker paradox and, in
addition, exhibit perfect correlations, then they can extract secure key, even
if they do not trust the quantum devices. This is the first operational
protocol that directly implements the fundamental feature of Nature: the
information gain vs. disturbance trade-off.
  For sake of proof we exhibit a new Bell's inequality which is interesting in
itself. The security is proved not by exploiting strong violation of the
inequality by quantum mechanics (as one usually proceeds), but rather by
arguing, that quantum mechanics cannot violate it too much.
