  Quantum cryptography shows that one can guarantee the secrecy of correlation
on the sole basis of the laws of physics, that is without limiting the
computational power of the eavesdropper. The usual security proofs suppose that
the authorized partners, Alice and Bob, have a perfect knowledge and control of
their quantum systems and devices; for instance, they must be sure that the
logical bits have been encoded in true qubits, and not in higher-dimensional
systems. In this paper, we present an approach that circumvents this strong
assumption. We define protocols, both for the case of bits and for generic
$d$-dimensional outcomes, in which the security is guaranteed by the very
structure of the Alice-Bob correlations, under the no-signalling condition. The
idea is that, if the correlations cannot be produced by shared randomness, then
Eve has poor knowledge of Alice's and Bob's symbols. The present study assumes,
on the one hand that the eavesdropper Eve performs only individual attacks
(this is a limitation to be removed in further work), on the other hand that
Eve can distribute any correlation compatible with the no-signalling condition
(in this sense her power is greater than what quantum physics allows). Under
these assumptions, we prove that the protocols defined here allow extracting
secrecy from noisy correlations, when these correlations violate a Bell-type
inequality by a sufficiently large amount. The region, in which secrecy
extraction is possible, extends within the region of correlations achievable by
measurements on entangled quantum states.
