  A crucial task for secure communication networks is to determine the minimum
of physical requirements to certify a cryptographic protocol. A widely accepted
candidate for certification is the principle of relativistic causality which is
equivalent to the disallowance of causal loops. Contrary to expectations, we
demonstrate how correlations allowed by relativistic causality could be
exploited to break security for a broad class of multi-party protocols (all
modern protocols belong to this class). As we show, deep roots of this dramatic
lack of security lies in the fact that unlike in previous (quantum or
no-signaling) scenarios the new theory "decouples" the property of extremality
and that of statistical independence on environment variables. Finally, we find
out, that the lack of security is accompanied by some advantage: the new
correlations can reduce communication complexity better than the no-signaling
ones. As a tool for analysis of this advantage, we characterize relativistic
causal polytope by its extremal points in the simplest multi-party scenario
that goes beyond the no-signaling paradigm.
