In the field of Security, process equivalences have been used to char- acterize various information-hiding properties (for instance secrecy, anonymity and non-interference) based on the principle that a protocol P with a variable x satisfies such property if and only if, for every pair of secrets s1 and s2 , P [s1 /x ] is equivalent to P [s2 /x ]. We argue that, in the presence of nondeterminism, the above principle relies on the assumption that the scheduler “works for the benefit of the protocol”, and this is usually not a safe assumption. Non-safe equivalences, in this sense, include trace equivalence and bisimulation. We present a formalism in which we can specify admissible schedulers and, correspondingly, safe ver- sions of these equivalences. We then show safe bisimulation is still a congruence. We conclude showing how to use safe equivalences to characterize information- hiding properties.