Information hiding is a general concept which refers to the goal of preventing an adversary to infer secret information from the observables. Anonymity and Information Flow are examples of this notion. We study the problem of information hiding in systems characterized by the coexistence of randomization and concurrency. It is well known that the presence of nondeterminism, due to the possible interleavings and interactions of the parallel components, can cause unintended information leaks. The most established approach to solve this problem is to fix the strategy of the scheduler beforehand. In this work, we propose a milder restriction on the schedulers, and we define the notion of strong (probabilistic) information hiding under various notions of observables. Furthermore, we propose a method, based on the notion of automorphism, to verify that a system satisfies the property of strong information hiding, namely strong anonymity or non-interference, depending on the context. Through the paper, we use the canonical example of the Dining Cryptographers to illustrate our ideas and techniques.