When users run a group key exchange (GKE) protocol, they usually extract the key from some auxiliary (ephemeral) secret information generated during the execution. Strong corruptions are attacks by which an adversary can reveal these ephemeral secrets, in addition to the possibly used long-lived keys. Undoubtedly, security impact of strong corruptions is serious, and thus specifying appropriate security requirements and designing secure GKE protocols appears an interesting yet challenging task -- the aim of our paper. We start by investigating the current setting of strong corruptions and derive some further refinements such as opening attacks that allow to reveal ephemeral secrets of users without their long-lived keys. This allows to consider even stronger attacks against honest, but "opened" users. Further, we define strong security goals for GKE protocols in the presence of such powerful adversaries and propose a Tree Diffie-Hellman protocol immune to their attacks. Our ...