In stream authentication protocols used for large-scale data dissemination in autonomuous systems, authentication is based on the timing of the publication of keys, and depends on trust of the receiver in the sender and belief on whether an intruder can have prior knowledge of a key before it is published by a protocol. Many existing logics and approaches have successfully been applied to specify other types of authentication protocols, but most of them are not appropriate for analysing stream authentication protocols. We therefore consider a fibred modal logic that combines a belief logic with a linear-time temporal logic which can be used to analyse time-varying aspects of certain problems. With this logical system one is able to build theories of trust for analysing stream authentication protocols, which can deal with not only agent beliefs but also the timing properties of an autonomous agent-based system.