In one embodiment, a processor includes: a core to execute instructions, the core including a plurality of mailbox storages and a trust table to store a trust indicator for each of the plurality of mailbox storages; a first core perimeter logic coupled to the core and including a first storage to store state information of the core when the core is in a low power state; and a second core perimeter logic coupled to the first core perimeter logic and the core, the second core perimeter logic including a second storage to store the state information of the core when the first core perimeter logic is in a low power state. Other embodiments are described and claimed.