Over the years, various security notions have been proposed and analyzed in the classical cryptographic model. Even though the study of implication relations among security notions has been very prolific, important open problems remain. For example, for a decade it was not known [Lindell-FOCS-2003] whether the notions of specialized simulator universally composable security and 1-bit specialized simulator universally composable security are equivalent or not. Another open problem [Halpern-Pass-ICS-2010] is finding strong security notions equivalent with appropriately defined game-theoretic notions. In this work, we give an answer to each of these open problems.