Secure instant group messaging applications such as WhatsApp, Facebook Messenger, Matrix, and the Signal Application have become ubiquitous in today's internet, cumulatively serving billions of users. Unlike WhatsApp, for example, Matrix can be deployed in a federated manner, allowing users to choose which server manages their chats. To account for this difference in architecture, Matrix employs two novel cryptographic protocols: Olm, which secures pairwise communications, and Megolm, which relies on Olm and secures group communications. Olm and Megolm are similar to and share security goals with Signal and Sender Keys, which are widely deployed in practice to secure group communications. While Olm, Megolm, and Sender Keys have been manually analyzed in the computational model, no symbolic analysis nor mechanized proofs of correctness exist. Using mechanized proofs and computer-aided analysis is important for cryptographic protocols, as hand-written proofs and analysis are error-prone and often carry subtle mistakes. Using Verifpal, we construct formal models of Olm and Megolm, as well as their composition. We prove various properties of interest about Olm and Megolm, including authentication, confidentiality, forward secrecy, and post-compromise security. We also mechanize known limitations, previously discovered attacks, and trivial attacker wins from the specifications and previous literature. Finally, we model Sender Keys and the composition of Signal with Sender Keys in order to draw a comparison with Olm, Megolm, and their composition. From our analysis we conclude the composition of Olm and Megolm has comparable security to the composition of Signal and Sender Keys if Olm pre-keys are signed, and provably worse post-compromise security if Olm pre-keys are not signed.

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