Backbone uses the official JIRA REST API in order to access both JIRA projects which it integrates. If you provide a HTTPS URL when you setup the synchronization, Backbone will use this encrypted channel in order to communicate with JIRA.
Backbone uses PGP encryption for both file and email protocols. PGP uses a combination of symmetric and asymmetric encryption algorithms. It generates a symmetric encryption key and uses it to encrypt the real data to transport (because symmetric encryption is faster than an asymmetric one). This symmetric key will then be encrypted with an asymmetric encryption algorithm.
During the setup of an synchronization, Backbone exchanges the public keys for the asymmetric encryptions so that they can be used later. The algorithms used in Backbone are RSA with 3000bit key length and AES with 128bit key length.