Advise ideas on secure storage and file synchronization

I'm going to do a project on the topic of secure storage and file synchronization.

I want to write a program to store files in an encrypted form, as it does TrueCrypt, synchronization of files on P2P, as bittorrent sync or Syncthing and version control files. As well as synchronization and storage of some files in the encrypted form on cloud services such as DropBox.

Unfortunately, this is not enough for the project, the teacher wants the project to contain the math.

Advise any algorithms or ideas on the topic.