Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data and all computations done inside the nodes.

Is it still possible under these conditions to ensure by mathematical means that this secret service cannot find out who communicates with whom, if one designs internet protocols in a suitable way?

My feeling is that the answer is likely "yes", but I am not working in cryptography. -- Probably a cryptographer can tell more.