Side channel attacks
This sort of thing has been known about for a long time.
For example, you can analyse encrypted chips inside secure tamper proof packages as they perform encryption operations to recover the secure keys inside. To reduce this sort of attack, you have to design the chip to use equal amounts of power for all operations, no matter how simple or complicated.
In a similar theme, the Germans realised that patterns of encrypted messages could be used to deduce attack plans, and commanded that all outposts send a set number of messages of fixed content per day. As one operator got lazy, and just sent a page of the same character each day, this helped break the enigma cypher, as one flaw in Enigma is that a character is never encoded to itself.
Both these solutions trade efficiency with security. If you wanted to avoid these attacks, you would have to send a fixed pattern of packets with random data at all times between end points, replacing the random data with real information as needed. This is not terribly efficient, but is more secure.
Security is always a trade against functionality.