Would we be able to detect any suspicious activity on our closed source operating system, or would a backdoor implemented specifically for intelligence agencies allow undetectable communication between hosts?
NSA wouldn't need something to contact their data centres and leak the information, because presumably they can tap some fiber connection or other your data ends up travelling to. All they'd need would be a slow leak of your keys. Even, say, just a few of tweaked data per packet, or tacked on to a file format that is resistant to it, so that they can pick up key information together with the data stream, and they're good.
Or even just intentionally introducing a specific way of generating keys, known to the with sufficient precision to allow them to reduce the key space enough that brute forcing the remaining bits is feasible. For a conceptual example, look up the old Netscape SSL vulnerability (Netscape's SSL used to rely on factors like the host time and pid, that especially on a typical Unix-y system also running services like mail servers that often would include the pid of of the delivery agent in headers etc., was extremely easy to narrow down)
It's difficult to do this in a way that can't/won't sooner or later be discovered, but given the number of likely unintentional mistakes that have been made that still have taken in some cases years to be discovered (publicly at least), it's plausible that there are intentional flaws in at least some software to make it possible for specific parties to be intended to be able to break it.
That's what's bothering me. How feasible would it be to implement something like that on a massive scale and people not finding out? And how about using your router to deny incoming and outgoing traffic unless it's whitelisted? Is there some way for the router to identify the service that sent out a packet and deny or allow it or at least "scan" the content and make sure nothing malicious is being received or sent?