If I gave someone the encryption key to my router could they hack my computer?
It Depends.
(Don’t you hate when people do that?)
Really, though, it does depend on what programs your computer is running at the time. If you don’t have any programs that listen to the outside world, then you really can’t get compromised barring a serious bug in the OS itself. The bug that enabled WinNuke to exist is an example, but WinNuke would only force a reboot; it couldn’t be used to get access to files on the afflicted computer. That’s bad, sure, and it can cost you a lot of time, money, and grief, but subverting a system implies being able to access the data on that system in some unauthorized fashion, and WinNuke didn’t enable that.
Even if you are running programs that listen for connections from the outside world, you could still be completely safe if none of them have serious bugs which would allow an outsider to use them to do malicious things.
Note that just crashing the program isn’t necessarily considered a seriously malicious act. It’s annoying, and it shouldn’t happen, but if that program’s crash is all that happens, very little actual harm is done. The dangerous bugs are the ones that let an attacker leverage an unusual situation (a bug, in other words) into access they’re not supposed to have.
So, obviously, what an attacker really wants is something on the inside, some program that’s already running on your machine which has the unknown-to-you feature of being a gateway into the private parts of your system. That is why running software from parts unknown is a bad idea: any one of them could be that convenient gateway.
So it’s likely fine if you take a really close look at the programs you’re running, but don’t do it if your friend insists on installing this really cool game you’ve never heard of on your system first.
Your computer can be controlled remotely, but this is not easy and requires a program installed on your side. Many apple products (I saw this work with two Apple 3gs phones) can “sync” (or was it “mirror”) with a desktop. This allows the desktop to appear on the apple device and be used just like you are sitting at the computer.
Also note that the two methods above do not require access to your router. It works as long as there is an internet connection.
No matter what method is used though, your computer is safe if it is off and cannot be turned on remotely.
You’d think that, wouldn’t you?
Here’s the money quote, which is a bit buried:
It’s not exactly common, not exactly rare, and I don’t know how relevant it is to the average SoHo (small office home office) or personal laptop or desktop system.
Yeah, I would assume people would turn that on/off in BIOS if they want to make sure their computer can/can’t be turned on remotely. However, it is true that most laptops will not allow access to the BIOS.
Wake on LAN is turned off in the BIOS by default, for obvious reasons. Even if it’s on, it’s not easy to get it to work.