The sophons are not that powerful and have very limited capabilities. Humans perceived as being very powerful because we doesn't understand how they work and they are being used to frighten us. It's like showing a gun to someone who doesn't know what a gun is. The limitations such line of sight, range, limited ammo are not immediately obvious. It looks as though you have a god-like ability to strike anyone dead by wishing it.

The sophons aren’t particularly powerful, though. They interfere with particle physics experiments by actually interacting with the particle beam, but they can’t do much else. It’s mentioned at some point that the humans are worried they could interfere with computers, but clearly they can’t even do that (if they could, Lou Ji’s plan probably wouldn’t work).

How would you use that private key without it 'leaving' your mind?

The same way you use a private key in a computer without transmitting it?

So how are you intending to do those calculations in your mind?

There is a section in the novel Cryptonomicon where the protagonist is in a jail cell and monitored 24/7. He communicates with another person via a deck of cards, with all of the cryptography performed in the two people's minds. It's annoying to do but should certainly be possible if the fate of the world is at stake.

By being a super-advanced space alien?

I mean, or just practicing. Humans who put in the time can do pretty amazing stuff.