If it's a matter of having the computing resources to crack a key, then people's computers are too easily compromised to act as bots to solve any such problem. Our computing infrastructure is simply too vulnerable. To avoid this scenario, you'd have to secure every operating system in the world in a way that they could not be compromised, even in principle. That's as close to a proof as should be needed, unless there's a compelling counterargument to suspect some assumption is false. I have yet to hear such an argument.