We have cryptographic algorithms that enable us to communicate securely over untrusted channels. Do we have similar techniques for storing/operating on data on untrusted hardware? Would such a thing even be possible?
Whitebox cryptography isn't enough in this case. Since it protects the keys you'd be able to keep them from being used but it wouldn't protect your data itself while you're encrypting it. To really be secure against something like this you'd need to be using Homomorphic encryption and processing your data without ever decrypting it.