There is a software solution called blinding. The idea is to 1) scramble your input (message, private key) by multiplying with a random blinding factor, 2) do the crypto math and 3) divide out the random blinding factor.

Since all the crypto math happens with what is now essentially random numbers, you gain no knowledge from power analysis.

Downside of this method is that it requires additional operations and your crypto algorithm needs to be suitable for this. There is literature on this method, but don't be confused by "blind signatures" which is something different.

Any crypto wizz around that can tell me if secp256k1 ecdsa can be blinded?