Usual signatures do not allow homomorphic operations on the signed values. You will have to use special fully homomorphic signatures (FHS) for that.

Suppose you did this and encrypted + signed your values with FHE and FHS as Sign(Encrypt(x)) and Sign(Encrypt(y)). Now you want to add x and y. For that, you will have to run the homomorphic addition of Encrypt(·) values using homomorphic operations of Sign(·), i.e. you do Encrypt(x) + Encrypt(y) inside Sign(·), not just x + y.

This means that the overheads introduced by FHE and FHE do not just add up, they get composed (as in composition of functions). This will be a terrible blow-up in performance. I guess, for this reason people try to build authenticated FHE that will have the properties of FHE + FHS but as a one, more efficient block.

Then your input is signed. And now what? How does this authenticate the output?

I suppose you can just rerun the function? Most cases I know don’t have a large nested set of operations.

Then you didnt need to do the remote operation... Some operations are verifiable though in less time than doing the whole computation.

Maybe in the way the operation affects the signature?