Maybe in some cases that is possible, but it is also possible that your function requires as much time to verify as the FHE evaluator would spend on (honestly)

Maybe in some cases that is possible, but it is also possible that your function requires as much time to verify as the FHE evaluator would spend on (honestly) computing the result. In other words it would make outsourcing pointless; this is briefly discussed in the related work section of this paper: https://eprint.iacr.org/2014/202.pdf https://eprint.iacr.org/2014/202.pdf You also have to be careful to ensure that the canary cannot be identified in the plaintext; otherwise the evaluator can homomorphically identify the canary (i.e. it can compute the canary values honestly and cheat everywhere else).