Composability of function in Homomorphic encryption

FHE Questions
1

As we know in FHE server computes:

Eval[f, Dec(c)]
= Dec(f(c))

That shows f and Dec are commutative.

Is it a limitation of FHE that a function should be commutative.

What is the proof of above scenario.

2

You can think of Dec as an identity function. Dec preserves the message as is; it simply removes the mask. So one can apply Dec at any point during the computation.

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3

Thanks!

Is there any mathematical proof ?
or it’s limitation that we have to consider Dec as an identity function.

4

Each exact homomorphic encryption scheme, such as BGV or BFV, requires correctness proofs, i.e., it shows that the decrypted result matches the result obtained in the clear as long as the noise is below the appropriate threshold. So the correctness proof is the main way to argue this.

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