# difference between function f in eval and dec of homomorphic authenticated encryption

In the homomorphic authenticated encryption, as described 1, there are Eval and Dec PPT algorithms:

In Eval, f is the function to be performed on the encrypted data. However, I need to know why it is used in Dec algorithm?

The key-word "authenticated" mean that Dec is has not just a decrypting functionality. It should also check the validity of the computation made. that's why it takes as input the function $$f$$. Read more in details the paragraph "4.4 Authenticity".