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

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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".

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