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Recall the definition: a cipher is AE-secure iff it is secure against chosen ciphertext attacks and has ciphertext ingegrity. Try going through the attack games with $(E_1,D_1)$ and $(E_2,D_2)$: if the adversary succeeds, can he succeed for $(E,D)$? It's fairly easy to see that $(E_1,D_1)$ and $(E_2,D_2)$ are CPA-secure. If the adversary can distinguish ...

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A cryptographic MAC is always symmetric By Definition. For those interested in its security properties, one may want to know Does data authenticity always, implicitly, provide data integrity? The tool that provides authenticity asymmetrically is known as Digital Signature. Digital Signatures are asymmetric By Definition; they guarantees the integrity, ...

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Why is the comment needed that encrypted MAC part is constant? With the scheme in the question, in a cryptogram with the correct MAC, the last block of the encrypted message is a constant. Importantly, that goes both ways: if the last block of the encrypted message is that constant, then the MAC is correct and whatever plaintext is deciphered will pass the ...

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First of all, in a digital signature scheme that follows the hash-and-sign paradigm $Enc_{sk}(hash(m))$ the hash procedure is essential to "fit" the message $m$ in the public-key scheme domain: you cannot encrypt messages of any size with RSA for example. The second important point is that a digital signature scheme is a bit closed to public-key encryption:...

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