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With this Elgamal variant defined below, we have covered in lecture but I do not still understand how a CCA could work against it. Is the use of the homomorphic nature of the encryption something you could use?

$c=\langle g^r, \mathcal{H}(y^r)\oplus m \rangle$

What could be a CCA against this type?

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    $\begingroup$ Xor some string to the second component. Does it invalidate the ciphertext? Constructing a CCA2 adversary from that should be easy. $\endgroup$
    – DrLecter
    Dec 4 '16 at 17:37
  • $\begingroup$ This basically falls against the same attacks as unauthenticated stream ciphers do. $\endgroup$
    – SEJPM
    Dec 4 '16 at 20:01
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I refer to you to a nice explanation of the basic attack on the "native" ElGamal scheme: How does chosen ciphertext attack on Elgamal work?

The difference between the native and this hashed scheme you outlined above is that the hashed elgamal version allows to encrypt a m-bit string message for any m since you use XOR, as long as the hashing function you use can output m bits. In the native ElGamal scheme, you have to map m into a group element, which may or may not be OK depending on which group you use. See Boneh's paper: section 4.1. For example, it is not straightforward when using Ed25519 with the special encoding, but solutions exists such as the elligator mappings.

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