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Is it possible to make ElGamal IND-CCA2 using OAEP or OAEP+?
(OAEP+ from: "OAEP Reconsiderd" by Shoup)

The reason I ask is that I recently answered this question and it came to my mind that OAEP or OAEP+ might be possible solutions.

Note this isn't a practical question at all. There's no intention in implementation, I'm just asking if it would be secure if I would implement it.

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  • $\begingroup$ Any reason not just to use Cramer-Shoup? That's a CCA2 extension of ElGamal and doesn't even require random oracles. $\endgroup$ – Bristol Apr 14 '15 at 12:38
  • $\begingroup$ @Bristol This isn't a practical question at all. In practice I'd use DHIES or ECIES, but thanks for the reference. $\endgroup$ – SEJPM Apr 14 '15 at 18:55
  • $\begingroup$ OAEP is interesting in case the homomorphic properties of ElGamal are required. $\endgroup$ – Maarten Bodewes May 27 '17 at 16:45
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After some thought, I think the answer is in fact NO, even for IND-1-CCA* and even for Shoup's OAEP+.

RSA-OAEP/OAEP+ work by taking a message $m$, producing a padding $p(m,r)$ and then encrypting this, so $c = f(p(m,r))$ where $f$ is RSA encryption, and $f(u) = u^e \pmod{N}$ is deterministic. In fact, the whole point of OAEP(+) is to inject some entropy into ciphertexts which is required for IND-CPA and higher security.

ElGamal encryption is already randomised. If we try ElGamal-OAEP(+) we get $c = (g^r, y^r \cdot p(m, r'))$ where $y$ is the public key. Since ElGamal is homomorphic, this is obviously not even CCA1: consider an adversary who picks $m_0, m_1$, asks for a challenge ciphertext $c = (u, v)$ and then sets $c' = (u \cdot g^s, v \cdot y^s)$ for randomly chosen $s$. This is still a valid OAEP(+) ciphertext whatever the padding $p$ is (since we're only changing the "outer" randomness $r \mapsto r + s$) so the IND-1-CCA game will happily decrypt this and return $m_0$ or $m_1$ as desired.

This is assuming of course that you can map your padding function's range into the group over which you're doing ElGamal --- for ECC, this should be fine, for $\mathbb Z^\times_p$ groups it's harder. As an alternative one could consider hashed ElGamal-OAEP+ with $c = (g^r, H(y^r) \oplus p(m, r'))$ where $H$ is independent of the hash functions used in the OAEP+ padding $p$. My intuition is that this is still not CCA1, even though it doesn't have the homomorphic property anymore. Certainly if $H$ has some homomorphic properties itself then one should be able to do something like the above counterexample.

IND-1-CCA: Is standard IND-CCA2 where you only get 1 decryption query after seeing the challenge ciphertext instead of polynomially many.

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    $\begingroup$ Your argument is fine, but only for CCA2 security. (For CCA1, you don't have a decryption oracle after you get the challenge ciphertext.) Also, there are no good CCA1 attacks against ElGamal on its own, and the padding doesn't change that. $\endgroup$ – K.G. Jul 25 '15 at 11:08
  • $\begingroup$ I've briefly considered the hashed ElGamal case, and I can't immediately see a security proof. For certain groups, there are CCA2 attacks, but for other groups, I can't find good attacks. $\endgroup$ – K.G. Jul 25 '15 at 11:25
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    $\begingroup$ @K.G. : you're right, I confused IND-CCA1 and IND-1-CCA (where you get only a single decryption query, but it may be after you've seen the challenge ciphertext). $\endgroup$ – Bristol Jul 27 '15 at 8:07

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