In ElGamal scheme we have message $M$; $p$, $g$ and $y=g^x \bmod p$ as public key where $x$ is unknown private key.
Encrypted message $(c,d)$, where $c=g^k \bmod p$ and $d=M \cdot y^k \bmod p$.
Signature $(r,s)$, where $r=g^k \bmod p$ and $s=(M - xr)\cdot k^{-1} \bmod p-1$

If $c=r$ then message encrypted and signed with the same $k$.
Is there a possibility to obtain private key $x$?


Is there a possibility to obtain private key $x$?

No; here's the proof.

Suppose we had a black box that, given $c = g^k, d = M \cdot y^k, r = g^k$ and $s = (M - xr) \cdot k^{-1} \bmod p-1$ (and we'll throw in $M$, and $z : y = g^z$), is able to give us $x$.

Then, here's how we can find the private key given an ElGamal signature.

We have $M, r = g^k$ and $s = (M - xr) \cdot k^{-1} \bmod p-1$ (as that's the ElGamal signature and message being signed).

That also gives us $c$. To get $d$, we select a random $z$ and compute $d = M \cdot c^z = M \cdot y^k$.

We now have everything the Oracle expects; we pass in everything, and we recover $x$.

As we believe that deriving the ElGamal private key from a signature is infeasible, we believe there cannot be such a black box.

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  • $\begingroup$ Shouldn't the private decryption key and the private signature key be the same? That is $y=g^x$ and therefore $x = z$? $\endgroup$ – K.G. Oct 24 '17 at 19:36
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    $\begingroup$ @K.G. well, typically, if you are doing a sign-and-encrypt operation, you're signing with your own private key, and encrypting with someone else's public key. Apart from homomorphic schemes, it rarely makes sense to encrypt with your own public key; if you want something where the encryptor and decryptor are the same, there's little reason not to use symmetric crypto... $\endgroup$ – poncho Oct 24 '17 at 19:39
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    $\begingroup$ I agree 100% with your reasoning, but the question says $y=g^x$, which I should have said to begin with. $\endgroup$ – K.G. Oct 24 '17 at 19:52

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