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I am trying to do some ElGamal encryption but having a different encryption formula. For that I doing the following steps:

The key generator:

  1. Choosing value $p = 107$ and $a = 2$
  2. Random number $d = 67$, and $b = a^d \bmod p$ where $b = 2^{67} \bmod 107 = 94$
  3. $k_{priv} = 67$ and $k_{pub} = (p,a,b) = (107,2,94)$

Encryption

  1. Random value $v = 45$ and $C_1 = a^v \bmod p = 2^{45} \bmod 107 = 28$
  2. We have the message $m = 66$; $C_2 = m \cdot b^v \cdot a^v \bmod 107 = 66 \cdot 94^{45} \cdot 2^{45} \bmod 107 = 38$
  3. Finally, $C = (C_1, C_2)$

My problem comes when I try to decrypt the message, maybe I am totally wrong. But I am doing:

$C_1 = a^v$
$C_2 = m \cdot a^v \cdot (a^d)^v$ $C_2 = m \cdot C_1 \cdot (a^d)^v$

I trying to do that is where I am a little bit lost. If someone can help me with a clue to decrypt the message I would be nice

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  • $\begingroup$ What is the source of this question? $\endgroup$ – kelalaka Oct 28 '20 at 18:10
  • $\begingroup$ You forgot to use the $d$, it is your secret, right? $\endgroup$ – kelalaka Oct 28 '20 at 19:35
  • $\begingroup$ my question is how i can decrypt the message. I not sure if my approach is the best $\endgroup$ – padenLeonardo Oct 29 '20 at 8:32
  • $\begingroup$ I'm assuming this is homework question there we only provide hints. Since you show some effort I'll direct you. Use the fact that $b=a^d$ since the message sent to you and $d$ is your secret. $\endgroup$ – kelalaka Oct 29 '20 at 9:57
  • $\begingroup$ Thanks, this helped me. I didn't solved yet. I don't know how to solve the $(a^d)^v$ $\endgroup$ – padenLeonardo Oct 30 '20 at 19:34
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Finally, the decryption is: $m = C_2 \cdot C_1^{-1} \cdot C_1^{-d}$

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    $\begingroup$ That was easy, right? Now accept and close. have fun! $\endgroup$ – kelalaka Oct 30 '20 at 21:44

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