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I have a question about the security proof. Suppose that I proved the following relations. That is, I proved that $Adv_A \le Adv_A^{O} \le Adv_B$, where A,B are some cryptographic schemes and O is an oracle. I also suppose that the oracle O has a very strong property. For example, the O can solve DLP in polynomial time.

On the other hand, there is an attacker $\mathcal{A}$ which can break the scheme A. That is, $Adv_A$ is not negligible in the security parameter.

Even though $Adv_B$ is also non-negligible, but we do not have the oracle O. In this setup, how to break the scheme B using $\mathcal{A}$?

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You have to look in more details the proof of the inequality $Adv^{O}_A \leq Adv_B$.

The proof will give you explicitly how to construct $\mathcal{B}$ from $\mathcal{A}^O$. It should be written how to simulate the experiment with $A$ and the powerful oracle $O$, by only having interaction with the experiment with $B$. And how to use the output of $\mathcal{A}$ to win the game against $B$.

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  • $\begingroup$ You mean that it depends on how to construct the game? I do not have the concrete example. I am just curious about this case. $\endgroup$ Dec 16 '21 at 13:25
  • $\begingroup$ No it depends about your proof about the inequality work. $\endgroup$
    – Ievgeni
    Dec 16 '21 at 13:29
  • $\begingroup$ Thanks. As I understood, sometimes B is broken, and sometimes B is still secure. Is there an example? $\endgroup$ Dec 16 '21 at 13:37
  • $\begingroup$ No, if A is broken then B is also broken, but the way to Break B depends about the proof of the inequality (and the way to break A). $\endgroup$
    – Ievgeni
    Dec 16 '21 at 13:53
  • $\begingroup$ Thanks. I will find some examples. $\endgroup$ Dec 17 '21 at 1:28

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