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I have to construct a key recovery attack on symmetric key encryption using a publicly known permutation $\Pi$ in $O(2^\frac{n}{2})$ time using $2^\frac{n}{2}$ queries to an encryption oracle.

The encryption is done as $ \Pi ( m \oplus K) \oplus K $, where $K$ is the key. Both $m$ and $K$ belong to ${0, 1}^{n}$

I do not know how I can use the queries to do the key recovery attack in that time. I can check my guesses against the output of the queries for $2^\frac{n}{2}$ of them. But how will I recover the key successfully ?

Really need some help here.

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    $\begingroup$ Is this homework? $\endgroup$
    – poncho
    Jul 15, 2021 at 23:28
  • $\begingroup$ @poncho One of the practice problems. Not exactly homework. $\endgroup$
    – Swagata
    Jul 15, 2021 at 23:34

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Well, you said that this was a "practice problem"; hence you're supposed to learn from it, so I won't give you the answer.

I will give you a hint: you have $E_k(M) = \Pi(M \oplus K) \oplus K$; suppose you defined $F_k(M) = E_k(M) \oplus E_k(M \oplus 1)$, and further defined $G$ such that $F_k(M) = G( M \oplus K )$ (and yes, such a $G$ exists independent of $K$). How could you use that to do key recovery?

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