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Let $k$ be uniformly sampled from $\{0,1\}^\lambda$, $F$ be a secure PRP with block length $\lambda$ and let $Enc(k, m)$ be such that it returns $c = (F(k,r), r \oplus F(k,m))$ with $r$ uniformly sampled from $\{0,1\}^\lambda$. How to prove that this scheme is not CCA secure? In particular, how can an attack be thought of?

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  • $\begingroup$ You mean scheme. $\endgroup$
    – DannyNiu
    Aug 21, 2019 at 8:50
  • $\begingroup$ Thanks. I think that also for these reasons there is an 'edit' button clickable by everyone :) $\endgroup$ Aug 21, 2019 at 11:15

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I won't give the answer; here are some hints:

  • Suppose we have a valid cipher text $(A, B) = F(k, r), r \oplus F(k, m)$. What happens if we modify the ciphertext to be $(A, B \oplus C)$ and get that decrypted?

  • How can we use this? What can we do if we have two ciphertexts that encrypt the same message $m$?

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  • $\begingroup$ Thinking about the decryption algorithm, can be $F^{-1}(k, F^{-1}(k,A) \oplus B)$, but now I can't see which $C$ can be useful in this scenario. I'll think about it $\endgroup$ Aug 20, 2019 at 18:32

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