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Suppose Alice shares a secret block cipher key, $K_{AB}$ with Bob, and a different secret block cipher key, $K_{AC}$ with Charlie. Describe a method for Alice to encrypt an m block message such that it can only be decrypted with the cooperation of both Bob and Charlie. The ciphertext should only be a constant size greater than m blocks. You may assume that Bob and Charlie have a pre-established secret channel on which to communicate.

The answer given is:

Alice can first pick a random key $K$ to encrypt the message $M$, and then she encrypts the key $K$ with both $K_{AB}$ and $K_{AC}$. In other words, Alice does the following:

$$E_{K_{AB}} [E_{K_{AC}}(K)] || E_K(M)$$

May I know if in this case, that Bob wants to decrypt $E_{K_{AC}}(K)$, can he complete this solely without the help from Alice? Thanks for the answer.

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Since Bob already knows $K_{AB}$ he can run $D_{K_{AB}}(E_{K_{AB}}(E_{K_{AC}}(K)))$. Which will leave him with $E_{K_{AC}}(K)$. And to decrypt that any further (in order to get $K$) he would need either Alice or Charlie's help.

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    $\begingroup$ "can he completes it solely without the help from Alice?" You answer yes, and then "to decrypt that he would need either Alice or Charlie's help". So the answer and the explanation seem to be at odds. $\endgroup$ – Maarten Bodewes Aug 12 '15 at 21:46
  • $\begingroup$ Yup. Clarified. $\endgroup$ – StackzOfZtuff Aug 13 '15 at 4:20

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