# Secret Sharing Encryption

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.

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.
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.