Designing a key expander out of ciphers Is it possible to compose or combine $n$ ciphers ($cipher_0, cipher_1, ... cipher_n$) in such a way that they generate $n$ ciphertexts from the same key, $k$, and that recovering the key, $k$, from the $n$ ciphertexts is equivalent to breaking all $n$ ciphers. 

Note that the size of the key, $k$, is the same as the size of the $n$ ciphertexts.

$$ ||k|| = ||cipher_i(k, k)||$$
$$ n \times |k| = ||cipher_0(k, k)||+||cipher_1(k, k)||+...+||cipher_n(k, k)||$$

I'm not asking about finding the pre-image of the $n$ ciphertexts, but rather the original key that was used. 

This is related to this answer ( Changing algorithms during encryption ).
I think you can just generate all the plaintexts but one randomly, and make the last one the XOR of the key and the other plaintexts. To recover the key, one must recover all the plaintexts and XOR them. Even every plaintext but one gives you no information about the key.