# Can I decrypt the message having it encrypted with different keys?

I have some encrypted binary data I want to decrypt. I know nothing about it, but I can get it encrypted with as many different (random, unknown) symmetric keys as I want. The keys are long enough to not get repeated. Something like this:

$A_1 = Data ⊕ Key_1\\ A_2 = Data ⊕ Key_2\\ A_3 = Data ⊕ Key_3\\ …\\ A_n = Data ⊕ Key_n$

Is it possible to get $Data$ just having the $A$s?

Thanks

-
Maybe this question can help you: crypto.stackexchange.com/questions/12281/…. Also, if you want to encrypt A using only the same cipher, then that does not increase the time to encrypt the text with brute force. Because for most ciphers, if $A_1 = E(E(A_0, K_0), K_1)$, it can be reduced to $A_1 = E(A_0, K')$ (E is the cipher, $K_x$ the keys), so you don't search for two keys, you still just need one. If someone thinks I am wrong (I am doubting myself at the moment), then please prove me wrong. –  Sirac May 17 '14 at 10:48
@Sirac : $\;\;\;$ The DES permutations are a non-empty subset of the group of all permutations of $\{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^{\hspace{-0.02 in}64}$. $\:$ Since that group is finite, composing a non-identity element of that group with itself sufficiently many times gives $\:$ (continued ...) $\;\;\;\;\;\;\;$ –  Ricky Demer May 17 '14 at 13:29
(... continued ) $\:$ that element's inverse; and then one more time gives the identity element. $\:$ Thus if DES had the property you describe, then the DES permutations would be a group under composition. $\:$ However, it is known that DES is not a group. $\;\;\;$ –  Ricky Demer May 17 '14 at 13:30
No. As long as only 1 of the 3 elements is known, $Data$ is not immediately recoverable… you’ld need either $A_n$ or $Key_n$ for that. Yet, this encoding could be trivially broken nevertheless if it weren’t for the fact that you wrote The keys are long enough to not get repeated. That’s called a one-time-pad. It may be interesting for you to check related Q&As and to learn about OTP security properties. –  e-sushi May 17 '14 at 13:31
@Wizardy : $\;\;\;$ How much is the situation like what you wrote? $\:$ In particular, are the cipher texts really equal to [data xor key]? $\;\;\;\;\;\;\;$ –  Ricky Demer May 17 '14 at 13:33