# Would a symmetric cipher with a keylength a big as the data length be information theoretically secure?

One-Time-Pad is information theoretically secure as long as the random number stream is evenly long or longer than the data stream it encrypts, for a "decyphered" message could have been any message with the same length as the given with the same probabillity. Does the same apply to symmetric ciphers, too?

For instance if I have 1024 bits to encrypt, break it into chunks 128 bits and encrypt it with AES-128, each with a different (assume: true) random key, will that be information theoretically secure just as OTP? Or would a hypothecial prime factorization algorithm (let's assume it would be as quick as the encryption function) impact the (theoretical) security?

In other words: Does using a symmetric encryption algorithm (such as AES) lower the probabillity of a ciphertext being originated from a specific plaintext even if the used key has "OTP-length" and is completely unknown?

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I don't have a proof but my guts tell me the answer is "no", because it does not seem esufficient that the family of functions $f_k$ used by the generalization of the OTP be all bijective, they also have to satisfy the criteria that $\{ f_k(x) \mid k \in \Sigma \}$ be a permutation of $\Sigma$ for all $x \in \Sigma$ (where $\Sigma$ is the relevant alphabet, e.g. the set of all 256-bit strings). XOR, modular addition and other simple constructs trivially satisfy this through symmetry, but AFAIK it is unknown whether there exist $k_1$, $k_2$ such that $AES_{k_1}(x) = AES_{k_2}(x)$ for some $x$. – Thomas Mar 12 '14 at 23:06
I could be way off, though, because I just confused myself. Anyway good question, looking forward to the answers – Thomas Mar 12 '14 at 23:08
To be fair, though, at the point where you have a truly random (not simply from a seeded CSPRNG), why bother with the overhead of a symmetric cipher like AES instead of simply XOR? – Stephen Touset Mar 12 '14 at 23:22
It's about a QKD solution. The question is, as the QKD keys applied with OTP are theoretically secure, would they also be with IPsec? – Marste Mar 12 '14 at 23:27
Do you mean Rijndael? $\:$ (AES's block size is 128 bits.) $\;\;\;\;$ – Ricky Demer Mar 12 '14 at 23:39