Question

Perfect secrecy implies that key-length is at least as long as message length. Why?

Answer 1

Perfect secrecy is the notion that, given an encrypted message (or ciphertext) from a perfectly secure encryption system (or cipher), absolutely nothing will be revealed about the unencrypted message (or plaintext) by the ciphertext.

The "for perfect secrecy, the key should be as long as the message" statement is correct, if applied to the encryption based on One-time pad - used by most symmetric key encryption schemes. If the key is as long as the message, each bit of the message can be XORed with a bit of the key to produce the ciphertext. The ciphertext could decode to any plain text message depending on the key (pick the key to be the XOR of the ciphertext and the desired plain text), so there is absolutely no way to decode what was originally encrypted without the key. If the key is longer than the message, the extra bits are unused. At first I thought there was absolutely no benefit to using a longer key, but I suppose if you want to hide the length of the message, you would need extra random bits in the key to append to the end of the ciphertext.

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