Definition: (Vernam Cipher) Given a plaintext (message you want to encode, represented by the numbers through ) of length characters and a suitably random key of characters, the ciphertext (encoded message, also represented by the numbers through ) is generated by
where denotes the value of the th character of a string .
Definition: Let be the probability of event . Let be the probability that both and occur and be the probability that occurs given that does.
Problem: Prove that the Vernam Cipher is secure – that is, the probability of being a certain character given is the same as the probability of being that character not given .
Solution: Let be an arbitrary characters. We establish that and are independent events because is arbitrarily generated (here the suitable randomness comes into play). So .
as desired. So knowing the key will not help at all in determining the plaintext, resulting in a secure cipher. QED.
Comment: Problems with this cipher lie in the suitable randomness. Creating such keys (and distributing them) proves to be a more difficult task than it sounds.
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