# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Vernam Cipher. Topic: Cryptography. May 5th, 2006

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 .

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Definition: Let be the probability of event . Let be the probability that both and occur and be the probability that occurs given that does.

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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 .

But then

as desired. So knowing the key will not help at all in determining the plaintext, resulting in a secure cipher. QED.

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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.

### 5 Responses to “Vernam Cipher. Topic: Cryptography.”

1. freshmenz=] Says:

this is hard… how come there’s only one question in the AMC level?

There are actually 12, but AMC problems are no fun; that’s why.

3. QC Says:

Nobody needs Jeffrey to talk about AMC problems XD

4. freshmenz=] Says:

AMC.. no fun?? wut’s fun of it when i cant even understand them??
and QC, aren’t you forgettin someone??.. i donno, maybe the FRESHMENZ?

5. #H34N1 Says:

I’ll try to make one of those codes