Given the plaintext, ciphertext and key space {0,1}, the Vernam cipher works as:

Question

 Given the plaintext, ciphertext and key space {0,1}, the Vernam cipher works as:          c = k (+) m   Where the key k must be chosen at random 0 or 1 with probability 1/2 of a 1. That is, with probability 1/2 the ciphertext equals the plaintext, else the ciphertext is the  complement of the plaintext.  We suppose there was a eavesdropper that saw the ciphertext, and we are considering all ways that eavesdropper can conclude with more or less certainty what is the plaintext.  In class we supposed to decision procedures by the eavesdropper: (1) guess always which of plaintext 0 or 1 is more likely; or (2) guess that the plaintext is equal to the ciphertext.  We showed that it is possible that often (1) is better than (2) - that is ignoring the  ciphertext is more likely to give a correct answer than basing your answer on the cipher text.  Note: In these problems, do not assume that the probability distribution on the message space is uniform.   1) Suppose the eavesdropper guesses the plaintext by ignoring the ciphertext and flipping a BIASED coin, and guessing that the plaintext equals the coin outcome. What is the probability that the eavesdropper will correctly guess the plaintext. 

Explanation / Answer

head=1; tail=0

for an unbiased coin,

let p be the probability of getting success

1-p is the probability of getting failure. ie; result of coin outcome and plaintext not equal.

The probability of having exactly m successes in n independent Bernoulli trials with success probability p is C(n,m) pm (1-p)n-m

here n is the length of the text. in order to correctly guess the plaintext, there should be 'n' success outcomes.

therefore, C(n,n) pn (1-p)n-n = pn

for an unbaised coin it is 0.5n

for a biased coin, p is not known

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