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Let's start by explaining how Playfair works normally to encrypt a message. First, you create a 5x5 table by writing the keyword letter-by-letter across the top of the table, from left to right, skipping duplicate letters; you then fill in the remaining characters in alphabetical order after the keyword (combining i&j or j&k into a single box). ...

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a) The question seems to be about a comparison of the size of the key spaces. The hint already shows the key spaces for Vigenère with a 10-letter key ($26^{10}$) and for simple substitution ($26!$). Simple substitution has the much bigger key space. b) Using frequency analysis simple substitution is much easier to solve. Basic frequency analysis does not ...

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To prove an encryption scheme to be perfectly secure, we need to prove: $$P[M=m|C=c]=P[M=m]$$ where $c$ is a cipher text and $m$ is a plain text. From Bayes theorem, we have: $$P[M=m|C=c]=\frac{P[C=c|M=m] \cdot P[M=m]}{P[C=c]}$$ It is noteworthy that: $$P[C=c|M=m]=P[K=k]$$ where $K$ is the key space and $k$ is a particular key. Now: ...

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