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Imagine that you have a ciphertext: Perfect secrecy means, that without knowing the key, any plaintext has to be a possible preimage. Because otherwise the ciphertext would give you information about the plaintext. Encryption is an injective function, because otherwise it could not be reversed. That means, for a given key and ciphertext you have at most ...

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Here's a more "down to earth" example. The following cryptosystem with plaintext space $\mathcal{M} = \{a,b,c,d\}$, keyspace $\mathcal{K} = \{1,2,3,4\}$ and ciphertext space $\mathcal{C} = \{A,B,C,D\}$ has perfect secrecy: \begin{array}{c|c c c c} & 1 & 2 & 3 & 4 \\ \hline a & A & B & C & D \\ b & B & C & D ...

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If the key would be smaller than the plaintext then you could brute force the cipher by using less than $N$ steps, where $N$ represents the amount of possible messages. Of course the brute force approach is an upper bound to what can be tried. If there are attacks on the cipher (that are less complex than brute force) then the plaintext may be recovered ...

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