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# testsol - 1(10 points What is the number of possible keys...

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1. (10 points) What is the number of possible keys of the following cryptosystems: A. Shift cipher over 26 letter alphabet 26 B. Affine cipher over 26 letter alphabet 26 · 12 C. Substitution cipher over 26 letter alphabet 26! D. One time pad for sending an n bit long message 2 n 2. (10 points) Explain what would go wrong and why, if someone would try to use the Affine cipher with a = 6, and an arbitrary value b . (That is, the encryption e ( x ) = ax + b mod 26, with a = 6.) Notice that for e to be a valid encryption function, e must be one-to-one, that is, for every y ∈ Z 26 , there must be at most one x such that e ( x ) = y . However, the given function with a = 6 is not one-to-one. For each x ∈ Z 26 , x and x + 13 will encrypt to the same value. For the function e ( x ) = ax + b mod 26 to be one-to-one, gcd( a, 26) must be 1. However, notice that gcd(6 , 26) = 2. 3. (10 points) Alice needs to send an element x Z 100 to Bob sometime in the future, and they don’t like mods. Instead they pick a random key K Z 100 , and Alice plans to send x + K to Bob (without the mod). Prove or disprove: this achieves perfect secrecy. If you cannot prove your answer, give intuition. No, perfect secrecy is not achieved. For example, if Oscar sees the ciphertext y = 198, he will know exactly what was the value of both x and K .

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