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That's an optimization for the attack. It would work without it, but slower. To do a Meet-in-the-middle attack, we need to encrypt a known plaintext with every possible key and save the resulting text (with the used key) in a list. Now we decrypt a known ciphertext with every possible key and look if we got the resulting text in our list. We want to know ...


1

For 1) you need to show that $S(X)$ is linear with respect to xor. We can define $S(X)$ as: $S(X) = (x_{7}, x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6})$ And define $A \oplus B$ as: $A \oplus B = (a_0 \oplus b_0, a_1 \oplus b_1,a_2 \oplus b_2,a_3 \oplus b_3,a_4 \oplus b_4,a_5 \oplus b_5,a_6 \oplus b_6,a_7 \oplus b_7,)$ So we have: $S(A \oplus B) = ...



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