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I figured I should try to learn about linear cryptanalysis with a super simple cipher with only one Sbox. The problem is that with that approach the last subkey bits seem to be involved in the linear approximation.

It looks like I need at least 2 Sboxes, keeping the last one to bring randomness in case of a bad guess on the bits of the last subkey. So obviously, the last one should also not appear in the linear approximation.

Is that understanding of Linear Cryptanalysis correct?

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If you include a key addition layer ($K_1$) at the output as well the key addition layer ($K_0$) at the input of the Sbox then you can perform linear cryptanalysis on this simple cipher. You shall have access to $P,C$ pairs but no keys, of course, and the $P/C$ relationship is $$C=K_1\oplus(S(P\oplus K_0))$$ and so your linear bias equations become something like $$ \textrm{bias}(A,B)=\sum_{P} (-1)^{B \cdot (C\oplus \hat{K_1}) + A \cdot P} $$ where $A$ is the input mask, $B$ is the output mask, $P$ plaintext, $C$ resulting ciphertext and you choose the most likely key guess $\hat{K_1}$ which maximizes the observed bias. Before this, you choose the best mask pairs $(A,B)$ with $A\neq 0$ which give the best bias for the Sbox linear characteristics defined below $$ \sum_{X} (-1)^{B \cdot S(X) + A \cdot X} $$ where the sum ranges over the input set of the Sbox, and perform the attack for that $(A,B).$

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