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If the AES S-Box is replaced with a linear or affine transformation, for instance the identity mapping $\sigma(x)=x$, does the cipher become entirely affine and hence trivially weak?

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does the cipher become entirely affine and hence trivially weak?

Yes; the AES sbox is the only source of nonlinearity (the ShiftRows and MixColumns are linear, and AddRoundKeys for a fixed key is affine), and so if you replace it with a linear/affine one, the entire cipher becomes affine.

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