In my introductory cryptography class. There is a line describe the construction of S-box in AES "The S-box of AES is constructed by combining a function $h(x)=x^{254}$ defined on $GF(256)$ with an invertible affine transformation." However, when I look into the Advanced Encryption Standard (AES) document , I can not find where the $h(x)=x^{254}$ is used. I understand that S-box is constructed by $a_{i,j}\to a_{i,j}^{-1}(mod\,x^8+x^4+x^3+x+1)$ and subsequent affine transformation. Does $h(x)=x^{254}=x^{-1}$ is used during the inverse proccess?
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In the field $GF(2^8)$, $x^{254} = x^{-1}$ (except for $x=0$, as $0^{-1}$ doesn't exist; for AES, that's treated as 0), and so it's two ways of describing the same thing.
When we talk about AES, we typically use the $x^{-1}$ nomenclature; for whatever reason, your class decided to go with the $x^{254}$ one.
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$\begingroup$ Probably a silly question, but don't we require a modulus for the field multiplication? $\endgroup$– holaFeb 19, 2020 at 18:55
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$\begingroup$ @yyyy0000: one way to implement $GF(2^8)$ field multiplication using the representation that AES uses is, in fact, treat the values are polynomials on $GF(2)$, and to multiply two values, you would multiply the two polynomials, and then reduce the product modulo $x^8+x^4+x^3+x+1$. However, when we're working the field (or ring or group), we typically just use the $\times$ or $+$ operations as the field operations, and don't explicitly write out how it could be implemented every time. $\endgroup$– ponchoFeb 19, 2020 at 19:20