I asked this at crypto.se months ago, but never got any feedback.
What I am actually looking for is a bit of nuance. In cryptography, and AES specifically, isomorphic transforms and affine transforms are used in hardware implementations. The affine transform is used to assist with non-linearity and masking, whereas the isomorphisms are generally used to transform some bytes into a format that is easier for a hardware implementation. You do an isomorphism, some squaring math and then an inverse isomorphism. In both cases, if you do the affine followed by the inverse, and then the isomorphism followed by the inverse, you will get the same data you put in as an input. My question is:
What are the mathematical assumptions I should make when "affine" is used in a paper when compared to an "isomorphism"? As far as I can tell, in the context of GF(2), they are the same thing.
Addendum: The origin of this question is that I wrote an AES S-Box hardware implementation guide. It wrote it in the notation of Vincent Rijmen, whose work became AES with Joan Daemen because I am not a cryptographer and did not want to make any assumptions of language. In my AES document, the to specific items are the Affine definition: \begin{align} \label{eqn:aff} Af(a) = \left( \begin{array}{@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}@{\hskip2pt}c@{\hskip2pt}} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \end{array} \right) \times \begin{pmatrix} a_7 \\ a_6 \\ a_5 \\ a_4 \\ a_3 \\ a_2 \\ a_1 \\ a_0 \end{pmatrix} \oplus \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} \end{align} and then the isomorphism definition: \begin{equation} \label{matrix:isomorph} \delta \times i = \begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} i_7 \\ i_6 \\ i_5 \\ i_4 \\ i_3 \\ i_2 \\ i_1 \\ i_0 \end{pmatrix} \end{equation}
Ultimately, they both become XOR logic circuits. As far as I can tell, when you push through on pen and paper, they are the affine and the isomorphism are structurally the same thing.