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I have seen various papers to use this notation $\mathbb{F}_{2^{-}}$ to describe the diffusion layer. As for example the permutation in AES (MixColumns). So, we have a field with order of 2, what does the minus sign means?

I have tried to search for the term in finite fields lectures and papers but none explained what it actually means.

Here are couple of references: (and generally i have seen it used when they discuss about the diffusion layer)

On sbox affine equivalent page2

Progress in Cryptology

The new codebreakers

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    $\begingroup$ Can you please link to a couple of these papers? $\endgroup$ – SEJPM Mar 8 '17 at 22:31
  • $\begingroup$ Sorry about that, I thought it was some common term. I have added some references. I wasn't sure if i should had included it but I have seen it coupled with linear/affine permutation $\endgroup$ – Anton Paragas Mar 8 '17 at 23:31
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You've been confused by the placement of the "-".

It's not $\mathbb F_{2^-}$ (Latex: $\mathbb F_{2^-}$), this does not exist. What was actually meant is $\mathbb F_2$-adjective (Latex: $\mathbb F_2$-adjective), where the hyphen binds the field and the adjective (eg "linear" or "affine") to bring more specific meaning to the adjective.

Example:

[...]with an $\mathbb F_2$-affine permutation[...]

You would have read "with a permutation that is affine in $\mathbb F_{2^-}$" as opposed to the intended meaning of "with a permutation that is affine in $\mathbb F_2$".

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  • $\begingroup$ Thanks for the clarification. I figured that this could be the case, but the references to that was consistently used with -, so i figured that is some kind of notation. $\endgroup$ – Anton Paragas Mar 8 '17 at 23:43

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