# What does this notation means $\mathbb{F}_{2^{-}}$?

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

• Can you please link to a couple of these papers? – SEJPM Mar 8 '17 at 22:31
• 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 – Anton Paragas Mar 8 '17 at 23:31

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.
[...]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 $\mathbb F_2$".