Abstract
In this paper, we classify <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(q,q)$ </tex-math></inline-formula> -biprojective almost perfect nonlinear (APN) functions over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {L}\times \mathbb {L}$ </tex-math></inline-formula> under the natural left and right action of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathop {\mathrm {GL}}\nolimits (2, \mathbb {L})$ </tex-math></inline-formula> where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {L}$ </tex-math></inline-formula> is a finite field of characteristic 2. This shows in particular that the only quadratic APN functions (up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathsf {CCZ}}$ </tex-math></inline-formula> -equivalence) over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {L}\times \mathbb {L}$ </tex-math></inline-formula> that satisfy the so-called subfield property are the Gold functions and the function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa: \mathbb {F}_{64} \to \mathbb {F}_{64}$ </tex-math></inline-formula> which is the only known APN function that is equivalent to a permutation over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {L}\times \mathbb {L}$ </tex-math></inline-formula> up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathsf {CCZ}}$ </tex-math></inline-formula> -equivalence as shown in Browning et al. (2010). Deciding whether there exist other quadratic APN functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathsf {CCZ}}$ </tex-math></inline-formula> -equivalent to permutations that satisfy subfield property or equivalently, generalizing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa $ </tex-math></inline-formula> to higher dimensions was an open problem listed for instance in Carlet (2015) as one of the interesting open problems on cryptographic functions.
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