Classification of (q, q)-Biprojective APN Functions
Faruk Göloğlu · IEEE Transactions on Information Theory · 2022
In this paper, we classify$(q,q)$-biprojective almost perfect nonlinear (APN) functions over$\mathbb {L}\times \mathbb {L}$under the natural left and right action of$\mathop {\mathrm {GL}} olimits (2, \mathbb {L})$where$\mathbb {L}$is a finite field of characteristic 2. This shows in particular that the only quadratic APN functions (up to${\mathsf {CCZ}}$-equivalence) over$\mathbb {L}\times \mathbb {L}$that satisfy the so-called subfield property are the Gold functions and the function$\kappa: \mathbb {F}_{64} \to \mathbb {F}_{64}$which is the only known APN function that is equivalent to a permutation over$\mathbb {L}\times \mathbb {L}$up to${\mathsf {CCZ}}$-equivalence as shown in Browning et al. (2010). Deciding whether there exist other quadratic APN functions${\mathsf {CCZ}}$-equivalent to permutations that satisfy subfield property or equivalently, generalizing$\kappa $to higher dimensions was an open problem listed for instance in Carlet (2015) as one of the interesting open problems on cryptographic functions.