Classification of (q, q)-Biprojective APN Functions

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}}\nolimits (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.