Abstract
-In this paper, we classify (q, q)-biprojective almost perfect nonlinear (APN) functions over L x L under the natural left and right action of GL(2, L) where L is a finite field of characteristic 2. This shows in particular that the only quadratic
APN functions (up to CCZ-equivalence) over L x L that satisfy the so-called subfield property are the Gold functions and the function κ : F64 RIGHTWARDS ARROW F64 which is the only known APN function that is equivalent to a permutation over L x L up to CCZequivalence as shown in (Browning, Dillon, McQuistan, and
Wolfe, 2010). Deciding whether there exist other quadratic APN functions CCZ-equivalent to permutations that satisfy subfield property or equivalently, generalizing κ to higher dimensions was an open problem listed for instance in (Carlet, 2015) as one of the interesting open problems on cryptographic functions.