On the APN-ness and differential uniformity of some classes of $ (n,n) $-functions over $ \mathbb{F}_2^n $

We study the almost perfect nonlinearity (APNness) and differential uniformity of $ (n, n) $-functions when viewed over the vector space $ \mathbb{F}_2^n $ (without the structure of the finite field $ \mathbb{F}_{2^n} $). In this framework, we first prove that symmetric $ (n, n) $-functions (defined as commuting with any permutation of the input/output coordinates) cannot be APN and we propose several super-classes of the symmetric function class, within which our non-existence result does not work, and which can be investigated for finding APN functions. We study in particular rotation symmetric (RS) functions (defined as commuting with the cyclic shift of the input/output coordinates), among which are known APN functions, obtained from power APN functions by decomposing their input and their output over a normal basis. We show necessary conditions that RS functions (and more generally functions commuting with some permutation of the coordinates) must satisfy for being APN. We deduce a general result on RS APN functions which, when the RS function is specifically a power function whose input and output are decomposed over a normal basis, gives Dobbertin's result saying that any APN power function for $ n $ odd is a permutation and for $ n $ even is 3-to-1 over $ \mathbb F_{2^n}^* $. In a second part of the paper, we derive a lower bound on the differential uniformity of monotone $ (n, n) $-functions, implying that no function in this large class (of size asymptotically equivalent to $ (2^{2^n})^{\sqrt{\frac {2n}{\pi }}} $) can be APN for $ n\geq 8 $. As we did with symmetry, we propose weakened conditions possibly allowing APNness. We state several open questions throughout the paper.