Abstract For functions over the finite field $$\mathbb {F}_{p^n}$$ , we study a generalization of affine equivalence. Two functions of algebraic degree d are equivalent if, when ignoring monomials of algebraic degree $$d-1$$ or less, they coincide with two affine equivalent functions. This equivalence appears naturally for functions defined by the properties of their derivatives of order $$d-1$$ . In order to test this equivalence, as usual, invariants can be used as a first approach. We describe several such invariants, based on the kernel/image of the derivatives of order $$d-1$$ and on the set of vectors orthogonal to those images (similar to the orthoderivatives used as invariants for quadratic APN functions). We also define a canonical form with respect to left composition with linear transformations, which decreases the computational effort of testing for equivalence. We then apply these techniques and computer search to classify all GAPN (generalized APN) functions of algebraic degree 3 over $$\mathbb {F}_{3^3}$$ . We determined that there are exactly 31 equivalence classes; for each class we list, firstly, a representative in multivariate algebraic normal form which is in canonical form with respect to left composition with linear transformations; secondly, we also list a representative with a shortest univariate representation. Moreover, our computations show that all of the GAPN functions of algebraic degree 3 over $$\mathbb {F}_{3^3}$$ have optimal second-order differential uniformity. While this is not always the case for other values of p and n , we show that the GAPN functions of algebraic degree p over $$\mathbb {F}_{p^n}$$ of the form $$x^{p-1+p^{\ell }}$$ with $$\gcd (\ell ,n)=1$$ do have optimal order- $$(p-1)$$ differential uniformity.

Abstract APN functions offer optimal resistance to differential attacks and are instrumental in the design of block ciphers in cryptography. While finding APN functions is very difficult in general, a promising way to construct APN functions is through symmetric matrices called Quadratic APN matrices (QAM). It is known that the search space for the QAM method can be reduced by means of orbit partitions induced by linear equivalences. This paper builds upon and improves these approaches in the case of homogeneous quadratic functions over $$\mathbb {F}_{2^n}$$ with coefficients in the subfield $$\mathbb {F}_{2^m}$$ . We propose an innovative approach for computing orbit partitions for cases where it is infeasible due to the large search space, resulting in the applications for the dimensions $$(n,m)=(8,4)$$ , and $$(n,m)=(9,3)$$ . We find and classify, up to CCZ-equivalence, all quadratic APN functions for the cases of $$(n,m)=(8,2)$$ , and $$(n,m)=(10,1)$$ . Also, we show that exhaustive searches for $$(n,m) = (10,2)$$ , and $$(n,m)=(8,4)$$ are infeasible for the QAM method using currently available means, following partial searches for these cases.