  Extended Affine (EA) equivalence is the equivalence relation between two
vectorial Boolean functions $F$ and $G$ such that there exist two affine
permutations $A$, $B$, and an affine function $C$ satisfying $G = A \circ F
\circ B + C$. While the problem has a simple formulation, it is very difficult
in practice to test whether two functions are EA-equivalent. This problem has
two variants: {\em EA-partitioning} deals with partitioning a set of functions
into disjoint EA-equivalence classes, and \emph{EA-recovery} is about
recovering the tuple $(A,B,C)$ if it exists.
  In this paper, we present a new algorithm that efficiently solves the
EA-recovery problem for quadratic functions. Although its worst-case complexity
occurs when dealing with APN functions, it supersedes, in terms of performance,
all previously known algorithms for solving this problem for all quadratic
functions and in any dimension, even in the case of APN functions. This
approach is based on the Jacobian matrix of the functions, a tool whose study
in this context can be of independent interest.
  The best approach for EA-partitioning in practice mainly relies on class
invariants. We provide an overview of the known invariants along with a new one
based on the \emph{ortho-derivative}. This new invariant is applicable to
quadratic APN functions, a specific type of functions that is of great
interest, and of which tens of thousands need to be sorted into distinct
EA-classes. Our ortho-derivative-based invariant is very fast to compute, and
it practically always distinguishes between EA-inequivalent quadratic APN
functions.
