  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 a priori simple, it is very difficult in practice to test
whether two functions are EA-equivalent. This problem has two variants:
EA-testing deals with figuring out whether the two functions can be
EA-equivalent, and EA-recovery is about recovering the tuple $(A,B,C)$ if it
exists.
  In this paper, we present a new efficient algorithm that efficiently solves
the EA-recovery problem for quadratic functions. Though its worst-case
complexity is obtained when dealing with APN functions, it supersedes all
previously known algorithms in terms of performance, even in this case. This
approach is based on the Jacobian matrix of the functions, a tool whose study
in this context can be of independent interest.
  In order to tackle EA-testing efficiently, the best approach in practice
relies on class invariants. We provide an overview of the literature on said
invariants along with a new one based on the \emph{ortho-derivative} which 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 both very fast to compute,
and highly discriminating.
