Two New Infinite Families of APN Functions in Trivariate Form

We present two infinite families of APN functions in trivariate form over finite fields of the form
${\mathbb F}_{2^{3m}}$
. We show that the functions from both families are permutations when
$m$
is odd, and are 3-to-1 functions when
$m$
is even. In particular, our functions are AB permutations for
$m$
odd. Furthermore, we observe that for
$m = 3$
, i.e. for
${\mathbb F}_{2^{9}}$
, the functions from our families are CCZ-equivalent to the two bijective sporadic APN instances discovered by Beierle and Leander. We thus generalize these sporadic instances into an infinite family of APN functions. We also perform an exhaustive computational search for quadratic APN functions with binary coefficients in trivariate form over
${\mathbb F}_{2^{3m}}$
with
$m \le 5$
and report on the results.