Constructing APN Functions Through Isotopic Shifts

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, mathematics and information theory. In this paper we deduce a new method for constructing APN functions by studying the isotopic equivalence, concept defined for quadratic planar functions in fields of odd characteristic. In particular, we construct a family of quadratic APN functions which provides a new example of an APN mapping over
${\mathbb F}_{2^{9}}$
and includes an example of another APN function
$x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$
over
${\mathbb F}_{2^{8}}$
, known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions.