Constructing APN Functions Through Isotopic Shifts
Lilya Budaghyan, Marco Calderini, Claude Carlet, Robert S. Coulter, Irène Villa · IEEE Transactions on Information Theory · 2020
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$x^{9}+ \mathop {\mathrm {Tr}} olimits (x^{3})$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.