Constructing APN Functions through Isotopic Shifts
Lilya Budaghyan, Marco Calderini
Abstract
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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{9}}$ </tex-math></inline-formula> and includes an example of another APN function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{9}+ \mathop {\mathrm {Tr}}\nolimits (x^{3})$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{8}}$ </tex-math></inline-formula> , known since 2006 and not classified up to now. We conjecture that the conditions for this family are satisfied by infinitely many APN functions.