A new almost perfect nonlinear function which is not quadratic

Following an example in [13], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that the approach can be used to construct “non-quadratic” APN functions. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. 1 Preliminaries In this paper, we consider functions F : F n 2 → F n 2 with “good” differential and linear properties. Motivated by applications in cryptography, a lot of research has been done to construct functions which are “as nonlinear as possible”. We discuss two possibilities to define nonlinearity: One approach uses differential properties of linear functions, the other measures the “distance” to linear functions. Let us begin with the differential properties. Given F : F n 2 → F n 2 , we define ∆F (a, b) := |{x : F (x+ a)− F (x) = b}|. We have ∆F (0, 0) = 2 , and ∆F (0, b) = 0 if b 6= 0. Since we are working in fields of characteristic 2, we may replace the “−” by + and write F (x+a)+F (x) instead of F (x−a)−F (x). We say that F is almost perfect nonlinear (APN) if ∆F (a, b) ∈ {0, 2} for all a, b ∈ F n 2 , a 6= 0. Note that ∆F (a, b) ∈ {0, 2} if F is linear, hence the condition ∆F (a, b) ∈ {0, 2} identifies functions which are quite different from linear mappings. Since we are working in characteristic 2, it is impossible that ∆F (a, b) = 1 for some a, b, since the values ∆F (a, b) must be even: If x is a solution of F (x + a)− F (x) = b, then x + a, too. In the case of odd characteristic, functions F : F n q → F n q with ∆F (a, b) = 1 for all a 6= 0 do exist, and they are called perfect nonlinear or planar. In the last few years, many new APN functions have been constructed. The first example of a non-power mapping has been described in [26]. Infinite series are contained in [5, 10, 11, 12, 13, 16, 17]. Also some new planar functions have been found, see [15, 22, 36]. There may be a possibility for a unified treatment of (some of) these constructions in the even and odd characteristic case. In particular, we suggest to look more carefully at the underlying design of an APN function, similar to the designs corresponding to planar functions, which are projective planes, see [29]. Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium. The research is supported by the Interuniversitary Attraction Poles Programme-Belgian State-Belgian Science Policy: project P6/26-Bcrypt. Department of Mathematics, Otto-von-Guericke-University Magdeburg, D-39016 Magdeburg, Germany An equivalent function has been found independently by Brinkmann and Leander [7]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one