A new family of semifields with 2 parameters

A new family of commutative semifields with two parameters is presented. Its left and middle nucleus are both determined. Furthermore, we prove that for any different pairs of parameters, these semifields are not isotopic. It is also shown that, for some special parameters, one semifield in this family can lead to two inequivalent planar functions. Finally, using similar construction, new APN functions are given

Non-Boolean almost perfect nonlinear functions on non-Abelian groups

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.Comment: 17 page

Crooked maps in F2n

AbstractA map f:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n} is an affine hyperplane for every fixed a∈F2n∗ (where F2n is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set {xd+γ(x+a)d:x∈Fpn} contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn

On Some Properties of Quadratic APN Functions of a Special Form

In a recent paper, it is shown that functions of the form $L_1(x^3)+L_2(x^9)$, where $L_1$ and $L_2$ are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and sufficient conditions for such functions to be APN