Two Classes of Quadratic Crooked Functions

It is known that almost perfect nonlinear (APN) functions have many applications in cryptography, and a quadratic function is crooked if and only if it is APN. In this paper, we introduce two infinite classes of quadratic crooked multinomials on fields of order 22m. One class of APN functions constructed in [8] is a particular case of the one we construct in Theorem 1. Moreover, we prove that the two classes of crooked functions constructed in this paper are EA inequivalent to power functions and conjecture that CCZ inequivalence between them also holds.

Crooked Functions, Bent Functions, and Distance Regular Graphs

Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.