Year: 2012
Type: preprint
Abstract: We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial $f(x)\in \F_{2^n}[x]$, where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on $\F_{2^n}$. The projective plane $Π$ obtained from $D$ in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $Π$ to be a presemifield plane. We also prove that a function $f$ on $\F_{2^n}$ with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any $x,y\in\F_{2^n}$.
Source: arXiv (Cornell University)
Author: Zhou, Yue
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: 多种密码学性质良好的密码函数的研究, G8360773726
OpenAlex ID: W2951752014