| Abstract: |
| A function $f:{\mathbb F}_2^n\{\mathbb F}_2^n$ is called almost perfect nonlinear (APN)
if $f(x)+f(y)+f(z)+f(x+y+z)\ne 0$
for all $x,y,z$ such that $|\{x,y,z,x+y+z\}|=4$.
The motivation to study these functions comes from
cryptography, since the defining property is somehow
opposite to linearity. My talk will
be a survey about APN functions. In particular, I will
discuss the following three problems:
\begin{itemize}
\item The big open problem on APN functions
is the question whether there are APN permutations
if $n$ is even. There is only one example known if $n=6$.
\item What is a good lower bound on the number of inequivalent
APN functions? Very recently, Christian Kaspers and Yue Zhou
showed that the members of a large family
of APN functions are pairwise inequivalent, which provides the best
known lower
bound on the number of inequivalent functions.
\item Most of the known APN functions $f$ are quadratic (which means that
the derivatives $x\mapsto f(x+a)+f(x)$ are linear). Find more
{\em nonquadratic} APN functions.
\end{itemize} |
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