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+%
+% bytes.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Bytes}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Endlicher Körper}
+1 Byte = 8 bits: $\mathbb{F}_{2^8}$
+mit Minimalpolynom:
+\[
+m(X) = X^8+X^4+X^3+X+1
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Inverse $a^{-1}$}
+Mit dem euklidischen Algorithmus
+\[
+\begin{aligned}
+sa+tm&=1
+&&\Rightarrow&
+\uncover<3->{
+a^{-1} &= s}
+\\
+&
+&&&
+\uncover<4->{
+\overline{a}
+&=
+\begin{cases}
+a^{-1}&\; a\ne 0\\
+0     &\; a =  0
+\end{cases}}
+\end{aligned}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Vektorraum}
+$\mathbb{R}_{2^8}$
+ist ein $8$-dimensionaler $\mathbb{F}_2$-Vektorraum
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{S-Box}
+$S\colon a\mapsto A\overline{a}+q$ mit
+\begin{align*}
+\only<1-7>{\phantom{\mathstrut^{-1}}A}
+\ifthenelse{\boolean{presentation}}{}{\only<8>{A^{-1}}}
+&=\only<1-7>{\begin{pmatrix}
+1&0&0&0&1&1&1&1\\
+1&1&0&0&0&1&1&1\\
+1&1&1&0&0&0&1&1\\
+1&1&1&1&0&0&0&1\\
+1&1&1&1&1&0&0&0\\
+0&1&1&1&1&1&0&0\\
+0&0&1&1&1&1&1&0\\
+0&0&0&1&1&1&1&1
+\end{pmatrix}}
+\ifthenelse{\boolean{presentation}}{}{
+\only<8->{
+\begin{pmatrix}
+0&0&1&0&0&1&0&1\\
+1&0&0&1&0&0&1&0\\
+0&1&0&0&1&0&0&1\\
+1&0&1&0&0&1&0&0\\
+0&1&0&1&0&0&1&0\\
+0&0&1&0&1&0&0&1\\
+1&0&0&1&0&1&0&0\\
+0&1&0&0&1&0&1&0
+\end{pmatrix}}
+}
+\\
+q&=X^7+X^6+X+1
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Inverse $S$-Box}
+\vspace{-10pt}
+\[
+S^{-1}(b) = \overline{A^{-1}(b-q)}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup