From c8a276fee03f789becc6b412d29e2358903152a2 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 15 Apr 2021 11:13:29 +0200
Subject: add devide and conquor slides

---
 vorlesungen/slides/a/dc/prinzip.tex | 86 +++++++++++++++++++++++++++++++++++++
 1 file changed, 86 insertions(+)
 create mode 100644 vorlesungen/slides/a/dc/prinzip.tex

(limited to 'vorlesungen/slides/a/dc/prinzip.tex')

diff --git a/vorlesungen/slides/a/dc/prinzip.tex b/vorlesungen/slides/a/dc/prinzip.tex
new file mode 100644
index 0000000..c75af61
--- /dev/null
+++ b/vorlesungen/slides/a/dc/prinzip.tex
@@ -0,0 +1,86 @@
+%
+% prinzip.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzieren $\mod p$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Berechne $a^n\in\mathbb{F}_p$ für grosses $n$
+\end{block}
+\uncover<2->{%
+\begin{block}{Mengengerüst}
+\(
+\log_2 n > 2000
+\)
+\\
+\uncover<3->{%
+RSA mit $N=pq$: Exponenten sind $e,d$, $e$ klein, aber 
+\(
+ed\equiv 1 \mod \varphi(N)
+\)}
+\end{block}}
+\uncover<4->{%
+\begin{block}{Naive Idee}
+\verbatiminput{../slides/a/dc/naiv.txt}
+Laufzeit: $O(n) \uncover<5->{= O(2^{\log_2n})}$%
+\uncover<5->{, d.~h.~exponentiell in der Bitlänge von $n$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Idee 1: Exponent binär schreiben}
+\vspace{-12pt}
+\[
+n = n_k2^k + n_{k-1}2^{k-1} + \dots  +n_12^1 + n_02^0
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Idee 2: Potenzgesetze}
+\vspace{-12pt}
+\[
+a^n
+=
+a^{n_k2^k}
+a^{n_{k-1}2^k}
+\dots
+a^{n_12^1}
+a^{n_02^0}
+\uncover<8->{=
+\prod_{n_i = 1}
+a^{2^i}}
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<9->{%
+\begin{block}{Idee 3: Quadrieren}
+\vspace{-10pt}
+\begin{align*}
+a^{2^i}
+&=
+a^{2\cdot 2^{i-1}}
+\uncover<10->{=
+(a^{2^{i-1}})^2}
+\\
+&\uncover<11->{=
+(\dots(a\underbrace{\mathstrut^2)^2\dots)^2}_{\displaystyle i}}
+\end{align*}
+\end{block}}
+\vspace{-18pt}
+\uncover<12->{%
+\begin{block}{Laufzeit}
+Multiplikationen: $\le 2 \cdot(\log_2(n) - 1)$
+\\
+\uncover<13->{Worst case Laufzeit: $O(\log_2 n)$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
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