From 62ae2096fa6642b396d21ccf6799bd12bb8748e7 Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Sun, 10 Jun 2012 18:03:12 +0200 Subject: [PATCH] UE5 Aufgabe 25-28 Sollte passen --- UE/lib.py | 2 -- UE/ue5.pdf | Bin 145242 -> 149838 bytes UE/ue5.tex | 26 +++++++++++++++++++++++++- 3 files changed, 25 insertions(+), 3 deletions(-) diff --git a/UE/lib.py b/UE/lib.py index 245f758..068f0f8 100644 --- a/UE/lib.py +++ b/UE/lib.py @@ -364,5 +364,3 @@ def sqmult_tex(x,k,func,e): print " ", x, "&" , y , "&" , k, "&" , k%2, "\\\\" print "\\end{array}" return y - - diff --git a/UE/ue5.pdf b/UE/ue5.pdf index 331ada58905101e7b1d3d36ea40c55366deb69a7..dba635b2147e7677e7e2e122ae80d40c312f073f 100644 GIT binary patch delta 71679 zcmY(pV|XB4(=8m^wr$(CZEIq?V@+&MoJ_2VZEIrNnwXQE`+lDH`_6m*)K$H!di9Uq zSM61+_WCu$7%iY9pRmD&((|r;N&}8Ln6<|)szM>-I#wasLQ}s+ zklWn!ZAeS6F7B7#nemMZg{7?|FK7s+;_bnd|GkEhG2@SJaCK0aI!-NZW4iAjvoMR5 z9eUct7uw~_URAS`5e3_SjH|v)GiT)2ZVUUJ&@p$3qC4-3@>J2|n(Bp-V6+PC_bfLmWV5Io90O)Qn5Uv3l@ 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Wie kann man ihre Stellenanzahl am einfachsten berechnen und was sind ihre $3$ Endziffern?} - +Die Anzahl der Stellen einer Zahl können mit $\log_{10}$ berechnet werden. Daher ist $\log_{10} 2^{43112609} = 43112609\cdot\log_{10} 2 \approx 12978188.5003$\\ +Für die letzten 3 Stellen berechnet man nun: \begin{align} 2^{43112609} \in (\Z_{1000}, * \mod 1000, 1) \text{, d.h. }x=2, k=43112609 \end{align} @@ -24,6 +25,29 @@ lib.sqmult_tex(2,43112609,lambda x,y:(x*y)%1000,1) \begin{align} p=2^{43112609} \mod 1000 &\equiv 512 \Rightarrow 2^{43112609}-1 \mod 1000 \equiv 511 \end{align} +\newpage\noindent +Oder aber man Überlegt sich vorher eine Sinnvolle Zerlegung des Exponenten und Verkürzt so den SM Algorithmus:\\ +Die letzten drei Ziffern entsprechen $2^{43112609} -1 \mod 1000$. Mithilfe von $\varphi(1000) = 400$ lässt sich nun der Exponent zerlegen. +\begin{align} +43112609 &= \alpha 400 + \beta\\ +&= 107781\cdot 400 + 209\\ +\end{align} +Da aber $a^{\varphi(n)}\equiv1\mod n$ ist, gilt: +\begin{align} + 2^{43112609} -1 \mod 1000&=2^{107781\cdot 400 + 209} -1 \mod 1000\\ + &=2^{209} -1 \mod 1000 +\end{align} +$2^{209}-1 \mod 1000$ lässt sich nun einfach mit dem Square and Multiply Algorithmus berechnen: +\begin{center} +$\begin{python} +import lib +lib.sqmult_tex(2,209,lambda x,y:(x*y)%1000,1) +\end{python}$ +\end{center} +Daraus Folgt nun +\begin{align} + p=2^{209} \mod 1000 &\equiv 512 \Rightarrow 2^{43112609}-1 \mod 1000 \equiv 511 +\end{align} \aufgabe{26} -- 2.47.3