From 862e89e11f7aa28ecfe19cf6771989f911b0338b Mon Sep 17 00:00:00 2001 From: Paul Date: Fri, 18 Sep 2020 18:52:36 +0200 Subject: [PATCH] Hashfunktionen completed --- Zusammenfassung_DC.aux | 12 +++ Zusammenfassung_DC.fdb_latexmk | 93 ++++++++----------- Zusammenfassung_DC.fls | 28 ++---- Zusammenfassung_DC.log | 165 +++++++++++++++------------------ Zusammenfassung_DC.pdf | Bin 2599764 -> 2631355 bytes Zusammenfassung_DC.synctex.gz | Bin 183533 -> 216356 bytes Zusammenfassung_DC.tex | 3 - Zusammenfassung_DC.toc | 11 +++ chapters/Hashfunktionen.tex | 124 ++++++++++++++++++++++++- 9 files changed, 267 insertions(+), 169 deletions(-) diff --git a/Zusammenfassung_DC.aux b/Zusammenfassung_DC.aux index b48d682..bf630f5 100644 --- a/Zusammenfassung_DC.aux +++ b/Zusammenfassung_DC.aux @@ -130,3 +130,15 @@ \@writefile{toc}{\contentsline {chapter}{\numberline {7}Hashfunktionen}{31}\protected@file@percent } \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} 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zVI(eTau67$rBWIjtt)fHOP`62zI){7TL?k!XL7VK5p3S)Vv||2W%BKUP-**jr87RQ5+luq}!N(hX)z6ovX1lLj<4gq8dvcc=VVk9*eUM@%x-m0%EOi z&aISy9YO;bZLJHzL;W@w~H6D2Q`W4Q<*C%zN zbc0xw8w_jv{nIY{@WQIKm6m5n+p`}KDHQv$;>96bKURE}@KYr`J61dh#IfRD!xP7f zw0!d){^9J0fBp6EzkMZJ`}*Kf@$WzV_MadB__sg*_RG(|e*N diff --git a/Zusammenfassung_DC.tex b/Zusammenfassung_DC.tex index e2c95ad..7d886a1 100644 --- a/Zusammenfassung_DC.tex +++ b/Zusammenfassung_DC.tex @@ -27,9 +27,6 @@ leftsub = \glq,% rightsub = \grq% ]{dirtytalk} -%bash functions -\usepackage{bashful} - %opening \title{Zusammenfassung DC} \author{Paul Lödige \\ Matrikel: 15405036} diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc index bb63a11..829470d 100644 --- a/Zusammenfassung_DC.toc +++ b/Zusammenfassung_DC.toc @@ -95,3 +95,14 @@ \contentsline {section}{\numberline {6.6}AES (Advanced Encryption Standard)}{29}% \contentsline {subsection}{\numberline {6.6.1}AES-128}{29}% \contentsline {chapter}{\numberline {7}Hashfunktionen}{31}% +\contentsline {section}{\numberline {7.1}schwache Kollisionsfreiheit}{31}% +\contentsline {section}{\numberline {7.2}\texttt {MessageDigest}-Instanzen: Hashfunktionen in Java}{31}% +\contentsline {section}{\numberline {7.3}Anwendungsbeispiele}{32}% +\contentsline {subsection}{\numberline {7.3.1}Anwendungsbeispiel: Passwortdatei}{32}% +\contentsline {subsubsection}{\numberline {7.3.1.1}Anwendungsbeispiel: Passwortdatei mit Salt und Iteration Count}{32}% +\contentsline {subsection}{\numberline {7.3.2}Anwendungsbeispiel: Integritätsschutz von Dateien}{32}% +\contentsline {subsection}{\numberline {7.3.3}Anwendungsbeispiel: Integritätsschutz bei einem Dateidownload}{32}% +\contentsline {section}{\numberline {7.4}Brute-Force-Angriffe auf Hashfunktionen}{32}% +\contentsline {subsection}{\numberline {7.4.1}Brute-Force-Urbildsuche}{32}% +\contentsline {subsection}{\numberline {7.4.2}Brute-Force-Kollisionssuche}{33}% +\contentsline {section}{\numberline {7.5}Konstruktionsverfahren von Hashfunktionen}{33}% diff --git a/chapters/Hashfunktionen.tex b/chapters/Hashfunktionen.tex index b5a65e1..d0ab074 100644 --- a/chapters/Hashfunktionen.tex +++ b/chapters/Hashfunktionen.tex @@ -6,5 +6,125 @@ wird als Hashfunktion bezeichnet, falls gilt: \begin{enumerate} \item $H$ ist eine \say{Einwegsfunktion} (es gibt keinen effizienten Algorithmus zum Auffinden eines Urbilds) - \item $H$ ist \say{kollisionsfrei} (es gibt keinen effizienten Algorithmus zum finden weiterer Nachrichten mit dem gleichen Hashwert) - \end{enumerate} \ No newline at end of file + \item $H$ ist \say{kollisionsfrei} (es gibt keinen effizienten Algorithmus zum Finden weiterer Nachrichten mit dem gleichen Hashwert) + \end{enumerate} + Die beiden Eigenschaften stehen in keiner Verbindung zueinander (Einwegsfunktion $\not\Leftrightarrow$ kollisionsfrei). + \vspace{2mm} + Hashfunktionen haben üblicherweise die folgenden Eigenschaften: + \begin{itemize} + \item Die Berechnung der Hashwerte ist effizient möglich und die Berechnungzeit ist im Wesentlichen proportional zur Nachrichtenlänge ($O(n)$) + \item $\mathscr M$ ist eine Teilmenge von $\mathscr Z ^{\ge 0}$ ($\mathscr Z = {\mathbb{Z}_2}^8$), + wobei die maximale Länge der in $\mathscr M$ vorkommenden Bytefolgen durch eine sehr große Zahl $N\in\mathbb{N}$ beschränkt ist ($\mathscr M=({\mathbb{Z}_2}^8)^{\le N}$). + In der Regel ist $N$ so groß, dass alle Nachrichten eine kleinere Länge haben. + \item Die Bitlänge $l$ des Hashwertes ist ein Vielfaches von 8. + \end{itemize} + Hashfunktionen sind wie die meisten Verschlüsselungsverfahren (Ausnahme OTP (siehe \ref{otp})) nicht \textbf{beweisbar} sicher. + Beispielsweise wurden für MD5 und SHA1 Algorithmen gefunden, die effizienter als durch Brute-Force eine Kollision finden. + + \section{schwache Kollisionsfreiheit}\label{schwache Kollisionsfreiheit} + Eine Hashfunktion $H:\mathscr M \rightarrow {\mathbb{Z}_2}^l$ ist schwach-kollisionsfrei, falls es keinen effizienten Algorithmus gibt, der zu einem beliebigen $m\in \mathscr M$ ein $\tilde{m}\in \mathscr m$ findet, für das gilt: + $$\tilde{m}\ne m \text{ und }H(\tilde{m})=H(m)$$ + Sie unterscheidet sich von der starken Kollisionsfreiheit darin, dass $m$ vorgegeben wird. + + \section{\texttt{MessageDigest}-Instanzen: Hashfunktionen in Java} + siehe Skript 2 Kapitel 4.2 (Seite 56(62)). + + \section{Anwendungsbeispiele} + \subsection{Anwendungsbeispiel: Passwortdatei} + Damit ein IT-System erkennt, ob ein vom Nutzer genanntes Passwort korrekt ist muss es Informationen über dieses Passwort abspeichern. + Am Einfachsten wäre es alle Passwörter in Plain-Text in einer Datenbank abzuspeichern. + Sollte diese Datenbank allerdings gehackt werden könnte der Hacker auf alle Nutzerkonten zugreifen. + Um dies zu verhindern müssen die Passwörter auf eine verifizierbare Art und Weise verschlüsselt (gehashed) werden. + Dies schützt allerdings nicht vor Wörterbuchangriffen, da es einem Angreifer möglich ist eine sortierte Liste der Hashwerte der häufig verwendeten Passwörter zu erstellen und abzugleichen. + + \subsubsection{Anwendungsbeispiel: Passwortdatei mit Salt und Iteration Count} + Zur Abwehr von Wörterbuchangriffen ist es üblich das Passwort $p_i$ mit einem für jeden Nutzer unterschiedlichen Salt-Wert $s_i$ zu verrechnen. + Dieser Saltwert wird ebenfalls in der Datenbank abgespeichert und führt dazu, dass jeweils $H(p_i||s_i)$ statt $H(p_i)$ bestimmt wird. + Eine Attacke ist somit nur noch Nutzerspezifisch durchführbar. + Um dies weiter zu erschweren wird die Hashfunktion häufig mehrmals hintereinander angewandt: + $$H^c(p_i||s_i)=\underbrace{H(H(...H(}_{c}p_i||s_i)...))$$ + Die Zahl $c$ wird als Iteration Count bezeichnet. + Hierdurch verzögert sich sowohl die reguläre Überprüfung von Passwörtern als auch die Laufzeit eines Angriffs um den Faktor $c$. + Für die Überprüfung ist dies meist kaum relevant (Verfahren wird einmal durchgeführt), während es für den Angriff einen großen Mehraufwand bedeutet (Verfahren wird \textbf{sehr} häufig durchgeführt). + + \subsection{Anwendungsbeispiel: Integritätsschutz von Dateien} + Um ein Rechnersystem vor der Ausführung von Schadsoftware zu schützen ist es Praxis, die Hashwerte der zur Ausführung zugelassenen Programme in einer Whitelist zu speichern. + Ein Angreifer kann Schadsoftware nur dann ausführen, wenn sie den gleichen Hashwert wie ein Programm auf der Whitelist hat. + + \subsection{Anwendungsbeispiel: Integritätsschutz bei einem Dateidownload} + Auf Webseiten wird häufig ein Hashwert für eine zu downloadenende Datei angegeben. + Dieser lässt sich dann nach dem Download mit der selbst aus der gedownloadeten Datei errechneten Hashwert überprüfen. + Dies bietet allerdings nur minimalen Schutz, da ein Angreifer, der die Webseite übernommen hat ebensogut den angegebenen Hashwert ändern kann. + Dies lässt sich erreichen, falls die Hashwerte der Datei auf mehreren Webseiten angegeben werden. + + \section{Brute-Force-Angriffe auf Hashfunktionen} + Ein Brute-Force-Angriff sucht nach einem Konflikt oder einem Urbild, indem alle möglichen Nachrichten durchprobiert werden. + + \subsection{Brute-Force-Urbildsuche} + Ein Angriff auf die Urbildresistenz stellt einen Angriff auf die schwache Kollisionsfreiheit (siehe \ref{schwache Kollisionsfreiheit}) dar. + D.h. es wird nach einem zweiten Urbild für ein bekanntes ($m,H(m)$)-Paar gesucht. + Hierfür werden die Hashwerte für alle $\tilde{m}\in \mathscr M\setminus\{m\}$ berechnet und mit $H(m)$ verglichen. + Um zu bestimmen, wie viele Versuche $r$ notwendig sind um mit einer Wahrscheinlichkeit $p$ ein $\tilde{m}$ kann die folgende Formel verwendet werden: + $$\begin{aligned} + r\ge&\frac{\ln(1-p_0)}{\ln(1-\frac{1}{n})}\\ + r\ge&|\ln(1-p_0)|\cdot n\\ + \end{aligned}$$ + mit $p\ge p_0$, $n=2^l\ge 2$. + Diese Formel ergibt für die folgenden Beispiele: + $$\begin{aligned} + r&\approx ln(2)\cdot 2^l &\approx 0,7\cdot 2^l \hspace{5mm} &\text{für } p=0,5\\ + r&\approx ln(10)\cdot 2^l &\approx 2,3\cdot 2^l \hspace{5mm} &\text{für } p=0,9\\ + r&\approx ln(100)\cdot 2^l &\approx 4,6\cdot 2^l \hspace{5mm} &\text{für } p=0,99\\ + \end{aligned}$$ + + \subsection{Brute-Force-Kollisionssuche} + Ein Angriff auf die starke Kollisionsfreiheit (siehe \ref{schwache Kollisionsfreiheit}), bei der zwei verschiedene Elemente $m_1,m_2\in \mathscr M$ gesucht werden, für die gilt: + $$H(m_1)=H(m_2)$$ + Hierbei werden alle $m\in \mathscr M$ ausprobiert und die zugehörigen Hashwerte $H(m)$ solange in eine Tabelle geschrieben, bis ein Hashwert doppelt vorkommt. + Hierdurch ist die Anforderung an den verfügbaren Speicher nicht zu vernachlässigen. + Auch hierfür lässt sich ebenfalls eine Formel errechnen, die vorgibt, wie viele Versuche $r$ zu erwarten sind, bis sich ein Hashwert $H$ mit einer Wahrscheinlichkeit $p$ wiederholt: + $$r\ge\sqrt{-2\cdot \ln(1-p_0)\cdot n+\frac{1}{4}}+\frac{1}{2}\hspace{10mm}\text{für } c(n,r)\ge p_0, r\le \sqrt{2n}$$ + Dies ergibt beispielhaft für die Kollisionswahrscheinlichkeit $p=c(n,r)=\frac{1}{2}$ die zwei mögliche Ungleichungen: + $$\begin{aligned} + r&\ge \sqrt{\ln(4)}\cdot\sqrt{n}+1\\ + r&\ge \sqrt{\ln(4)\cdot n+\frac{1}{4}}+\frac{1}{2} + \end{aligned}$$ + + \section{Konstruktionsverfahren von Hashfunktionen} + Im allgemeinen Bestehen Hashfunktionen aus 2 Teilen: + \begin{enumerate} + \item eine iterative Anwendung einer \textbf{Kompressionsfunktion} $f:{\mathbb{Z}_2}^{r+s}\rightarrow{\mathbb{Z}_2}^s$ auf $r$ Bits von $m$ und ein internes Zustandsregister $s$ + \item eine einmalige Ausführung einer \textbf{Ausgabefunktion} $g:{\mathbb{Z}_2}^s\rightarrow{\mathbb{Z}_2}^l$ + \end{enumerate} + Bei vielen Verfahren ist $l=s$ und es ist keine Ausgabefunktion nötig.\\ + Mithilfe von $f$, $g$ und einem Initialisierungswert $t_0\in{\mathbb{Z}_2}^s$ für das Zustandsregister wird $H(m)$ wie folgt berechnet: + \begin{enumerate} + \item $m$ wird durch ein Padding-Verfahren so erweitert, dass für $l(m||\text{pad})=l(m)+l(\text{pad})$ gilt: $r|l(m||\text{pad})$ + \item $m$ wird in Blöcke $m_1,...,m_n$ zerlegt, die alle $r$ Bit lang sind. + \item der Wert des Zustandsregisters $t$ wird durch iteratives Anwenden der Kompressionsfunktion $f$ auf den jeweiligen Block $m_i$ berechnet: + $$t_i:=f(m_i||t_{i-1})\hspace{10mm}\text{für }i=1,...,n$$ + \item der Hashwert wird ausgegeben: $H(m):=g(t_n)$ + \end{enumerate} + Die unterschiedlichen Hashfunktionen unterscheiden sich in der Bitlänge der Hashwerte, Blöcke und Zustandsregister: + \begin{center} + \begin{tabular}{c|c|c|c|c} + $H$ & $l=l(H(m))$ & $r=l(m_i)$ & $s=l(t_i)$ & $\min\{l(\text{pad})\}$\\ + \hline + \hline + MD5 & 128 & 512 & 128 & 65\\ + \hline + SHA-1 & 160 & 512 & 160 & 65\\ + \hline + SHA-224 & 224 & 512 & 256 & 65\\ + SHA-256 & 256 & 512 & 256 & 65\\ + SHA-512/224 & 224 & 1024 & 512 & 129\\ + SHA-512/256 & 256 & 1024 & 512 & 129\\ + SHA-384 & 384 & 1024 & 512 & 129\\ + SHA-512 & 512 & 1024 & 512 & 129\\ + \hline + SHA3-224 & 224 & 1152 & 1600 & 4\\ + SHA3-256 & 256 & 1088 & 1600 & 4\\ + SHA3-384 & 384 & 832 & 1600 & 4\\ + SHA3-512 & 512 & 576 & 1600 & 4\\ + \end{tabular} + \end{center} \ No newline at end of file

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