From 2030ae0d26df555a75f964bbfbf6034e5704150b Mon Sep 17 00:00:00 2001 From: finn Date: Sun, 24 Jul 2022 17:00:38 +0200 Subject: [PATCH] Ausarbeitung --- Ausarbeitung/Ausarbeitung.bib | 165 +++++++++++++++- Ausarbeitung/Ausarbeitung.tex | 264 ++++++++++++++++++++++++- Ausarbeitung/Barchart.png | Bin 0 -> 43718 bytes Ausarbeitung/BarchartNorm.png | Bin 0 -> 41492 bytes Ausarbeitung/MD2Checksum.png | Bin 0 -> 82808 bytes Ausarbeitung/MD2HashBerechnung.png | Bin 0 -> 125595 bytes Ausarbeitung/Padding.png | Bin 0 -> 19412 bytes Ausarbeitung/Performance5GB.png | Bin 0 -> 52113 bytes Ausarbeitung/PiechartRuntime.png | Bin 0 -> 17894 bytes Ausarbeitung/benchmarks-error-in-4.png | Bin 0 -> 32986 bytes 10 files changed, 415 insertions(+), 14 deletions(-) create mode 100644 Ausarbeitung/Barchart.png create mode 100644 Ausarbeitung/BarchartNorm.png create mode 100644 Ausarbeitung/MD2Checksum.png create mode 100644 Ausarbeitung/MD2HashBerechnung.png create mode 100644 Ausarbeitung/Padding.png create mode 100644 Ausarbeitung/Performance5GB.png create mode 100644 Ausarbeitung/PiechartRuntime.png create mode 100644 Ausarbeitung/benchmarks-error-in-4.png diff --git a/Ausarbeitung/Ausarbeitung.bib b/Ausarbeitung/Ausarbeitung.bib index ce2deb3..ebea4aa 100644 --- a/Ausarbeitung/Ausarbeitung.bib +++ b/Ausarbeitung/Ausarbeitung.bib @@ -1,10 +1,163 @@ @manual{intel2017man, - author = {{Intel Corporation}}, - title = {Intel 64 and IA-32 Architectures Software Developer’s Manual}, - year = {2016}, - month = apr, + author = {{Intel Corporation}}, + title = {Intel 64 and IA-32 Architectures Software Developer’s Manual}, + year = {2016}, + month = apr, organization = {Intel Corporation}, - note = {\url{http://www.intel.com/content/www/us/en/processors/architectures-software-developer-manuals.html}, - visited 2017-08-19} + note = {\url{http://www.intel.com/content/www/us/en/processors/architectures-software-developer-manuals.html}, + besucht am 2017-08-19} +} +@manual{md2man, + author = {{B. Kaliski}}, + title = {The MD2 Message-Digest Algorithm}, + year = {1992}, + month = apr, + organization = {RSA Laboratories}, + note = {\url{https://datatracker.ietf.org/doc/html/rfc1319}, + besucht am 16.07.2022} +} +@manual{cmsman, + author = {{R. Housley}}, + title = {Cryptographic Message Syntax (CMS)}, + year = {2009}, + month = sep, + organization = {Vigil Security}, + note = {\url{https://datatracker.ietf.org/doc/html/rfc5652#section-6.3}, + besucht am 18.07.2022} } +@manual{md2berechnung, + title = {Kommunikation und verteilte Systeme, Kapitel 2.3 Hash Functions}, + year = {2007}, + organization = {RWTH Aachen}, + note = {\url{http://www-i4.informatik.rwth-aachen.de/content/teaching/lectures/sub/sikon/sikonSS07/05_Hash_1P.pdf}, + besucht am 20.07.2022} +} + +@manual{pidigits, + author = {{mikeazo}}, + year = {2014}, + note = {\url{https://crypto.stackexchange.com/questions/11935/how-is-the-md2-hash-function-s-table-constructed-from-pi}, + besucht am 20.07.2022} +} + +@manual{preimage, + author = {{S. Thomsen}}, + title = {An improved preimage attack on MD2}, + year = {2008}, + organization = {Technical University of Denmark}, + note = {\url{https://eprint.iacr.org/2008/089.pdf}, + besucht am 20.07.2022} +} + +@manual{notoneway, + author = {{F. Muller}}, + year = {2004}, + title = {The MD2 Hash Function Is Not One-Way}, + note = {\url{https://www.iacr.org/archive/asiacrypt2004/33290211/33290211.pdf}, + besucht am 20.07.2022} +} + +@manual{collisionattack, + author = {{L. Knudsen et al.}}, + year = {2010}, + title = {Cryptanalysis of MD2}, + organization = {Technical University of Denmark}, + note = {\url{https://link.springer.com/content/pdf/10.1007/s00145-009-9054-1.pdf}, + besucht am 23.07.2022} +} + +@manual{md5, + author = {{X. Wang et al.}}, + year = {2005}, + title = {How to Break MD5 and Other Hash Functions}, + organization = {Shandong University}, + note = {\url{http://merlot.usc.edu/csac-f06/papers/Wang05a.pdf}, + besucht am 23.07.2022} +} + +@manual{sha256, + author = {{S. Turner}}, + year = {2011}, + month = {March}, + title = {MD2 to Historic Status}, + organization = {Independent Educational Consultants Association}, + note = {\url{https://datatracker.ietf.org/doc/html/rfc6149#section-7}, + besucht am 23.07.2022} +} + + +@manual{sha265secure, + author={National Institute of Standards and Technology}, + title={Secure Hash Standard (SHS)}, + year={2015}, + month={August}, + note = {\url{https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.180-4.pdf}, + besucht am 23.07.2022} +} + + + +@book{hashing-in-computer-science, + author = {Alan G. Konheim}, + title = {Hashing in Computer Science : Fifty Years of Slicing and Dicing}, + year = {2010}, + month = jul +} + +@book{hashing-techniques, + author = {Lianhua Chi, Xingquan Zhu}, + title = {Hashing Techniques: A Survey and Taxonomy}, + year = {2018}, + month = jan +} + +@INPROCEEDINGS{salt-password-hashes, + author={Gauravaram, Praveen}, + booktitle={2012 International Conference on Advanced Computer Science Applications and Technologies (ACSAT)}, + title={Security Analysis of salt||password Hashes}, + year={2012}, + volume={}, + number={}, + pages={25-30}, + doi={10.1109/ACSAT.2012.49}} +} + +@ARTICLE{what-is-blockchain, + author={{Di Pierro, Massimo}}, + journal={Computing in Science & Engineering}, + title={What Is the Blockchain?}, + year={2017}, + volume={19}, + number={5}, + pages={92-95}, + doi={10.1109/MCSE.2017.3421554} +} + +@manual{pgp-message-formats, + author={D. Atkins}, + title={PGP Message Exchange Formats}, + year={1996}, + month = aug, + note = {\url{https://www.rfc-editor.org/rfc/rfc1991}, + besucht am 21.07.2022} +} + +@manual{md4, + author={{Bert den Boer, Antoon Bosselaers}}, + title={An Attack on the Last Two Rounds of MD4}, + year={1991}, + note = {\url{https://web.archive.org/web/20030523231212/http://dsns.csie.nctu.edu.tw/research/crypto/HTML/PDF/C91/194.PDF}, + besucht am 23.07.2022} +} + +@manual{mooreslaw, + author={{ Arjen K. Lenstra }}, + title={Key Lengths}, + year={2006}, + organization={Citibank, N.A., and Technische Universiteit Eindhoven}, + note={\url{https://infoscience.epfl.ch/record/164539/files/NPDF-32.pdf}, + besucht am 24.07.2022} +} + + diff --git a/Ausarbeitung/Ausarbeitung.tex b/Ausarbeitung/Ausarbeitung.tex index b741214..433765e 100644 --- a/Ausarbeitung/Ausarbeitung.tex +++ b/Ausarbeitung/Ausarbeitung.tex @@ -1,5 +1,8 @@ % Diese Zeile bitte -nicht- aendern. \documentclass[course=erap]{aspdoc} +%\input{Ausarbeitung.bib} +\usepackage{listings} +\usepackage{float} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% TODO: Ersetzen Sie in den folgenden Zeilen die entsprechenden -Texte- @@ -9,7 +12,6 @@ \author{Dorian Zedler \and Finn Dröge \and Thomas Florian} \date{Sommersemester 2022} % Beispiel: Wintersemester 2019/20 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % Diese Zeile bitte -nicht- aendern. \title{Gruppe \theGroup{} -- Abgabe zu Aufgabe \theNumber} @@ -17,32 +19,278 @@ \maketitle \section{Einleitung} +Der Begriff \glqq Hashing\grqq heißt ursprünglich \glqq zerhacken und mixen\grqq. Das bedeutet, eine Hash-Funktion zerhackt und mischt Informationen und leitet daraus ein Hash-Ergebnis ab \cite{hashing-techniques}. +1989 hat Ron Rivest die Hash-Funktion Message-Digest 2 Algorithm (MD2) veröffentlicht. Für eine lange Zeit war der MD2-Algorithmus eine der meist verwendeten Hash-Algorithmen. Die Hauptanwendungsgebiete sind Passwortsicherung, Nachrichtenauthentifizierungscodes, digitale Signaturen und somit auch Zertifikate\cite{notoneway}. 2004 wurde die Hashfunktion im Paper +\glqq The MD2 Hash Function is Not One-Way\grqq von Frédéric Muller als unsicher erklärt \cite{notoneway}. + + +\subsection{Die Eigenschaften kryptografischer Hashfunktionen} +Sichere Hashfunktionen müssen 3 Kriterien erfüllen: + +\subsubsection{Preimage resistance} +Preimage resistance besagt, dass eine Hashfunktion nur sehr schwer umkehrbar sein darf. +Das heißt, dass für einen Hash die entsprechende Eingabe nur mit sehr hohem Rechenaufwand bestimmt werden kann.~\cite{notoneway} \\ +Mit gegebenem $y = H(x)$ sollte es schwer sein, ein $x' \neq x$ zu finden, sodass gilt: \\ +$H(x') = y$.~\cite{preimage} + +\subsubsection{Second preimage resistance} +Second preimage resistance ist dann gewährleistet, wenn es für eine gegebene Ein- und Ausgabe einer Hashfunktion sehr schwierig ist, eine weitere Eingabe zu finden, die dieselbe Ausgabe produziert.~\cite{notoneway} \\ +Mit gegebenem $x$ und $y = H(x)$ sollte es schwer sein, ein $x' \neq x$ zu finden, sodass gilt: +$H(x') = y$.~\cite{preimage} + +\subsubsection{Collision resistance} +Eine collision resistance trifft dann zu, wenn es nur sehr wenige gleiche Hashs für unterschiedliche Eingaben gibt. +Bei einer Hashfunktion mit einer hohen collision resistance ist die second preimage resistance auch hoch. +Wenn es nur wenige Kollisionen des Hashs gibt, ist es folglich auch schwerer eine Eingabe zu finden, die denselben Hash produziert.~\cite{notoneway} \\ +Es sollte schwer sein ein $x$ und $x'$ zu finden, sodass gilt: $x \neq x'$ und $H(x) = H(x')$.~\cite{preimage} + +\subsection{Angriffe auf Hashfunktion} +\label{section:geburtstagsparadoxon} +Zu jeder der oben genannten Kriterien gibt es einen entsprechenden Angriff. Der Preimage-Angriff und der second Preimage-Angriff kann über eine brute-force Implementierung mit einer Komplexität von $2^n$ durchgeführt werden, wobei n die Anzahl der Bits des Hashs entspricht. Da der MD2-Algorithmus einen 128 Bit großen Hash erstellt, hat dieser eine Komplexität von $2^{128}$. Aufgrund des Geburtstagsparadoxons befindet sich der brute-force Kollisionsangriff in der Komplexität $2^{n/2}$ (für MD2: $2^{64}$). +Ein Hash-Algorithmus gilt als sicher, wenn es keine effizienteren Algorithmen als die drei brute-force Algorithmen gibt. +~\cite{preimage} + +\subsubsection{Preimage-Angriff} +Bei einem Preimage-Angriff soll zu einem gegebenen Hash die dazu passende Nachricht gefunden werden. +Der aktuell schnellste Preimage-Angriff auf MD2 wurde im Jahr 2008 von Søren S. Thomsen veröffentlicht~\cite{preimage}. +Dort wird beschrieben, wie anhand eines entdeckten Musters in der Berechnung des Hashes, die vorherigen Zustände der Berechnungstabelle aus einem Hash wiederhergestellt werden können. Es kann durch die bijektive Eigenschaft von XOR aus zwei Elementen auf das dritte geschlossen werden ~\cite{collisionattack}. Dadurch gibt es weniger mögliche Permutationen, welche die Eingangsnachricht annehmen kann. Insgesamt reduziert dieses Verfahren die Komplexität des Preimage-Angriffs von $2^{128}$ auf $2^{73}$. + +\subsubsection{Kollisionsangriff} +Basierend auf ähnlichen Mustererkennungen kann auch ein Kollisionsangriff auf den MD2 Hash durchgeführt werden. +Somit verringert sich die Komplexität des oben genannten Geburtstagsparadoxons von $2^{64}$ auf $2^{54}$ mithilfe einer \glqq Collision Attack with Arbitrary Chaining Input\grqq ~\cite{collisionattack}. + + + +\subsection{Alternativen zu MD2} +Wegen der oben genannten Angriffen ist der MD2 Hashfunktion seit 2004 unsicher. Mögliche Alternativen wären die Nachfolger von MD2, die auch von Ron Rivest erfunden wurden: MD4 und MD5. MD4 wurde 1991 ~\cite{md4} nur ein Jahr nach Veröffentlichung bereits als unsicher erklärt und gilt daher nicht als Alternative. In 2005 wurde MD2 als unsicher gegen Kollisionsattacken deklariert ~\cite{md5}. +Gängige Alternativen sind die Secure Hashing Algorithms (SHA-1, SHA-2, SHA-256, etc.) ~\cite{sha256}. Insbesondere SHA-256 gilt als noch als secure (stand: August 2015)~\cite{sha265secure} und wird daher als weitverbreitete Alternative genutzt. + + + +\subsection{Anwendungen von kryptografischen Hashfunktionen} + +Das heutzutage wohl wichtigste Anwendungsgebiet von Hashes ist das Sichern von Passwörtern zur Authentifizierung an computergestützten Systemen \cite{salt-password-hashes}. +Die Passwörter werden gehasht. Falls die Datenbank mit den Passwörter-Hashes in die Hände Dritter gelangt, macht dieses Hashing es unmöglich, den Klartext der Passwörter festzustellen. +Um eine Wörterbuch-Attacke vorzubeugen, wird oft zusätzlich vor dem Hashing noch ein \glqq salt\grqq (zufällige Daten) an das Passwort angehangen, welches zusammen mit dem Hash abgespeichert wird. \\ +Eine weitere gängige Anwendung ist die digitale Signierung von Daten. Hierbei wird der aus den Daten gebildete Hash signiert \cite{pgp-message-formats}. So kann sichergestellt werden, dass jene Daten nach der Signierung nicht mehr verändert wurden, da jede kleine Änderung an den Daten den Hash stark verändern würde. \\ +Auch wenn es darum geht, eine Kette von Ereignissen kryptografisch zu fixieren, ist Hashing ein wichtiges Werkzeug. In einer sogenannten Blockchain enthält jedes Element (z.B. eine Überweisung) den Hash des vorherigen Elementes.\cite{what-is-blockchain} Dadurch ist es möglich, die gesamte Kette an Überweisungen zu verifizieren. Ein praktisches Beispiel dafür sind Kryptowährungen wie Bitcoin. \\ + + + +\subsection{Berechnung von Prüfsumme und Hashwert in der MD2 Hashfunktion} +\label{section:berechnung} +Die MD2 Hashfunktion lässt sich in drei Schritte gliedern: Padding hinzufügen, Prüfsumme berechnen und einer finalen Berechnung. + +\subsubsection{Padding} +\label{section:padding} +Damit eine Nachricht mit MD2 gehasht werden kann, muss diese die Länge der Nachricht zunächst ein Vielfaches von 16 sein. +MD2 benutzt dafür das Paddingverfahren PKCS\#7. Es fügt an eine Nachricht N Bytes mit dem Wert N an. \\ +Im folgenden Beispiel ist N = 5: + +\begin{figure}[H] + \centering + \includegraphics[width=1.0\columnwidth]{Padding.png} + \caption{Padding Beispiel} +\end{figure} + +Bei dem Verfahren muss mindestens ein Byte eingefügt werden. Somit wird, falls die Nachricht bereits ein Vielfaches von 16 ist, genau 16 Bytes mit dem Wert 16 angefügt. Damit stellt man sicher, dass das letzte Byte eindeutig zum Padding gehört. \\ +Außerdem wird so eine triviale Preimage-Angriff verhindert: \\ +Zur Vereinfachung wird angenommen, dass das erwähnte Paddingverfahren die Nachrichtenlänge auf ein Vielfaches von 8 bringt. \\ +Sei $m_1 = [01, 01, 01, 01, 01]$ die Eingabe, dargestellt im Hexadezimalformat, dann wird $m_1$ mit diesem Paddingverfahren zu $p_1 = [01, 01, 01, 01, 01, 03, 03, 03]$ erweitert.\\ +Wird nun $m_2 = [01, 01, 01, 01, 01, 03, 03, 03]$ gewählt, so ist das Padding davon $p_2 = [01, 01, 01, 01, 01, 03, 03, 03]$, ohne die oben genannte Bedingung. Dadurch kommt es zu $hash(m_1) = hash(m_2)$, wobei $m_1 \neq m_2$. +Mit der Bedingung ergibt sich $p'_2 = [01, 01, 01, 01, 01, 03, 03, 03, 08, 08, 08, 08, 08, 08, 08, 08]$. In der Regel gilt dann $hash(m_1) \neq hash(m_2)$ und deswegen ist kein trivialer Preimage-Angriff möglich. \\ +Beim MD2-Hash gilt zusätzlich, dass für den Fall einer leeren Nachricht ein Padding mit 16 Bytes angefügt wird. Die Prüfsumme und der Hash kann damit, ohne eine Fallunterscheidung gebildet werden~\cite{md2man},~\cite{cmsman}. + +\subsubsection{Substitutionsbox} +\label{section:substitution} +Zur weiteren Berechnung wird eine Substitutionsbox verwendet, die sich aus Nachkommastellen von $\pi$ zusammensetzen lässt. +Hierbei werden die Nachkommastellen mithilfe des Durstenfeld shuffle ausgewählt.~\cite{pidigits} +Die Zahlen der Box werden als Zufallszahlen angesehen und werden verwendet um das Ergebnis zu streuen. + +\subsubsection{Prüfsumme} +An die Padding-Bytes werden 16 weitere Bytes angehängt, die sogenannte Prüfsumme. Diese Bytes werden zunächst mit 0 initialisiert. +Für die Berechnung der Prüfsumme wird für jedes n-te Byte der Prüfsumme die XOR-Operation auf das Byte selbst mit einem Wert von der Substitutionsbox $\pi$ ausgeführt: +\[ Pr"ufsumme[n] = Pr"ufsumme[n] \oplus \pi [k] \] + +Das Element k der Substitutionsbox $\pi$ in dem i-ten Schleifendurchlauf ist das Ergebnis der XOR-Operation von dem (i*16 + n)-te Byte der Nachricht (inklusive Padding) mit dem vorherigen Byte der Prüfsumme: +\[ k = Nachricht[i*16 + n] \oplus Pr"ufsumme[n-1 \mod 16] \] + +Hierbei gilt es zu beachten, dass der Vorgänger vom 0-ten Byte der Prüfsumme das letzte Byte ist. Deswegen ist die Formel für den Vorgänger von n als $n-1 \mod 16$ definiert. \\ +In der folgenden Grafik wird die Berechnung des achten Bytes der Prüfsumme veranschaulicht: + +\begin{figure}[H] + \centering + \includegraphics[width=1.0\columnwidth]{MD2Checksum.png} + \caption{MD2 Prüfsumme ~\cite{md2berechnung}} +\end{figure} + +Beispielsweise wird im dritten Schleifendurchlauf das oben dargestellte achte Byte der Prüfsumme wie folgt berechnet: +\[ Pr"ufsumme[8] = Pr"ufsumme[8] \oplus \pi [k] \] +mit +\[ k = Nachricht[3*16 + 8] \oplus Pr"ufsumme[8-1 \mod 16] \] + + +\subsubsection{Finale Berechnung} +\label{section:mdupdate} +Zur finalen Berechnung werden zunächst weitere 48 Byte Speicher alloziert und mit dem Wert 0 initialisiert. +Im Folgenden werden diese 48 Bytes in 3 Blöcke unterteilt: Block A beschreibt die ersten 16 Byte, Block B die mittleren 16 Byte und Block C die letzten 16 Byte. +Die Daten des zuvor erzeugten Buffers aus Nachricht, Padding und Prüfsumme werden mithilfe von weiteren Operationen verarbeitet und in die entsprechenden Blöcke geschrieben. \\ +Als Erstes wird vom Buffer ein 16-Byte Block in Block B kopiert. +Anschließend wird in Block C das Ergebnis der XOR-Operation auf Block A und Block B geschrieben. +Danach werden alle Bytes der drei Blöcken 18 Mal durchlaufen. +Zur Berechnung eines Bytes wird die XOR-Operation auf ein Element der Substitutionsbox $\pi$ und das Byte selbst angewandt: +\[ Byte[k] = Byte[k] \oplus \pi [t] \] +Das ausgewählte Element der Substitutionsbox wird durch das vorherige Byte und den Schleifendurchlauf j bestimmt: +\[ t = (Byte[k-1] + j) \mod 256 \] +Somit ist jedes Byte von seinem Vorgänger abhängig. Es gilt zu beachten, dass das vorangegangene Byte vom ersten Byte ist das letzte: +\[ Byte[0] = Byte[0] \oplus \pi [t] \] +mit +\[ t = (Byte[47] + j) \mod 256 \] +Im ersten der 18 Schleifendurchläufe wird außerdem $t = 0$ gesetzt. + +Nach den 18 Iterationen steht der Hash in Block A. +Das beschriebene Verfahren lässt sich mittels Abbildung \ref{md2finalehashberechnung} darstellen. + + +\begin{figure}[H] + \centering + \includegraphics[width=1.0\columnwidth]{MD2HashBerechnung.png} + \caption{MD2 Finale Hash Berechnung ~\cite{md2berechnung}} + \label{md2finalehashberechnung} +\end{figure} \section{Lösungsansatz} -\subsection{V0 - Naive Implementierung} +\subsection{V0 - Basis Implementierung} +Die basis Implementierung wurde mithilfe des Pseudocodes, welcher auf in der RFC1319 Spezifikation \cite{md2man} zu finden ist, erstellt. Sie folgt genau dem oben beschriebenen Schema \ref{section:berechnung}. -\subsection{V1 - Optimierte Implementierung SIMD?} + +\subsection{V1 - Optimierte Implementierung SIMD} +In dieser Implementierung wird der Code mittels SIMD-Operationen optimiert. +Eine for-Schleife zur Berechnung des Hashes wird durch 5 Intrinsics-Operationen ersetzt. +Beispielsweise wird in der Schleife eine XOR-Operation auf jedes Element eines Buffers ausgeführt. +Diese vielen iterativen Berechnungen werden durch den Befehl \_mm\_xor\_si128(x,y) ersetzt. +\\ +Das Einsetzen des Paddings wird mithilfe einer LUT gemacht. Hierbei verbessert sich die Laufzeit jedoch nicht, weil man trotzdem ein memcpy() aufrufen muss. +\\ +Kleinere Optimierungen werden auch vorgenommen, wie das Ersetzen des modulo-Operators durch ein logisches UND. +\\ +Weitere SIMD-Optimierungen können nicht gemacht werden, da die Berechnungen von Elementen immer von der Berechnung ihres Vorgängers abhängen. \subsection{V2 - Partielles Einlesen der Datei} +In den anderen Implementierungen wird die gesamte Datei zu Beginn in den RAM geladen. Zusätzlich muss nochmals so viel Speicher alloziert werden, um Platz für das Padding und die Prüfsumme zu schaffen. Daher ist die Dateigröße auf maximal die Hälfte des Hauptspeichers abzüglich Padding und Prüfsumme beschränkt. Der Speicherbedarf ist also in $\mathcal{O}(n)$. \\ +In dieser Implementierung wird die Datei in 16-Byte Blöcken geladen, welche jeweils einzeln verarbeitet und danach wieder aus dem Hauptspeicher gelöscht werden. Daher ist der Speicherbedarf hier in $O(1)$ und es können beliebig große Dateien verarbeitet werden. +\\ +Bei der Implementierung dieser Optimierung konnten die Vorgaben aus der Aufgabenstellung nicht eingehalten werden, da sie das vollständige Laden der Datei in den Hauptspeicher voraussetzen. Auch die Funktionen +\begin{lstlisting} +void md2_checksum(size_t len, uint8_t* buf) +\end{lstlisting} und +\begin{lstlisting} +void md2_hash(size_t len, const uint8_t buf[len], uint8_t out[16]) +\end{lstlisting} konnten aus diesem Grund nicht so umgesetzt werden, da auch sie das Vorhandensein der gesamten Datei im Hauptspeicher voraussetzen. +\begin{figure}[H] + \centering + \includegraphics[width=0.8\columnwidth]{Performance10GB.png} + \caption{Vorteil der zweiten Implementierung} + \label{fig:impl2} +\end{figure} + +Die Vorteile dieser Implementierung zeigen sich in Abbildung \ref{fig:impl2}. Nachdem die Eingabegröße ca. 6 Gigabyte überschritten hat, funktioniert bei dem verwendeten Rechner nur noch diese Implementierung. \subsection{V3 - Threading} +Bei dieser Implementierung soll die Berechnung des Hashes in zwei Threads aufgeteilt werden. Ein Thread soll sich um die Berechnung der Prüfsumme durchführen, während der zweite den Hash für die Nachricht berechnen soll. \\ +Aus dem Main-Thread werden mithilfe der Funktion \texttt{pthread\_create\(\)} diese beiden Threads erstellt. Sobald diese mit ihrer Berechnung fertig sind, werden die Threads wieder mittels \texttt{pthread\_join\(\)} wieder in den Main-Thread zusammengeführt.\\ +Durch das Aufteilen wurde die Prüfsumme noch nicht mit in den Hash gerechnet. Dies geschieht in einem letzten Durchlauf der Hash-Berechnungs-Funktion. - -\subsection{V4 - Kombination aller Optimierungen} - +\subsection{V4 - Referenzimplementierung} +Als vierte Implementierung wurde die Referenzimplementierung aus der RFC1319 Spezifikation \cite{md2man} übernommen. Dies dient dazu, eine Referenz für die Geschwindigkeit der anderen Algorithmen zu haben. Außerdem ist es ein wichtiges Werkzeug, um die Korrektheit der anderen Implementierungen zu beurteilen. \section{Korrektheit} +Um die Korrektheit der Implementierungen zu testen, wurden zwei Strategien angewendet: \\ +Zuerst wurden Beispielwerte von der RFC1319 Spezifikation \cite{md2man} getestet. Zusätzlich wurden einige Dateien mit zufälligen Daten getestet und mit den Ergebnissen der Referenzimplementierung aus selbigem Dokument verglichen. \\ +Da schon kleine Fehler im Algorithmus zu sehr großen Abweichungen im Ergebnis führen würden, kann davon ausgegangen werden, dass die Implementierungen korrekt sind. \\ -\section{Performanzanalyse} +\section{Performance-Analyse} + +Getestet wurde auf einem System mit einem [CPU] Prozessor, [Takt], [RAM] Arbeitsspeicher, [OS], [Bit], [Kernel-Version]. Kompiliert wurde mit GCC[Version] mit der Option \texttt{-O3}. +Die Berechnung wurden mit Eingabegrößen von [Start] bis [Ende] jeweils [Wiederholungen] mal durchgeführt und das arithmetische Mittel für jede Eingabegröße wurde in folgendes Diagramm eingetragen: + + +Auch für die Performance-Analyse wurden zwei Strategien gewählt: \\ +Um einen groben Überblick über die Performance zu bekommen, wurde zunächst die Laufzeit der gesamten Berechnung des Hashes gemessen. \ref{fig:benchmarks1} \\ + +\begin{figure}[H] + \begin{minipage}[t]{0.5\linewidth} + \centering + \includegraphics[width=1.0\columnwidth]{Performance5GB.png} + \caption{Laufzeitanalyse $\mathcal{O}(n)$} + \label{fig:benchmark1} + \end{minipage} + \begin{minipage}[t]{0.5\linewidth} + \centering + \includegraphics[width=1\columnwidth]{Performance5GB.png} + \caption{Laufzeitanalyse $\mathcal{O}(n)$} + \label{fig:benchmark2} + \end{minipage} +\end{figure} + +In Abbildung \ref{benchmark1} ist zu erkennen, dass die asymptotische Laufzeit aller Algorithmen in $\mathcal{O}(n)$ liegt. Dies lässt sich dadurch begründen, dass in verschachtelten for-Schleifen maximal eine der Schleifen über n iteriert. Alle anderen Schleifen iterieren über eine konstante Zahl. \\ +In den Implementierungen lassen sich sehr leichte Unterschiede in der Performanz erkennen (siehe vergrößerte Analyse in Abbildung \ref{fig:benchmark2}). Beispielsweise sind Implementierung 1 und 3 schneller als die Basisimplementierung 0, weil sie Parallelisierung bzw. SIMD ausnutzen. \\ + + +Die größte Schwierigkeit bei der Optimierung des MD2-Algorithmus ist die zweite Schleife der finalen Berechnung. Diese Schleife lässt sich nicht optimieren und benötigt die meiste Zeit bei der Berechnung. Um diese Problematik genauer zu analysieren, wurde die zweite Performance-Analyse entwickelt. Hierbei wird die Laufzeit der Berechnung der Prüfsumme, der ersten Schleife im finalen Schritt und der zweiten Schleife im finalen Schritt, separat gemessen. \\ +Mit diesem Test ist feststellbar, dass die zweite Schleife im finalen Schritt den Großteil der gesamten Laufzeit ausmacht. Das erklärt, weshalb der Effekt der oben beschriebenen Optimierungen minimal ist. Das Verhältnis dieser verschiedenen Laufzeiten ist im folgenden Diagramm dargestellt: + +\begin{figure}[H] + \centering + \includegraphics[width=1.0\columnwidth]{PiechartRuntime.png} + \caption{Verhältnis der Teil-Laufzeiten (in Prozent) ~\cite{md2berechnung}} +\end{figure} + +Gemessen wurde mit 100 Megabyte Eingabedaten und dem Durchschnitt über zehn Wiederholungen. + +In der dritten Implementierung (Threads) kann also lediglich ein Speed-Up durch die Parallelisierung der Prüfsumme erreicht werden. Somit werden bei 100 Megabyte maximal 2,31\% der Geschwindigkeit des Programms optimiert. +In der ersten Implementierung wird die erste Schleife mithilfe von SIMD Operationen optimiert. Hier kann der Speed-Up maximal 1.56\% betragen. +Diese beiden Erkenntnisse lassen auch aus den Abbildungen \ref{fig:barchart1} und \ref{fig:barchart2} ablesen. + +\begin{figure} [H] + \begin{minipage}[t]{0.5\linewidth} + \centering + \includegraphics[width=1\columnwidth]{BarchartNorm.png} + \caption{Normierte Laufzeitanalyse} + \label{fig:barchart1} + \end{minipage} + \begin{minipage}[t]{0.5\linewidth} + \centering + \includegraphics[width=1\columnwidth]{Barchart.png} + \caption{Vergrößerte Laufzeitanalyse} + \label{fig:barchart2} + \end{minipage} +\end{figure} + +Im ersten Balkendiagramm ist der Speed-Up kaum zu erkennen. Dies lässt sich mit der unoptimierten zweiten Schleife begründen. Beim Vergrößern des Diagramms (siehe Abbildung \ref{fig:barchart2}) wird der oben beschriebene, geringe Speed-Up in Implementierung 3 erkennbar. Implementierung 1 hat einen noch kleineren Speed-Up aufgrund der Tatsache, dass nicht die vollen 1.56\% optimiert werden können. + + +\\ + + +%Die Referenz-Implementierung hatte ab ca 4000MB eine deutlich kürzere Laufzeit als alle anderen Implementierungen. Das ließ sich darauf zurückführen, dass in dieser Implementierung für die Länge der Daten der Typ \code{unsigned int} verwendet wurde. Daher kam es bei rund 4,3 Gigabyte zu einem Überlauf und es wurden nicht mehr alle Daten verarbeitet. Nach dem Lösen dieses Problems entsprach das Ergebnis unseren Erwartungen \ref{fig:benchmarks}. +%% + +%In Abbildung \ref{fig:benchmarks} ist zu erkennen, dass die korrigierte Referenzimplementierung am langsamsten ist. Das lässt sich damit begründen, dass nicht in-place direkt auf dem Array gearbeitet wird. Stattdessen wird in jeder Iteration der Hauptschleife ein neuer Zwischenspeicher angelegt, welcher am Ende der Iteration in den Buffer kopiert wird. Durch diese vielen Kopiervorgänge ändert sich die Laufzeitklasse nicht, da sie eine konstante Laufzeit haben. Allerdings erhöht sich durch diesen Overhead trotzdem die tatsächliche Laufzeit. DAS MÜSSEN WIR STREICHEN... ES GEHT NICH MEHR AUS DEN BENCHMARKS HERVOR :'( \section{Zusammenfassung und Ausblick} +Der MD2-Algorithmus wurde im Jahr 1989 entwickelt und war lange ein Standard-Hashverfahren für kryptografische Anwendungsbereiche. Er wurde 2004 als aufgrund von Attacken als unsicher deklariert. +Im Projekt hat sich herausgestellt, dass sich der Algorithmus nur schwer optimieren lässt. Kleine Optimierungen können bei sehr großen Datenmengen mittels Threads umgesetzt werden. Diese Schwerfälligkeit des Optimierens bringt aber auch Vorteile für den Algorithmus, denn alle Angriffe müssen Permutationen ausprobieren, bis das gewünschte Ergebnis erreicht wird. Ist es zeitintensiv, den Hash zu berechnen, werden auch Angriffe sehr aufwändig.\\ +Weitere Optimierungsmöglichkeiten, welche in diesem Projekt nicht umgesetzt wurden, könnten möglicherweise aus dem Vorgehen der Angriffe resultieren. Vor allem wäre es möglich das, in den Angriffen erkannte, Muster als Anlaufstelle für weitere Parallelisierungen im Code zu benutzen.\\ +Durch Moore's Law kann davon ausgegangen werden, dass sich die Kosten eines Angriffes auf ein Kryptosystem alle 18 Monate halbieren~\cite{mooreslaw}. 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