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Van Houtven}, - title = {Crypto 101}, - year = {2016}, - note = {\url{https://www.crypto101.io}}, -} + Author = {L. Van Houtven}, + Note = {\url{https://www.crypto101.io}}, + Title = {Crypto 101}, + Year = {2016}} @misc{ref4, - author = {Whitfield Diffie und Martin Hellman}, - title = {New Directions in Cryptographie}, - year = {1976}, - note = {\url{https://www-ee.stanford.edu/\%7Ehellman/publications/24.pdf}}, -} + Author = {Whitfield Diffie und Martin Hellman}, + Note = {\url{https://www-ee.stanford.edu/\%7Ehellman/publications/24.pdf}}, + Title = {New Directions in Cryptographie}, + Year = {1976}} @misc{ref5, - author = {R.L. Rivest und A. Shamir und L. Adleman}, - title = {A Method for Obtaining Digital Signatures and Public-Key - Cryptosystems}, - year = {1978}, - note = {\url{https://people.csail.mit.edu/rivest/Rsapaper.pdf}}, -} + Author = {R.L. Rivest und A. Shamir und L. Adleman}, + Note = {\url{https://people.csail.mit.edu/rivest/Rsapaper.pdf}}, + Title = {A Method for Obtaining Digital Signatures and Public-Key Cryptosystems}, + Year = {1978}} @misc{ref6, - author = {Wikipedia}, - title = {{D}iffie-{H}ellman-{S}chlüsselaustausch --- {W}ikipedia{,} - {D}ie freie {E}nzyklop{\"a}die}, - year = {2016}, - note = {\url{https://de.wikipedia.org/wiki/Diffie-Hellman-Schl%C3%BCsselaustausch}}, -} + Author = {Wikipedia}, + Note = {\url{https://de.wikipedia.org/wiki/Diffie-Hellman-Schl%C3%BCsselaustausch}}, + Title = {{D}iffie-{H}ellman-{S}chl{\"u}sselaustausch --- {W}ikipedia{,} {D}ie freie {E}nzyklop{\"a}die}, + Year = {2016}} @misc{ref7, - author = {Wikipedia}, - title = {{H}ybride {V}erschlüsselung --- {W}ikipedia{,} - {D}ie freie {E}nzyklop{\"a}die}, - year = {2016}, - note = {\url{https://de.wikipedia.org/wiki/Hybride_Verschl%C3%BCsselung}}, -} + Author = {Wikipedia}, + Note = {\url{https://de.wikipedia.org/wiki/Hybride_Verschl%C3%BCsselung}}, + Title = {{H}ybride {V}erschl{\"u}sselung --- {W}ikipedia{,} {D}ie freie {E}nzyklop{\"a}die}, + Year = {2016}} + +@misc{ref8, + Author = {Manuel P{\"o}ter}, + Note = {\url{http://www.manuel-poeter.at/tutorials/FBA-V1.0.PDF}}, + Title = {Kryptographie - Public-Key Verfahren am Beispiel von RSA}, + Year = {2001/2002 }} + +@misc{ref9, + Author = {Prof. Johannes Buchmann}, + Note = {\url{http://www.spektrum.de/magazin/faktorisierung-grosser-zahlen/823255}}, + Title = {Faktorisierung grosser Zahlen}, + Year = {1996}} + +@misc{ref10, + Author = {Adi Shamir, Eran Tromer}, + Note = {\url{http://www.wisdom.weizmann.ac.il/~tromer/papers/twirl.pdf}}, + Title = {Factoring Large Numbers with the TWIRL Device}, + Year = {2003}} diff --git a/main.pdf b/main.pdf index 0041ec5c89dc1f2d847eb73f17729f21b2bf04bf..387472d4db29786f01f1e2dfe07bf102b4e861b6 100644 GIT binary patch delta 217999 zcmeFXbyU=E+b;?TA`L@>NH}zNNHZcsr*wCB3JeA@5)#slbhk7p-5nAl-5@0);QkK& z-o2l_-@Vp8XRr0V=btlc!GgK#y07?Lbr-+G2-(Dg(W^pa*||CRG3m#qy5}(YXt-#c z&FnBmMKL)weO)YRICWIa>@3YaFgZ0m%{=bEk#%-+#N?EgcJ`ss=K|()bKUQ2e`xlyc{u51rU{9$S{x?+pyJh(=RQ+dJ{-x^w zq;5P27XKSL_vBwJf(Id`|18e`BZ<8Kg(M>I;s1g}f&W4h5nA zXOFn`1bWOPIy=FMO>X6nDe?&el4rupN734NjB1{>ce8@#heugdHf zOGP3wVHd_p8tWkjV&%azRTl)TZ$}6@S3bGlMd+AvjC1Et((jquG`l2C%ooJd7b)T7 z6^5017-@nfWUH@7Mpb_OKIk_+{HZzVE6do|eZk>)jJ1SefcD>=eB8V%7U{o2#Xqru zkO1%B*x(>(P}G@j}XNI?M<4W{=mwStn;=Yp&8}&zN)V zN+1qtC8H7TrjK&RR~U{)Dxr)=qq)jSU}?JyY-dmpcGV(7i2T=g$|1A;^w+9Z zpUk4jpVy!}#8mt^Arim#Pm908{zDdMKpsf~V|JaYGKpa*aslh$<&!CsdlKQCoNUUI zC2NJBc&**7^K1;Lz4jR_&Dyg{W{@5PZ-g+JQ>>w4>2cELN^S)TjF`NwbcOrB!eM2m z6Vf{SVXTMO^O%?2tHjv0Q-Rah)ufwEW|>>#ONEP|Usq5`%i(dj`LAsg*ANJ2X?@fr zEN}H4{e%GO>r7srAMV_2l5Wq(&K)+xt3TP>Mo6DbH*K;Sy7!EIuA^rlGvHz0nl^j0 z0^Pi-ttkvDqslTbb!S&!&>XTS*4PvmAJ(5TGJ*G zCdRN*64-_m2l^yJM}%Ah`fc9!s>iNs%w~*Qw9UqGJxDO{5x6_GG<(NUQg_)=ia& z`fc~PPH(u+N~1S8pc+$X@0ntil5uYwq?3l*zOY~P$wa0 z`U2v5yt>NqM~%WCO~^-DDXV6&6o)6lBzvr=g4vg>{pI@1>##^wi;ramW!?c}ieiUl zPwnI0JZ70Xp`I*tcwDm-|4bA@oDGw0ACr+>^|gm(j}!Gdok}Me#M_oMl#eNPk^2x( zd#PXg!`Dx9lD00Mix=^#lS%B66IG)oGTrKJc&m_s7MS*+Dp3P}7QCGE#xbm>!;V`X zZ6UR(_)^FYrB}N{L`nOXwEg5)HFBwbsG3S z8QyeSbGQ?#_97(ItE4VuAJZj5YB_(+UlYRj_JjmwTHE*Y^4Q^6-P$jS0o5wr=E6P9 zmxWFaDLfiIlWk6%dEi`DS?}I5Zv6GJ04(KFqbJjji#%i*HzC^OfZw*K{Znp46Z<=jlP5#fsm(KApZR0~u`xPubv{$#$6auHTeD7d z{^O?MFC-0fhWmkYXyTO?iIwYi-K$5Zd($-vmY{OCq6oS2&lp<*oRAoud^ZjyH$!?@ zuBg6ykIr@VPJL#smM3i2>s#imbB3XOf=j})O(B?GjRm`5RL0o;^ELF$GBh2dGr>0D ziOaSf-jRv-ZT^$b)7_7x&WMRsgf?wAF2FbZYa-_X`4grZpKZ9Y+t5<89lXi4ce0(M z&hPl5uhEGDTCzTH9dW07L}Hk9MOUr7mu@>Y#7$ApCp9$mBKt(NBT+e%{Zu$AKn{s+ z+3F1}ZzEs);b*lqnWV)js(kMLZc*p4VhWYFgAu7`Q%hEF3VJv3<5?xTgnH7L@`97h zA<}u4=wTlV>c51|w7C5Eob&p~8JQ)|FR7Awy^l)4*qcxHwJ(|Kc7xHkuU=9Ro43p3 z%VQC^GB-kDgKs;}o`f21j1Qt>aGX8c)R`&!0K>Z5DCx{MlHaU-ffMy(RWdhoeAEw> zLhZ|rVbDn&-@{*RqkMU&gq%!l8D3ef&+}v`y|Yi&U9m&XQjiU)ci!@95+|D#(QRxE zySA;GlJx{X;*Gic=o)kVTHMz!+|F{AonEk`r=#2s=O->YMdWi352SW4gF$*cn56(o ztq6wm2-7huNwi4Gb=;bNU@Lk}@ZfQjO=1&oK-|ooheOY=4)XT)!5OPYrAIm*PhMTv zwNZoLgcQ6s8yVZa@u= zVVE8F8yrTLZBGHZFeZxl=9gFJ^AN zpe*H=3l`_%90wjJ2Rx$Mi}HPRcZb$=?Cu&|<^7+BgF)(I%uz2tBm~qxmSb4-D5tt> zFdzcCu5oWfA33yqd(laiWH=nyKc1q15#DT3R@ayOs4ahRb)|uNy4Fg#OQnurbb

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z1?vbG;|v=C2|A?3Buqx}I+5xhHL9e`6V*|9oRnDQT-uI-h|Xh3)hu* z_74Fs6wg0&vQd|_VFa>~Te9)7n0)p}?y8C$El}c^_$Su?1qBdnVs5-rpokNNP{L4i z!0j+?X&M?q;~DAXpN!aoSoY}a;3JqcqSm!?whkHr)MKV8en-2>i|dtOVemIea0oUy z7-8{CnE6Ymr4v+=6Wo&%DxVWA4dD5hwM-ncf_*nk(8lRys?z@_Qk*b|3*H=MnBez=S#l!ncW*7ZAh)mF h$^Nl$qNnU)0ZD6DFeIp)+ 7 \\ - +\subsection{Konstruktion \textit{(e)}} +Wir bestimmen eine zu \textit{N} = 60 teilerfremde Zahl +\textit{e} < \textit{N} -> 7 \\ \begin{align*} Platzhalter \end{align*} -\subsection{Konstrktion d } -Um d zu Konstruieren müssen wir den erweiterten euklidischen Algorithmus mit m +\subsection{Konstruktion \textit{d} } +Um \textit{d} zu Konstruieren müssen wir den erweiterten euklidischen Algorithmus mit m anwenden -\begin{align*} -60 & = 8 * 7 + 4\\ - 7 & = 1 * 4 + 3\\ - 4 & = 1 * 3 + 1\\ - 3 & = 3 * 1 + 0\\ -\end{align*} +\begin{align} +60 & = 8 \cdot 7 + 4\\ % IH | frage an Ismail: wo her kommt die 8 und die 4? Da verstehe ich die Rechnung nicht... + 7 & = 1 \cdot 4 + 3\\ + 4 & = 1 \cdot 3 + 1 \label{eq:bspd1}\\ % IH | Labels können mit \ref{} dynamisch wieder aufgerufen werden + 3 & = 3 \cdot 1 + 0 +\end{align} -Danach verwenden wir die 2. letzte Zeile und formen die Gleichung nach 1 um +Danach verwenden wir Formelnr. \ref{eq:bspd1} und formen die Gleichung nach 1 um \begin{align*} -1 & = 4 - 1 * 3 \\ -1 & = 4 - 1 * ( 7- 1*4)\\ -1 & = 4 - 1 * 7 + 1 * 4\\ -1 & =(-1) * 7 + 2*4 \\ -1 & = (-1) * 7 + 2 * (60-8*7)\\ -1 & = 2* 60 -17 * 7 \ (mod)\\ -1 & = 43*7 mod 60 // (60 -17 = 43)\\ -1 & = d ^e mod n /// ( Umformen_ nach_ d)\\ -d & = 7^-1 mod 60 \\ - d & = 43\\ +1 & = 4 - 1 \cdot 3 \\ +1 & = 4 - 1 \cdot ( 7- 1 \cdot 4)\\ +1 & = 4 - 1 \cdot 7 + 1 \cdot 4\\ +1 & =(-1) \cdot 7 + 2 \cdot 4 \\ +1 & = (-1) \cdot 7 + 2 \cdot (60-8 \cdot 7)\\ +1 & = 2 \cdot 60 -17 \cdot 7 \, (mod)\\ +1 & = 43 \cdot 7 \, mod 60 \, (60 - 17 = 43)\\ +1 & = \textit{d} ^\textit{e} \, mod \, \textit{n} \, \slash \, ( \, Umformen \, nach \, \textit{d} \, )\\ +% IH | in Formeln werden leerzeichen ignoriert darum kann mit " \, " , " \: " , " \; " erstellt einen "schmalen", "mittleren" und "breiten" Leerschlag. Es geht auch mit \thinspace, \medspace oder \thickspace... +\textit{d} & = 7^{-1} \, mod \, 60 \\ + \textit{d} & = 43 \end{align*} \section{Entschlüsselung} Hallo Dies ist ein Test \section{Schwachstellen} -Hallo Dies ist ein Test +Obwohl schon einige verkündet haben die RSA Verschlüsselung geknackt zu haben ist es bisher noch niemandem gelungen einer Überprüfung stand zu halten. Es gibt aber durchaus realistische ideen wie der Code zerbrochen werden kann, nachgehend stellen wir die Wichtigsten Methoden vor. + +\subsection{Brut-force} +Die Methode alle möglichen Primzahlen von $\varphi=\pmineinsxqmineins$ auszuprobieren gilt als nicht einfacher als \textit{N} zu Faktorisieren. + +\subsection{Fakturierung durch die Kenntnis von \textit{N} } +Weil die Faktoren von \textit{N} den $\varphi$\textit{N} ermitteln lassen kann auch \textit{d} ermittelt werden. +Die Erfinder RSA selbst, berechneten anhand eines Algorithmus von Richard Schroeppel und der Annahme das ein Annäherungsschritt 1ms benötigt die Zerlegung von: + +\begin{center} + \begin{tabular}{| l | l | l |} % >{\( }c<{ \)} + \hline + \textbf{Zeichen} & \textbf{Operationen} & \textbf{Zeit} \\ + 50 & $1.4 \cdot 10^10$ & 3.9 Stunden \\ + 75 & $9.0 \cdot 10^12$ & 104 Tage \\ + 100 & $2.3 \cdot 10^15$ & 74 Jahre \\ + 200 & $1.2 \cdot 10^23$ & $3.8 \cdot 10^9$ Jahre \\ + \hline + \end{tabular} +\end{center} + +\begin{enumerate} + \item Diese Berechnungen der Entschlüsselungs-Zeiten sind überholt. (stand 1978) + + \item 1996 schreibt der Prof. Johannes Buchmann von der Universität Saarbrücken das ein Parallelisiertes Netz von 250 Rechnern auf dem Campusareal für eine 130 Stellige Zahl mehrere Wochen benötigt und sich mit mit drei zusätzlichen Dezimalstellen verdoppelt. + + \item 2003 veröffentlichte Adi Shamir und Eran Tromer einen technischen Report wie ein RSA Schlüssel von 1024 bit in unter einem Jahr gebrochen werden kann. \cite{ref10} % <-- bibtex Link +\end{enumerate} + +\textbf{Diese drei Beispiele zeigen auf wie unvorhersehbar die Standhaftigkeit eines Schlüssels in Bezug auf Zeit ist. } + +Die Formel zur Zerlegung von $\varphi$\textit{N} lautet: +\begin{equation} + \varphi=2 \cdot kgV\left(\frac{p-1} {2} , \frac{q-1}{2}\right) +\end{equation} + +\subsection{Berechnung von $\varphi$\textit{N} ohne Fakturierung von \textit{N} } +Natürlich lässt sich $\varphi$\textit{N} auch ohne Fakturierung von \textit{N} ermitteln wenn \textit{d} bekannt ist oder ermittelt werden kann. +Da \textit{d} jedoch ein multiplikator von $\varphi$\textit{N} ist, ist sein wert nicht leichter zu ermitteln als die Fakturierung von \textit{N} ist. +%IH | Ivan, check ob die Formel auch stimmt sobald Ismail die Section Konstruktion von "d" fertig hat! + +\subsection{zu kleine Multiplikator-Primzahlen} +Da die Sicherheit von RSA darauf beruht dass die Fakturierung von Primzahlen Zeit benötigt, ist sie auch nur +so stark wie die Grösse der Primzahl \textit{q} die Multipliziert mit \textit{p} den Modulus ergibt. Ist \textit{q} oder \textit{p} kleiner als 100 Stellen, wird daraus nicht ein Schlüssel $>10^200$ entstehen und damit die Verschlüsselung zwar schneller geschehen aber sie ist auch gefährdeter durch Brute-force Attacken oder Fakturierung zerlegt zu werden. + +\subsection{Gleiche $\varphi$ \textit{N} } + +\subsection{Riehmann hypotese } +Die Riehmann Hypothese beschreibt ein bisher ungelöstes Mathematisches Problem. +Sollte sich die Theorie der Reihmann Hypothese bewarheiten könnten daraus Primzahlen abgeleitet werden auf dessen Basis die Zerlegung von \textit{N} einfacher und schneller ausgeführt werden kann. + +\subsection{Social Engineering} +Durch das Abfangen einer Nachricht kann ein Angreifer damit noch nichts anfangen da sie mit dem Schlüssel des Empfängers Verschlüsselt ist. Möchte er diese nun entschlüsseln muss er an den Schlüssel des Empfängers kommen. Dazu kann er die Datei wiederum mit einem Ihm bekannten Schlüssel verschlüsseln und sie dem Empfänger erneut und gegebenenfalls unter Verschleierung seiner Identität zustellen. Dieser wird nun die Datei mit seinem Schlüssel entschlüsseln und nichts damit anfangen können da sie immer noch mit dem Schlüssel des Angreifers verschlüsselt ist. Bringt nun der Angreifer durch Geschick den Empfänger dazu Ihm diese entschlüsselte vermeintlich defekte Datei zuzusenden kann er sie mit seinem Schlüssel entschlüsseln und den Inhalt lesen. +% IH | Ivan finde heraus ob auch der Schlüssel gefunden werden kann... \newpage \section{Referenzen} diff --git a/math_rsa.cls b/math_rsa.cls index 34b08ac..f416a8a 100644 --- a/math_rsa.cls +++ b/math_rsa.cls @@ -22,11 +22,11 @@ \linespread{1.3} %Make whitespace in tables bigger -\renewcommand\arraystretch{1.5} +\renewcommand\arraystretch{1.2} % IH. von 1.5 auf 1.2 geändert. %Enable colours for tables \RequirePackage[table]{xcolor} - +%\RequirePackage{array} % formula rows in tables %Needed to rotate graphics and tables \RequirePackage{rotating} @@ -72,3 +72,6 @@ listing options={basicstyle=\ttfamily,}, title=\thetcbcounter #2, #1 } + +% Reference a Bibtext item +\RequirePackage{cite}