From 6b40a0fcee9d8ee14df738b390d315a574fd1192 Mon Sep 17 00:00:00 2001 From: Daniel Date: Sat, 6 Mar 2021 17:02:20 +0100 Subject: [PATCH] SQL Injection und RSA --- SQL-injection.odt | Bin 0 -> 141101 bytes SQL-injection.pdf | Bin 0 -> 139937 bytes rsa.org | 77 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ rsa.pdf | Bin 0 -> 203546 bytes 4 files changed, 77 insertions(+) create mode 100644 SQL-injection.odt create mode 100644 SQL-injection.pdf create mode 100644 rsa.org create mode 100644 rsa.pdf diff --git a/SQL-injection.odt b/SQL-injection.odt new file mode 100644 index 0000000000000000000000000000000000000000..0c713691c679dd04a500a3c4578efb3af3b4f52a GIT binary patch literal 141101 zcmb4p1ymhRvnL)PxVuBJ;DO-o8r<#T?k>TDyItHZxVyW%ySqcsyX61&dvEugZ{MCb zXZlP}_f&UvS9RB~rb|H@5()zh3>FMb#>YfacYp=q3m6#K-}F%gW^Hb5?Bs4|Y+z?+ zWezZKGPkv1bg?mFur+WrcVw`&Gqy3Z1vpz9+c+^e8M`?t{9lcI(EsO!^N|y_wJ|X_ zb$0lt8AldICtF)9LjwoK{~48um92r3@&Af`3D2 z000dKrj?uQ{1i02B4uF 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Was sind die Primfaktoren von \( 7608506433492111151 \)? + +- Plaintext \( m \) ist eine Ganzzahl \( \in [1, n-1] \subset \mathbb{Z} \) +- Cyphertext \( c \) ist eine Ganzzahl \( \in [1, n-1] \subset \mathbb{Z} \) +- \( n \) bezeichnet den Modulus. \( n = p \cdot q \text{, }p, q \in \mathbb{P} \) (Menge der Primzahlen) +- \( \phi(n) = (p-1) \cdot (q-1) \) +- /e/ -- Public Exponent, /d/ -- Private Exponent. \( e, d \in [1, \phi(n)] \) +- Euler'sche \phi-Funktion: \( \phi(n) := |\{ a \in \mathbb{N} | 1 \leq a \leq n \land ggT(a, n) = 1 \}| \). Zählt + - die Anzahl der der teilerfremden Zahlen: + - \( \phi(7) = |\{1,2,3,4,5,6\}| = 6 (= 7 - 1)\) → 7 ist Primzahl + - \( \phi(6) = \phi(2 \cdot 3) = \phi(2) \cdot \phi(3) = (2-1) \cdot (3-1) = 2 \) und tatsächlich \( \phi(6) = |\{1, 5\}| = 2 \) + - \( p, q \in \mathbb{P} \rightarrow \phi(p \cdot q) = (p-1) \cdot (q-1)\). Gleichtes gilt für alle Primfaktoren jeder Ganzzahl + - Und daher genauso die Anzahl der Elemente einer Gruppe \( \mathbb{Z}_{n}^{*}, n \in \mathbb{N} \) (der eigentliche Grund für die Verwendung in RSA) + + +** RSA Trapdoor Permutation +Der Grund, warum es funktioniert: +- /e, d/ sind invers: \( e \cdot d \mod \textcolor{red}{\phi(n)} = 1 \) +- Daher: \( c^{d} \mod n = (m^{e})^{d} \mod n= m^{e \cdot d} \mod n = m \) + +\pagebreak +** RSA Key Generation +1. Wähle zwei zufällige Primzahlen /p, q/: + - Sie müssen groß sein (jeweils mindestens irgendwo im Bereich \( 2^{1024} \)) damit \( p \cdot q \approx 2^{2048}\) und nicht zu nah bei einander liegen. +2. Generiere /public exponent e/: + - \( e = \text{ randprime}(\phi(n)) \) (zufällige Primzahl kleiner \( \phi(n) \)) +3. Generiere /private exponent d/: + - Inverse von /e/ bezüglich \( mod \phi(n) \) mithilfe des erweiterten euklidischen Algorithmus: + - mathematisch: \( e \cdot \textcolor{red}{d} + \phi(n) \cdot \textcolor{gray}{t} = ggT(e, \phi(n)) \) + - Pseudocode: (ggT, \textcolor{red}{d}, \textcolor{gray}{t}) ← \texttt{xgcd(e, }\phi(n)\texttt{)} + +\( \phi(n) \) muss offensichtlich geheim bleiben, da man sonst /d/ aus /e/ und \( \phi(n)\) mit =xgcd()= berechnen kann. Ebenso müssen \( p, q \) geheim bleiben, da so schnell \( \phi(n) \) berechnet werden kann. Modulus /n/ muss hingegen öffentlich sein, damit verschlüsselt werden kann. + +** RSA Encryption +\( c = m^{e} \mod n \) + +** RSA Decryption +\( m = c^{d} \mod n \) + +\pagebreak +** Angriffe gegen RSA /Verschlüsselung/ +*** RSA ist deterministisch +Gleiche Daten werden immer gleich verschlüsselt. Es sollte Padding auf eine Standardlänge genutzt werden. Das Padding sollte ungefähr 64 Bits an Randomness haben. +*** RSA ist "weich" +Man kann Cyphertext-Nachrichten mit einander multiplizieren und erhält wieder eine valide Cyphertext-Nachricht: +\begin{equation*} + c_{1} \cdot c_{2} \mod n = m_{1}^{e} \cdot m_{2}^{e} \mod n = (m_{1} \cdot m_{2})^{e} \mod n +\end{equation*} + +Aus vorhandenen Cyphertexten können neue Cyphertexte generiert werden. Das hilft bei /known cyphertext/ und /known plaintext/ Attacken. Es kann auch nach einem inversen Element für den Cyphertext gesucht werden, der bei der Multiplikation mit dem Cyphertext /1/ ergibt, wofür Plaintext und Cyphertext gleich sind. + +** Signieren mit RSA +→ Verschlüsseln einer Nachricht /m/ mithilfe des /privaten/ Schlüssels. + +Es ist möglich, eine Signatur mithilfe seines privaten Schlüssels zu erzeugen, doch davon wird abgeraten: Es ist sehr langsam und kann nur Daten (Ganzzahlen) signieren, die kleiner sind, als der Modulus /n/. + +Daher ist es eher üblich die Hashsumme der Daten zu bilden und diese Hashsumme zu signieren. So kann die Hashsumme auch bis zur Größe des Modulus (sha512sum und auffärts) herankommen, was sehr sicher ist. + +** Probleme mit RSA /Signaturen/ +- Known Plaintext Attack wird einem quasi geschenkt, wenn die Signatur unter die Nachricht kommt. +- \( 0^d \mod n = 0, 1^d \mod n = 1, (n-1)^d \mod n = n-1 \). Diese Signaturen kann jeder erzeugen. +- /Blinding Attack/: Sei /M/ eine Nachricht, die nicht signiert werden würde, aber \( R^eM \) schon ( \( MR^e \) geht auch). Signiert der Besitzer des /privaten/ Schlüssels \( R^eM \), kann auf folgende Art eine Signatur für /M/ erzeugt werden: + +\begin{equation*} + (R^eM)^d \mod n = R^{e \cdot d}M^d \mod n = RM^d \mod n +\end{equation*} +Jetzt muss die Signatur /S/ nur noch durch /R/ geteilt werden: +\begin{equation*} + S/R \mod n= \frac{RM^d}{R} \mod n = M^d \mod n +\end{equation*} diff --git a/rsa.pdf b/rsa.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2cf46c934bf3d1cee66f513d2f416e68e2257748 GIT binary patch literal 203546 zcma&NQ;;se(q`SZZQHi(?%g)uwrz8_ZQHhO?6z&&n)ClLGdJgAzPhT4yv>Mb<;o~> zB{2zlW(IZ`@_$PsYcT9YOhop^Rxo^gFpP4hcIGY?L@Z2fTtxr>fnk)ev~e+YB4U)V zF>)~#Gc~aXn8FAMz&N`&nHt%`cx+s2YuRr~AoZ35@&@(l=g;H!bVOc+ zydU8xU39n@p%j=rpMVDTr=aN2^GU$(@#Xz!=)R6ARWgK#RE!xzQe`sX;J6%h^lfY+ zIbrRDaPepntd`-+eMtQvd)gbOQMOl z^B|f*v0o?aFN|sfZ5vb>yOET`VT+%unh)D+Zk2x=->3imoaT35zS|2_$(A=T|`es5i6@<*P6km ziDW5s^MZFFEqUv{Y{O{C(vT@Dw0ERrv3R1wgkhQE4s3#FyDmxU5-CyRA@TcVU%rpW 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