From 2b6d6f80f159f33e23bf10b8aba69607738e6d9a Mon Sep 17 00:00:00 2001 From: Peter Gerwinski <peter.gerwinski@hs-bochum.de> Date: Thu, 9 Jun 2022 14:40:17 +0200 Subject: [PATCH] Notizen und Tafelbilder 9.6.2022 --- 20220519/ad-20220519.txt | 2 +- 20220609/ad-20220609.txt | 181 ++++++++++++++++++++++++++++++ 20220609/polynome-20220609.xcf.gz | Bin 0 -> 514621 bytes 3 files changed, 182 insertions(+), 1 deletion(-) create mode 100644 20220609/ad-20220609.txt create mode 100644 20220609/polynome-20220609.xcf.gz diff --git a/20220519/ad-20220519.txt b/20220519/ad-20220519.txt index 1403571..1b7b3f7 100644 --- a/20220519/ad-20220519.txt +++ b/20220519/ad-20220519.txt @@ -29,7 +29,7 @@ Damit sich die Teams nicht gegenseitig blockieren: - "Dummy"-Funktionen für 64-Bit-Zahlen schreiben, um damit schon mal die Schnittstelle festzulegen. - - Später dann die Dummy-FUnktionen gegen die richtigen austauschen. + - Später dann die Dummy-Funktionen gegen die richtigen austauschen. Beispiel: diff --git a/20220609/ad-20220609.txt b/20220609/ad-20220609.txt new file mode 100644 index 0000000..ff9a82a --- /dev/null +++ b/20220609/ad-20220609.txt @@ -0,0 +1,181 @@ +Fehlererkennung und Fehlerkorrektur +=================================== + +Parität, 09.06.2022, 11:05:40 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Nachricht enthält n Datenbits und ein Paritäts-Bit. + - Ungerade Parität: Die Gesamtzahl der Eins-Bits muß ungerade sein. + ungerade Anzahl von Daten-1-Bits --> Paritäts-Bit muß 0 sein. + gerade Anzahl von Daten-1-Bits --> Paritäts-Bit muß 1 sein. + - Gerade Parität: Die Gesamtzahl der Eins-Bits muß gerade sein. + +Anders formuliert: Die n+1-Bit-Zahl aus den Datenbits und dem Paritäts-Bit + ... muß insgesamt ungerade/gerade sein. + = ... muß bei Division durch 2 einen vorgegebenen Rest liefern. + +Nachteil: nur 1 Bit-Fehler erkennbar + +Verbesserungsidee: durch etwas anderes teilen + +CRC, 09.06.2022, 11:05:27 +~~~~~~~~~~~~~~~~~~~~~~~~~ +Nachricht enthält n Datenbits und m Paritäts-Bits. + +Wir betrachten die n+m-Bit-Zahl als ein Polynom über { 0, 1 } (mod 2). + +CRC: Bei Polynomdivision muß ein vorgegebener Rest bleiben. + (Siehe "Tafelbild".) + +--> sehr zuverlässige Fehlererkennung, + sehr leicht zu implementieren, auch in Hardware + +--> nicht geeignet gegen gezielte Manipulation + +Kryptographische Hash-Funktionen, 09.06.2022, 13:02:24 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Spezielle Funktionen: Nur 1 Bit anders im Input --> völlig anderer Output + +Für digitale Signaturen: + - nicht mehr empfohlen: MD5 + - aktuell empfohlen: SHA-2 (= SHA-224, SHA-256, SHA-384 und SHA-512) + NICHT empfohlen für Passwort-Hashes + +Passwort-Hashes: + - Wir speichern Passwörter nicht im Klartext, sondern als Hashes. + Es ist nicht möglich, vom Hash auf das Passwort zu schließen. + - Authentifizierung: Der Server kennt das Passwort nicht, sondern nur den Hash. + Der Benutzer gibt das Passwort ein. + Der Server berechnet den Hash und prüft, ob dieser stimmt. + Er merkt sich insbesondere nicht das Passwort. + - Salt: Wir speichern eine zusätzliche Zufallszahl auf dem Server + und bilden den Hash über das Passwort zusammen mit der Zufallszahl. + - Man sieht nicht sofort, wenn Passwörter (zufällig) gleich sind. + - Man kann keine Hashes "auf Vorrat" anlegen. + Beispiel: + - Alle Benutzer verwenden Passwörter von 8 Zeichen Länge. + Dies entspricht etwa 8 * 6 = 48 Bits Zufall. + - Ein Angreifer berechnet "auf Vorrat" die Hashes aller 48-Bit-Zahlen. + - Aus einer versehentlich öffentlich gewordenen Tabelle ("Leak") + lassen sich leicht die Passwörter rekonstruieren. + - Gegenmaßnahme: Salt, z.B. 8 Bit + --> Die "Vorratstabelle" müßte dann nicht mehr "nur" alle 48-Bit-Zahlen + erfassen, sondern alle 56-Bit-Zahlen. + --> Obwohl das Salt bekannt ist (Teil der Tabelle), + erhöht es die Sicherheit. + - Problem mit SHA-2 als Passwort-Hash: + SHA-2 läßt sich schnell berechnen, insbesondere mit GPUs. + --> Durchprobieren riesiger Mengen von Passwörtern geht relativ schnell, + auch mit Salt. + Trotzdem hilft auch hier das Salt, denn es verhindert, + daß man mehrere Passwörter gleichzeitig angreift. + - Für diesen Zweck geeignetere Hash-Funktionen: + bcrypt, scrypt, argon2, evtl. auch pbkdf2 + Nicht leicht auf GPUs berechenbar, insgesamt langsamer. + --> Möglicherweise daher für kryptographische Signaturen schlechter geeignet, + {weil dort größere Datenmengen ge"hash"t werden} + weil dort Hashes über größere Datenmengen gebildet werden. + --> Bei kryptographischen Signaturen besteht nicht das Problem + {mit der ge"leak"ten Nachricht nicht,} + mit der versehentlich öffentlich gewordenen Nachricht nicht, + denn die Nachricht ist ja absichtlich öffentlich + (Falls nicht, hilft Verschlüsselung.) + bzw. weil es ohnehin aussichtslos ist, anhand des kryptographischen Hashes + die Nachricht zu erraten. + +Fehlerkorrektur, 09.06.2022, 13:36:48 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Hamming-Code: siehe frühere Lehrveranstaltungen +https://www.cvh-server.de/~pgerwinski/net-2013ss.pdf, Seite 28 bis 30 + +Reed-Solomon-Code: Fehlererkennung und -Korrektur + + - Anwendung: QR-Codes, optische Datenträger, Raumsonde Voyager 2 + + - Prinzip: Funktionsgraphen von Polynomen + 3 Punkte --> eindeutig für Polynom 2. Grades + 4 Punkte --> Polynom 2. Grades überbestimmt + 4 Punkte, einer falsch --> Polynom 2. Grades paßt nicht --> Fehlererkennung + 5 Punkte --> auch Fehlerkorrektur möglich, notfalls durch Ausprobieren + (die richtige Parabel kommt häufiger vor) + --> "einstellbares" Fehlererkennungs- und korrekturverfahren: + nur 3 Punkte: keine Fehlererkennung und -korrektur, nur die Daten + 4 Punkte: Fehlererkennung + 5 Punkte: Fehlererkennung, 1 Fehler korrigieren + 6 Punkte: Fehlererkennung, 2 Fehler korrigieren + ... + + - In der Praxis: Polynom über endlichem Körper, z.B. mit 5 Elementen + Mathematik: Dies funktioniert alles genauso wie in den reellen Zahlen. + Z.B.: Man vereinbart vorher die Stützstellen (x-Koordinaten) + und übermittelt die Funktionswerte (y-Koordinaten). + + ( { 0, 1, 2, 3, 4 }, +, · ) (mod 5) + + Stützstellen: alle, also 0, 1, 2, 3, 4 + + Parabel übermitteln, z.B. x², + also a2 = 1, a1 = 0, a0 = 0, + also die Nachricht ( 1, 0, 0 ) + + Funktionswerte berechnen: + 0² = 0 + 1² = 1 + 2² = 4 + 3² = 4 + 4² = 1 + + Funktionswerte übermitteln, Übertragungsfehler + + x | y + ---+--- + 0 | 0 + 1 | 1 + 2 | 4 + 3 | 3 <-- falsch; richtig wäre: 4 + 4 | 1 + + Gleichungssystem aufstellen und lösen --> widersprüchliche Ergebnisse --> Fehler erkannt + + Gleichungssystem aufstellen und lösen für jeweils 3 der Punkte + + Parabel: a2 x² + a1 x + a0 + + 0: a2 · 0² + a1 · 0 + a0 = 0 \ + 1: a2 · 1² + a1 · 1 + a0 = 1 } a0 = 0, a2 + a1 = 1, a2 + a1 = 1 + 2: a2 · 2² + a1 · 2 + a0 = 4 / 4a2 + 2a1 = 4, 2a2 + a1 = 2 + 3: a2 · 3² + a1 · 3 + a0 = 3 ------------ + 4: a2 · 4² + a1 · 4 + a0 = 1 -a2 = -1, a2 = 1, a1 = 0 + --> Nachricht rekonstruiert + Auf diese Weise bekommen wir häufiger die richtige + Nachricht rekonstruiert als falsche, ... + + ... es sei denn, die Übertragung enthält mehr als 1 Fehler. + + In der Praxis häufig: Körper mit 8 oder 16 Elementen. + Diese sind NICHT { 0, 1, ..., 7 } bzw. { 0, 1, ..., 15 }. + + Zur Illustration: Körper mit 25 Elementen. + Dieser ist NICHT { 0, 1, ..., 24 }, sondern + { 0, 1, 2, 3, 4 } [ sqrt(2) ] ("adjungiert Wurzel 2"). Das bedeutet: + + - Wir haben die Zahlen { 0, 1, 2, 3, 4 } modulo 5 + und zusätzlich w = sqrt(2) mit der Eigenschaft w · w = 2 + und zusätzlich alle Kombinationen davon, also z.B. 3 + 2w. + + - Dann können wir multiplizieren, z.B.: + (3 + 2w) · (1 + w) + = 3 · 1 + 2w · 1 + 3w + 2w² + = 3 + 5w + 2 · 2 + = 3 + 0w + 4 + = 2 + + - Jede Zahl: x + y·w (ähnlich komplexen Zahlen: x + y·i) + + - Speicherung als 5-Bit-Zahl (0..31; wir verschwenden die Werte 25 bis 31) + oder als 6-Bit-Zahl mit jeweils 3 Bit für "normalen Teil" und "w-Teil" (schneller, + aber wir verschwenden mehr als 1 Bit, also mehr als die Hälfte aller Werte) + + Zur Illustration: Körper mit 4 Elementen: siehe + https://de.wikipedia.org/wiki/Endlicher_K%C3%B6rper#Der_K%C3%B6rper_mit_4_Elementen + + - In der Praxis in der Praxis: fertige Bibliothek diff --git a/20220609/polynome-20220609.xcf.gz b/20220609/polynome-20220609.xcf.gz new file mode 100644 index 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