From 22712060f7191880067f0d56cafd6855fad8daff Mon Sep 17 00:00:00 2001 From: fram3d Date: Tue, 16 May 2023 13:33:10 +0200 Subject: [PATCH] add rsa slides --- slides/rsa/dhke.png | Bin 0 -> 80915 bytes slides/rsa/dhke.xcf | Bin 0 -> 150311 bytes slides/rsa/github-verified.png | Bin 0 -> 41968 bytes slides/rsa/rsa.md | 241 ++++++++++++++++++++++++++++++ slides/rsa/tor-onion-services.png | Bin 0 -> 13917 bytes slides/rsa/tor-onion-services.xcf | Bin 0 -> 72338 bytes slides/rsa/tor.png | Bin 0 -> 20546 bytes slides/rsa/tor.xcf | Bin 0 -> 50342 bytes 8 files changed, 241 insertions(+) create mode 100644 slides/rsa/dhke.png create mode 100644 slides/rsa/dhke.xcf create mode 100644 slides/rsa/github-verified.png create mode 100644 slides/rsa/rsa.md create mode 100644 slides/rsa/tor-onion-services.png create mode 100644 slides/rsa/tor-onion-services.xcf create mode 100644 slides/rsa/tor.png create mode 100644 slides/rsa/tor.xcf diff --git a/slides/rsa/dhke.png b/slides/rsa/dhke.png new file 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z?-LYfOY*dJ|nt_{$x=W^;%Ya6UlbpKW{ak^F$vb1-B?!@ +![Diffie–Hellman](dhke.png) + + +# RSA +## Mala Fermaova teorema +Ako je $p$ prost broj, za svako $a$ vazi: + + +$a^{p-1} \equiv 1 \mod p$ + +\pause + +## Posledica +Ako su $p$ i $q$ prosti brojevi, za svako $a$ vazi: + + +$a^{(p-1)(q-1)}$\pause$\equiv ({a^{p-1}})^{q-1}$\pause$\equiv 1 \mod q$\pause + + +$a^{(p-1)(q-1)}$\pause$\equiv ({a^{q-1}})^{p-1}$\pause$\equiv 1 \mod p$\pause + + +$(a^{(p-1)(q-1)}-1)$ je deljivo i sa $p$ i $q$.\pause + + +$p$ i $q$ su prosti, pa mora da je deljivo i sa $p \cdot q$. + + +# RSA + +## Primecujemo +$a^{(p-1)(q-1)} \equiv 1 \mod pq$\pause + + +Takodje: +$a^{x(p-1)(q-1)}$\pause$\equiv ({a^x})^{(p-1)(q-1)}$\pause$\equiv 1 \mod pq$\pause + + +$a^{x(p-1)(q-1)+1} \equiv a \mod pq$\pause + + +## Trazimo +$e$ i $d$ tako da: + + +$({a^e})^d \equiv a^{ed} \equiv a^{x(p-1)(q-1)+1} \mod pq$\pause + + +Odnosno: +$ed \equiv 1 \mod (p-1)(q-1)$ \pause + + +$d$ je modularni inverz od $e$ pod modulom $(p-1)(q-1)$\pause + + +Mozemo koristiti `Produzeni Euklidov algoritam`.\pause + + +U buduce cemo oznacavati $n=pq$, a $\varphi(n)=(p-1)(q-1)$\pause + + +$a^{\varphi(n)} \equiv 1 \mod n$\pause + + +$a^{ed} \equiv a^{x\varphi(n)+1}$\pause$\equiv a \mod n$ + + +# RSA + +- Problem faktorisanja $n=pq$\pause +- $\varphi(n)=(p-1)(q-1)$ nije poznato bez $p$ i $q$\pause +- $d$ kao modularni inverz od $e$ nije poznat bez $\varphi(n)$\pause +- $d$ mozemo da cuvamo tajnim cak i ako objavimo $e$ i $n$ javno\pause + + +# RSA +- Generisanje kljuceva + - Nadjimo velike proste brojeve $p$ i $q$\pause + + + Testovi prostosti brojeva (Fermaov test)\pause + - Generisemo $n=pq$\pause + - Nadjimo $e$ koji je uzajamno prost sa $(p-1)(q-1)$\pause + - Nadjimo $d$ koriscenjem Produzenog Euklidovog algoritma\pause + - Zaboravimo $p$ i $q$, jer nam vise ne trebaju\pause +- Javni kljuc se sastoji od brojeva $e$ i $n$ + + + $m^e \equiv C \mod n$ + +\pause + +- Privatni kljuc se sastoji od brojeva $d$ i $n$ + + + $C^d \equiv m \mod n$ + +\pause + +- Digitalni potpis se postize sifrovanjem sa privatim kljucem + + + $m^d \equiv S \mod n$\pause + +- Provera digitalnog potpisa: + $S^e \equiv m \mod n$ + +# Prodruzeni Euklidov algoritam + +``` +def egcd(a, b): + if a == 0: + return (b, 0, 1) + g, y, x = egcd(b%a,a) + return (g, x - (b//a) * y, y) + +def modinv(a, m): + g, x, y = egcd(a, m) + if g != 1: + raise Exception('No modular inverse') + return x%m +``` + +# Napadi na RSA + +- Napadi na RSA + - Pogadjanje poruke, potrebno dopunjavanje poruke random podacima (padding)\pause + - Premali eksponent $e$, korenovanje sifrovanog teksta za male poruke (veliko $e$)\pause + - Koriscenje istog eksponenta za vise kljuceva, napad koriscenjem Kineske teoreme o ostatku (random izabrano $e$)\pause + - Desifrovanje sumnjivog teksta, $(x^e \cdot C)^d \equiv (x^e)^d \cdot C^d \equiv x \cdot m \mod n$ + +# Primena + +## GNU Privacy Guard +- 1999\. Werner Koch\pause +- Generisanje kljuca: `gpg --gen-key`\pause +- Lista javnih kljuceva: `gpg --list-keys`\pause +- Export privatnih kljuceva: `gpg --export-secret-keys --output backup.gpg`\pause +- Upload kljuceva: `gpg --send-key [KEYID]`\pause +- Sifrovanje poruke: `gpg -e file.txt`\pause +- Desifrovanje: `gpg -d file.txt`\pause +- Potpisivanje poruke ili fajla: `gpg -s file.exe`\pause +- Potpisivanje kljuca: `gpg --sign-key [KEYID]`\pause +- ASCII output: `gpg --armor -se file.txt`\pause +- GPG password manager: `gpg --armor -c passwords.txt` + +# Primena + +## Git +- Podesavanje kljuca: `git config --global user.signingkey [KEYID]`\pause +- Potpisivanje komita: `git commit -S`\pause +
    +![Github signed commits](github-verified.png) +
    + +# Primena + +## SSH +- Generisanje kljuca: `ssh-keygen [-f filename]`\pause +- Dodavanje kljuca na remote masinu: `ssh-copy-id [-i filename] user@hostname`\pause +- `~/.ssh/authorized_keys` + +# The Onion Router + +## Tor +- 1990\.-te United States Naval Research Laboratory (Paul Syverson,Michael G. Reed,David Goldschlag)\pause +- 20.9.2002. prva verzija Tor-a (javni projekat, anonimnosti u masi) + + +# The Onion Router + +
    +![How Tor works](tor.png) +
    + +# Onion hidden services + +
    +![How hidden services works](tor-onion-services.png) +
    + +# The Onion Router + +- Napadi na Tor + - Tor ne stiti od vremenske korelacije (pristup sa obe strane veze)\pause + - Slabosti u aplikacijama koje koriste Tor\pause + - Pogresno konfigurisane aplikacije\pause + - DNS Leak + +# Hvala + +Hvala na paznji! diff --git a/slides/rsa/tor-onion-services.png b/slides/rsa/tor-onion-services.png new file mode 100644 index 0000000000000000000000000000000000000000..c6b756ee35bf7451210e2a354b15780250de0299 GIT binary patch literal 13917 zcmajGby!qi^e;R#(m6Cp*N~D@(kabwC>gq?JEVs01_^0_0ftV6A*4Y-T2vZDKuS~u z?tFjuy??y_-1|J|JkMTxtzDnJ*4}HMvv#7st~xORJpl*=BG%MUF$94y0T2kI3Lgud zvkqz?M-vcz9b;9*@cQf4>5scNmA#7>=aXw|s|ktm|Nh+`9PGDGZmFxQ1qBCr#+8Sw z6}WXe5m#W9&KC?14@pW&US3`V{R>>2s{8$Wr=qfAXyr#=UoR#mW_elg#FvY{`2%Fr ztIWEgwIjS=!GY2F&GSbyx3`zsAj~Ygx|S~(QQeua*DOux;36mI-tPR-;j*2JZF6?! zFPr^F+EgwsPVt6Z=~H)08$0jE7xD}8o;J$)m>8EKI$PU278VwV>*s#`K7Hy)88+dg zE&ZzJOXr%kb{R~yLEiNbFR*v4D(?uMkNht&-&RoHj{mkTIS{b!J7w?GfTgf 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