1. Kriptografija sa asimetričnim ključem
Jelena Ignjatović
Symmetric Key Cryptography – Block Ciphers
• Common block and key sizes • Electronic code book mode (ECB) • Cipher block chaining mode (CBC)
Block Ciphers – Present and Future Standards
• Some popular block ciphers (DES, 3DES, IDEA, CAST, Blowfish, RC2, RC5) • Digital Encryption Standard (DES) – rounds of confusion and diffusion • Feistel networks • Triple DES • Advanced Encryption Standard (AES) • AES finalists (MARS, RC6, Twofish, Serpent, Rijndael)
Symmetric Key Cryptography – Stream Ciphers
• Linear feedback shift registers (LFSRs) • RC4 • Output feedback mode (OFB) – block ciphers used as stream ciphers • Counter mode (CTR) – block ciphers used as streams ciphers
Coverage by the textbook "Network Security"
• Chapter 3 – Secret Key Cryptography, pp. 59…94 • Chapter 4 – Modes of Operation, pp. 95…115

2. Asimetrični-ključ (javni ključ) šifrovanja
Osnovna ideja:
Korisnik ima dva ključa: javni ključ i privatni ključ.
Poruka može da bude šifrovana javnim ključem i dešifrovana privatnim ključem koji bi obezbedio sigurnost.
Poruka može da bude šifrovana privatnim ključem i dešifrovana javnim ključem koji bi obezbedio autentičnost.

3. Problem distribucije ključeva u gustim mrežama
U gusto-razgranatim mrežama u kojima dosta članova komunicira među sobom, zahteva se određeni broj tajnih ključeva u šifrovanju Kompleksnost algoritama raste kvadratno sa porastom broja učesnika obzirom da u potpuno-zatvorenoj, razgranatoj mreži svakom od n komunikacinih partnera mora da bude sigurno isporučeno (n-1) ključeva.

4. • Uzmimo primer široke komunikacione mreže sa 100 potpuno-razgranatih čvorova gde se ključ svake sesije menja svakog sata što rezultuje zahtevom da se distribuira oko ključeva svaki dan.
• Kao što se može videti, distribucija tajnih ključeva se slabo menja sa porastom broja učesnika. Zato su dugo vremena ljudi tražili neku altrnativu za uspostavljanje sigurnih veza. Efikasno rešenje je najzad pronađeno 1976 sa novim konceptom kriptosistema sa javnim ključem.

5. Problem sigurne distribucije ključeva
KAB, KAC, KAD, KAE, KAF
KAC, KBC, KCD,
KCE, KCF
KAB, KBC, KBD,
KBE, KBF
KAF, KBF, KCF,
KDF, KEF
KAE, KBE, KCE,
KDE, KEF
KAD, KBD, KCD, KDE, KDF
Alice
Bob
Carol
Dave
Edna
Fred
Key Distribution Problem in Dense Networks
• In densely-meshed networks where many parties communicate with each other, the required number of secret keys necessary when using symmetric encryption algorithms increases quadratically with the number of participants since in a fully-meshed network to each of the n communication partners (n-1) keys must be securely delivered.
• Take as an example a broadband communications network with 100 fully-meshed nodes were each session key is changed every hour, resulting in a requirement to safely distribute about 240‘000 keys each day.
• As can easily be seen, secret key distribution scales very badly with an increasing number of participants. Therefore for a long time people had been looking for alternative ways of establishing secure connections. A very efficient solution was finally found in 1976 with the novel concept of a Public Key Cryptosystem.
Sigurna distribucija n2 ključeva
Ključevi se moraju razmenjivati na siguran način

6. Sistem distribucije javnog ključaKA KB KC KD KE
Alice
Bob
Carol
Dave
Edna
Fred
javna distribucija n ključeva
Ključ se može poslati putem interneta
Javni
direktorijum
Alice : KA Bob : KB Carol : KC Dave : KD Edna : KE Fred : KF
Public Key Distribution System
• In a Public Key Cryptosystem each user or host possesses a single key pair consisting of a private key which is kept secret by the no and a matching public key which is published in a public directory (usually an LDAP or WWW server).
• If a user Alice wants to send an encrypted message to user Bob then Alice encrypts her message with Bob‘s public key KB fetched from the public directory and sends it to Bob. Since Bob is the only one in possession of the matching private key, he alone can decrypt the encrypted message sent to him.
• Since only the public key of the recipient is required, with n users only n distinct keys are required. Under the assumption that each user generates her own public/ private key pair locally, no secure channels are required for the distribution of the public keys, since the don‘t contain any secret and must be put into the public domain anyway.

18. Kriptografija sa javnim ključem
pronalazači
Whitfield Diffie i Martin Hellman 1976
Ralph Merkle 1978
Alice
Bob
PUBKB PRIVKB
C = fPUBKB(P)
Šifrovanje jednosmernom funkcijom
P = f-1PRIVKB(C)
Joe
P = f-1PUBKB(C)
Računanje inverzne funkcije je ekstremno skupo
Inventors of Public Key Cryptography
• The concept of a Public Key Cryptosystem was invented at around the same time by Whitfield Diffie, Martin Hellman and Ralph Merkle. Whereas the first two researchers published their invention in 1976 and got all the fame, Ralph Merkle had the misfortune that the printing of his paper got delayed by more than a year so that it got published not until Today it is generally recognized that all three scientists are the fathers of public key cryptography.
• Recently it became known that already in 1970, James Ellis, at the time working for the British government as a member of the Communications-Electronics Security Group (CESG), formulated the idea of a Public Key Cryptosystem. Several practical algorithms including one variant very similar to RSA and another one identical to the Diffie-Hellman key exchange were discovered within the CESG. Unfortunately the British researchers were not allowed to publish their results due to state security reasons.
Basic Principles of Public Key Cryptography
• All public key cryptosystems are based on the notion of a one-way function, which, depending on the public key, converts plaintext into ciphertext using a relatively small amount of computing power but whose inverse function is extremely expensive to compute, so that an attacker is not able to derive the original plaintext from the transmitted ciphertext within a reasonable time frame.
• Another notion used in public key cryptosystems is that of a trap door which each one-way function possesses and which can only be activated by the legitimate user holding the private key. Using the trapdoor, decryption of the ciphertext becomes easy.
• Many public key cryptosystems are based on known hard problems like the factoring of large numbers into their prime factors (RSA) or taking discrete logarithms over a finite field (Diffie-Hellman).
Jednosmerne funkcije se obično zasnivaju na dobro poznatim teškim matematičkim problemima
Faktorizacije
Diskretni logaritamski problem

19. RSA kriptosistem sa javnim ključem
Razvili su ga 1978 Rivest, Shamir and Adleman (RSA)
Ovo je najpopularniji kriptosistem sa javnim ključem
Zasniva se na matematičkom problemu „faktorizacije celih brojeva“ 143 = 11*13

20. RSA kriptosistem sa javnim ključem Algoritam za generisanje ključa
Korak 1: Slučajno biramo dva velika prosta broja p i q
Za maksimalnu sigurnost, biramo p i q približno jednake dužine, od bitova svaki.
Korak 2: Računamo proizvod n = p·q
Korak 3: Slučajno biramo broj e < (p-1)(q-1)
Brojevi e i (p-1)(q-1) moraju biti uzajamno prosti, tj. Ne smeju da imaju zajedničke proste faktore.
Korak 4: Računamo jedinstveni inverz d = e-1 mod (p-1)(q-1)
Jednakost d·e mod (p-1)(q-1) = 1 može se rešiti korišćenjem Euklidovog algoritma.

21. RSA kriptosistem Primer generisanja ključa
p = 3, q = 11: n = p·q = 33
(p-1)·(q-1) = 2 · 10 = 2 · 2 · 5 = 20
Javni eksponent e mora biti uzajamno prost sa (p-1)·(q-1) , tj. on ne može imati 2 i 5 kao faktore.
e d e·d e·d mod 20
Svi mogući izbori
eksponenata e i d

22. RSA kriptosistem Javni i privatni ključevi
Javni ključ: moduo n i javni eksponent e
n i e se objavljuju u javnom direktorijumu,
Privatni ključ: moduo n i privatni eksponent d
Privatni eksponent d je tvoj tajni ključ. On može biti zaštićen ili skladištenjem na čip kartici ili na disku uz šifrovanje simetričnom šifrom po tvom izboru.
Veliki prosti brojevi p i q koji su korišćeni za generisanje ključa više nisu potrebni i mogu biti izbrisani.

23. RSA kriptosistem Šifrovanje i dešifrovanje
Šifrovanje bloka x otvorenog teksta: y = xe mod n
Pošiljalac koristi javni ključ primaoca da šifruje x < n.
Dešifrovanje bloka y šifrata: x = yd mod n
Primalac koristi privatni ključ da otkrije
blok x otvorenog teksta.
Bez dokaza:
yd = (xe)d = xe·d = xm·(p-1)·(q-1) + 1 = x1 = x (mod n)
Šifrovanje i dešifrovanje su simetrične operacije
Redosled stepenovanja javnim eksponentom e i privatnim stepenom d može biti promenjen.

24. Generisanje ključeva

28. RSA kriptosistem Primer šifrovanja / dešifrovanja
Šifrovanje javnim ključem n = 33, e = 3
Binarni otvoreni tekst
Groupe od 5 Bitova
Decimalni plaintext
y = x
y = x3 mod
Dešifrovanje privatnim ključem n = 33, d = 7
Decimalni hipertext
x = y
x = y7 mod

29. RSA–576 izazova
Pokušaj (napor)
576 bitni broj (174 decimalne cifre)
Faktorisan 3. decembra posle 3 meseca rešavanja.
Linux Cluster: 144 PCs with 400 MHz Pentium II procesori.
Urađeno na Universitetu u Bonu, postprerađeno uz podršku BSIa.
= ? = *

30. Šifrovanje RSA javnim ključem
Cena: $200,000
Status: Nije faktorisan
Decimalne cifre : 617
Suma decimalnih cifara: 2738
Broj izazova
Cena ($US)
Status
Datum prijave
Podnosilac
RSA-576
$10,000
faktorisano
December 3, 2003
J. Franke et al.
RSA-640
$20,000
Nije faktorisano
RSA-704
$30,000
RSA-768
$50,000
RSA-896
$75,000
RSA-1024
$100,000
RSA-1536
$150,000
RSA-2048
$200,000
Not Factored