Network Security

1. Network Security Review

2. Secure channel • Communication security • Confidentiality • Message • Traffic • Authentication • Integrity • How to achieve? • Establish shared key • Encrypt • MAC • Left out: non-repudiation, etc.

3. Shared Key Establishment • “Trusted third party” • Kerberos • Tickets • Public key methods • SSL • IPSEC • “Out-of-band”

4. Public Key Crypto

5. Diffie-Hellman RSAN=pq; ed  1 (mod (N))Public:e,N;Private:d,N Encrypt M: CMemodN DecryptC:MCdmodN Sign M: SMdmod N VerifyS: Se M (modN) Public Key techniques p, g Alice Bob a b magamod p mbgbmod p ma mb mbamod p =gabmod p= mabmod p ? shared secret key! • Discrete log: • Given y,p,b • Find x: bxmod p = y • Factoring: • Given N=pq • Find p,q

6. Discrete log based schemes • DH (key establishment) • DSS/DSA (signatures) • El-Gamal (signatures, encryption) • Elliptic Curves Cryptography (ECC) • Why modulus (p) is so large? • Little-step/giant-step attack

7. Factoring based • RSA • Square Roots (=Factoring) • Rabin (Encryption, Signature) • Fiat-Shamir (ID scheme, Signature)

8. World mod N • How many objects?|Z*N|= (N); for all z Z*N, z (N) mod N=1 • If N=pq, then (N)= (p-1)(q-1)[If N=p, then (N)= p-1] • Blum integers: N=pq, pq3 (mod 4) • Thenx(p+1)/4mod p= y; y2x(p+1)/2x(p-1)/2 x±x mod p

9. Chinese Remainder Theorem (CRT) • Given y2=x mod p; z2=x mod q; N=pq;Find s: s2=x mod N • More generally:Given a,A, b,B;Find x: x=a mod A, x=b mod B • Let u, v be s.t. uA=1 mod B, vB=1 modAThen x=uAb+vBa[indeed: x mod A = uAb+vBa = vBa = a; x mod B = uAb+vBa = uAb = b] • How to find u,v?

10. Extended GCD • Euclid’s GCD algorithm(greatest common divisor):gcd(a,b) = gcd( b, a mod b) =…= gcd(a’,b’)=ca’=ib’+c, … , ax+by=c • If gcd(a,b)=1: ax=1 mod b

11. Summary • RSA • Given p,q; Can compute (N), for N=pq; • With Extended gcd, can compute e, d = 1/e mod (N); [ gcd(e, (N)) must be 1 ] • Rabin • Using Blum integers can compute SQRT mod p,q • Using CRT can combine them to SQRT mod N

12. Prime number generation • Why? • How? • Exhaustive search • Too long • Miller-Rabin • Little Fermat’s Theorem (again) • Prime Number Theorem

13. Efficiency for all • Exponentiation: Repetitive Squaring • bA mod N takes 1.5 lg A long multiplications • Cost of multiplication • quadratic in length • Optimization: mod N  mod p + mod q +CRT • Watch out!

14. Attacks on factoring • (N), N => factoring (quadratic equation) • Trick: • obtain x, s.t. x=0 mod p, x0 mod q • gcd(x, N)=p • SQRTmodN => Factoring • vy2mod N; zSQRTmodN(v) • If z  ±y, then x  y-z • Computing mod p + mod q + CRT • Random error mod p (or mod q) => factoring

15. Other Crypto Encryption Hashing MACs

16. Encryption • One time pad • Block cipher • DES • Feistel approach • AES/Rijndael • Modes of operation • EBC, CFB, CBC, etc. • Stream ciphers • RC-4 • Pseudo-random generators

17. Hashing • Hashing algorithms • MD-5 • SHA • Applications • Digital signatures • MAC

18. Systems Certificates SSL IPSEC Kerberos

19. Certificates • X-509 • CA’s • Trust infrastructure • Hierarchical • X.509 • Networks of Trust • PGP

20. SSL • TCP level secure channel • Establish Shared Secret • DH+Certificates [+signatures] • RSA+Certificates [+signatures] • Kerberos • Encrypt & MAC • Usually authenticates only server • Client authentication possible • Typical application: HTTPS

21. IPSEC • IP level secure channel • Similar tools to SSL • Some traffic confidentiality • Both ends authenticated • Tunneling • Typical application:VPN

22. Kerberos • Authentication Server • Ticket Granting Servers • Tickets • Realms

23. Other topics • Firewalls • Non-repudiation • SET

24. Final: Tuesday May 5 9-11am See you there! Best of Luck!!!