Mid-term Review

1
Mid-term Review

• Network Security

2
Secure channel

• SSL (and many others incl. IPSEC)
• Shared key establishing
• Trusted party (Kerberos, etc. - to be covered)
• Public key methods

3
Public Key techniques

• Diffie-Hellman

• RSA
• Npq ed ? 1 (mod ?(N))
• Public e,N Private d,N
• Encrypt(m) c?memodN
• Decrypt(c) m?cdmodN
• Sign(m) s?md modN
• Verify(s,m) se?m (modN)

p, g
Alice
Bob
a
b
ma?gamod p
mb?gbmod p
ma
mb
mbamod p
mabmod p
gabmod p
?
shared secret key!

• Discrete log
• Given y,p,b
• Find x bxmod p y

• Factoring
• Given Npq
• Find p,q

4
Discrete log based schemes

• DH, DSS (El-Gamal)
• Elliptic Curves Cryptography (ECC)
• Why modulus (p) is so large?
• Big-step/Little-step attack
• Pohlig-Hellman attack
• Beware of primes p with only small factors f(p)
• Safe primes p2q1 for some prime q

5
Factoring based

• RSA
• Square Roots (factoring)
• Rabin (Encryption,Signature)
• Fiat-Shamir (ID scheme, Signature)

6
World mod N

• How many objects?ZN ?(N) for all z? ZN,
  z ?(N) mod N1
• If Npq, then ?(N) (p-1)(q-1)If Np, then
  ?(N) p-1
• Blum integers Npq, p?q?3 (mod 4)
• Thenx(p1)/4 mod p y y2?x(p1)/2?x(p-1)/2 x?x
  mod p

7
Chinese Remainder Theorem (CRT)

• Given y2x mod p z2x mod q NpqFind s s2x
  mod N
• More generallyGiven a,A, b,BFind x xa mod
  A, xb mod B
• Let u, v be s.t. uA1 mod B, vB1 modAThen
  xuAbvBaindeed x mod A uAbvBa vBa a
  x mod B uAbvBa uAb b
• How to find u,v?

8
Extended GCD Inverses

• Euclids GCD algorithm(greatest common
  divisor)gcd(a,b) gcd(b, a mod b) c
• Extended GCD gives in addition x,y
• axbyc
• If gcd(a,b)1 ax (mod b) 1
• i.e., xa 1 in Zb

9
Summary RSA Rabin

• RSA
• Given p,q Can compute ?(N), for Npq
• 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

10
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!

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Attacks on factoring

• ?(N), N gt factoring (quadratic equation)
• Trick
• obtain x, s.t. x0 mod p, x?0 mod q
• gcd(x, N)p
• SQRTmodN gt 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) gt factoring

12
Key Establishing

• Diffie-Hellman or RSA
• Watch out for man-in-the-middle attack!!!
• Authentication (signatures)
• PKI
• Remember AKE authenticated key establishment
• Beyond AKE
• Ciphers
• MACs

13
Ciphers

• Block ciphers
• DES, AES, 3DES,
• Modes of operation EDE, OFB, CBC,
• Stream ciphers
• Pseudo-random pad

14
Later in the course

• Crypto
• Hashing
• MD5, SHA
• MAC
• Systems
• PKI
• Kerberos - key distribution (symmetric crypto)
• IPSec - security on another level
• Firewalls, IDS, etc.