Mid-term Review

Network Security

Secure channel

SSL (and many others:incl. IPSEC)

Shared key establishingTrusted party (Kerberos, etc. - to be covered)

Public key methods

Public Key techniques

Diffie-Hellman RSAN=pq; ed 1 (mod (N))Public:e,N;Private:d,N

Encrypt M: CMemodN

Decrypt C: MCdmodN

Sign M: S Mdmod N

Verify S: Se M (modN)

Alice Boba b

p, g

magamod p mbgbmod p

ma mb

mbamod p ma

bmod p=gabmod p=

shared secret key!•Discrete log:

•Given y,p,b

•Find x: bxmod p = y

?

•Factoring:

•Given N=pq

•Find p,q

Discrete log based schemes

DH, DSS (El-Gamal); Elliptic Curves Cryptography (ECC)

Why modulus (p) is so large?Have not seen attacks on it (yet!)

Factoring based

RSA

Square Roots (=factoring)Rabin (Encryption,Signature)

Fiat-Shamir (ID scheme, Signature)

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, pq3 (mod 4)

Thenx(p+1)/4 mod p= y; y2x(p+1)/2x(p-1)/2 x±x mod p

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?

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

Summary

RSAGiven 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 ]

RabinUsing Blum integers can compute SQRT mod p,q

Using CRT can combine them to SQRT mod N

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!

Attacks on factoring

(N), N => factoring (quadratic equation)

Trick: obtain x, s.t. x=0 mod p, x0 mod q

gcd(x, N)=p

SQRTmodN => Factoring

vy2mod N; zSQRTmodN(v)

If z ±y, then x y-z

Computing mod p + mod q + CRTRandom error mod p (or mod q) => factoring

Later in the courseCrypto

Hashing• MD5, SHA• MAC

Symmetric ciphers• DES, AES/Rijndael, RC-4• Modes of operation (for block ciphers)

SystemsIPSec - security on another levelKerberos - key distribution (symmetric crypto)Firewalls, IDS, etc.