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Comprehensive Review of Network Security: SSL, Public Key Methods, Discrete Log, RSA, Cryptographic Algorithms

This mid-term review covers various aspects of network security including SSL, shared key, public key methods like Diffie-Hellman and RSA, Discrete Log, RSA, and more. It delves into concepts like modulus, factoring, Chinese Remainder Theorem and offers insights on efficient exponentiation techniques, optimization strategies, and potential attacks. Additionally, it serves as a primer for broader topics within cryptography such as hashing, MAC, symmetric ciphers, IPSec and Kerberos systems.

kirestin-pollard
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Comprehensive Review of Network Security: SSL, Public Key Methods, Discrete Log, RSA, Cryptographic Algorithms

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  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. 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

  4. 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!)

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

  6. 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

  7. 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?

  8. 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

  9. 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

  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!

  11. 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

  12. Later in the course • Crypto • Hashing • MD5, SHA • MAC • Symmetric ciphers • DES, AES/Rijndael, RC-4 • Modes of operation (for block ciphers) • Systems • IPSec - security on another level • Kerberos - key distribution (symmetric crypto) • Firewalls, IDS, etc.

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