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Mid-term Review

Mid-term Review. Network Security. Secure channel. SSL (and many others:incl. IPSEC) Shared key establishing Trusted party (Kerberos, etc. - to be covered) Public key methods. Diffie-Hellman. RSA N=pq; ed  1 (mod (N)) Public: e,N; Private: d,N Encrypt M: CM e mod N

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Mid-term Review

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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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