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Unformatted text preview: Modern Cryptography Lecture 11 Yongdae Kim 2 Admin Stuff Email Subject should have [5471] in front, e.g. [5471] Project proposal CC TA and PostDoc: hkang@cs.umn.edu, aaram@cs.umn.edu Office hours Me: T 1:00 ~ 2:00 , Th 10:00 ~ 11:00 (and by appointment) TA: M 1:15 PM ~ 2:15 PM Work on projects Final Report due: May 6 (Firm Deadline: NO EXTENSION ) 6th assignment is due: 4/20 9:00 AM. Study Guide Come and talk to me and TA as much as possible. (Google chat is good!) Check Calendar 3 Recap Math Proof techniques Direct/Indirect proof, Proof by contradiction, Proof by cases, Existential/Universal Proof, Forward/backward reasoning Divisibility: a divides b (ab) if c such that b = ac GCD, LCM, relatively prime, existence of GCD Eucledean Algorithm d = gcd (a, b) x, y such that d = a x + b y. gcd(a, b) = gcd(a, b + ka) Modular Arithmetic a b (mod m ) iff m  ab iff a = b + mk for some k a b (mod m ), c d (mod m ) a+c ( b+d ) (mod m ), ac bd (mod m ) gcd(a, n) =1 a has an arithmetic inverse modulo n. Counting, probability, cardinality, Security Symmetric Key vs. Public Key, Hash function, MAC, Digital signature, Key management through SKE and PKE, certificate 4 Recap (cnt) Block Cipher Modes of operation and their properties: ECB, OFB, CFB, CBC, CTR Meetinthemiddle attack and the Double (triple) DES Feistal Cipher and DES Hash function and MAC Probability and Birthday paradox MerkleDamgard Construction, MD4: design and break MAC Advanced number theory CRT Euler theorem: If a Z n * , then a (n) =1 (mod n) Cor: if r s mod (n) and (a, n)=1, then a r a s (mod n) 5 Recap (cnt) Generator If ord n (a) = (n) then a is a generator of Z n *. a is a generator iff a (n)/p 1 mod n for all p  (n). Let a Z m * and ord(a) = h. Then ord(a k ) = h/gcd(h, k). RSA Encryption n = pq , ( n ) = ( p1)( q1), gcd( ( n ), e ) = 1, ed 1 mod ( n ) As public key is (n, e ); As private key is d Encryption: compute c = m e mod n Decryption: m = c d mod n Group Theory (G, ) is a group if it satisfies closedness, associativity, and has identity and every element has an inverse. (G, ) is cyclic if group generator. DLP: Given p , a generator g of Z p * , and an element y Z p * , find the integer x such that g x = y mod p . 6 Recap (cnt) DiffieHellman: Z p * = {1, 2, , p 1}, g generator A B : N A = g n1 mod p, B A : N B = g n2 mod p A : N B n1 = g n1n2 mod p, B : N A n2 = g n1n2 mod p Efficient and Secure Construction Z p * = {1, 2, , p 1}, g generator p = k q + 1 (p= 1024, q= 160), g = g k , ord p (g) = q G = <g> With Pollards rho: O(sqrt(q)) With Index calculus: Lp[1/3, c] ElGamal: prime...
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 Spring '08
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