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Unformatted text preview: Modern Cryptography Lecture 12 Yongdae Kim 2 Admin Stuff Email Subject should have [5471] in front, e.g. “[5471] Project proposal” CC TA and PostDoc: [email protected], [email protected] 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 㲇 b (mod m ) iff m  ab iff a = b + mk for some k a 㲇 b (mod m ), c 㲇 d (mod m ) a+c 㲇 ( b+d ) (mod m ), ac 㲇 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 φ (n) =1 (mod n) Cor: if r ´ s mod φ (n) and (a, n)=1, then a r ´ a s (mod n) 5 Recap (cnt) Generator If ord n (a) = φ (n) then a is a generator of Z n *. a is a generator iff a φ (n)/p ≠ 1 mod n for all p  φ (n). Let a ∈ Z m * and ord(a) = h. Then ord(a k ) = h/gcd(h, k). RSA Encryption n = pq , φ ( n ) = ( p1)( q1), gcd( φ ( n ), e ) = 1, ed ≡ 1 mod φ ( n ) A’s public key is (n, e ); A’s 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 Pollard’s rho: O(sqrt(q)) With Index calculus: Lp[1/3, c] ElGamal: prime...
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 Spring '08
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 Cryptography, TA, Publickey cryptography

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