lec8-4 - Admin Stuff E-mail Modern Cryptography Lecture 8...

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Modern Cryptography Lecture 8 Yongdae Kim 2 Admin Stuff E-mail 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 1:30 ~ 2:30, Th 10:00 ~ 11:00 (and by appointment) TA: M 1:15 PM ~ 2:15 PM Work on projects Interim Report due: Mar 30 (Firm Deadline: Read instruction) 4th assignment is due: 3/23 9:00 AM. 5 th assignment will be posted this week (programming). 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 (a|b) 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 | a-b iff a = b + mk for some k (mod ), c d (mod ) a+c ( b+d ) (mod ), ac bd (mod ) 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 Meet-in-the-middle attack and the Double (triple) DES Feistal Cipher and DES Hash function and MAC Probability and Birthday paradox Merkle-Damgard 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)
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5 Generator Let a Z n * . The order of a (ord n (a)) is the least positive t s.t. a t 1 (mod n) if t = φ (n) then a is said to be a generator of Z n * ord n (a) must divide φ ( n) If not, φ ( n) = ord n (a) k + r (ord n (a)>r>0). Then a r 1 (mod n) (*) If a v = 1 mod n, then ord n a | v. 6 Generator (cnt.) a is a generator iff a φ (n)/p 1 mod n for each prime divisor p of φ (n) Proof) ) Obvious, since a is a generator. ) Proof by contrapositive Suppose a is not a generator Let ord n (a) = k < φ (n). Then, k | φ (n). Since k is a proper-divisor of φ (n), k has to divide φ (n)/p for some p | φ (n). k q = φ (n)/p. a φ (n)/p = (a k ) q = 1 q = 1 mod n. 7 Generator (examples) Example: Z 7 *= {1,2,3,4,5,6}, φ (7) = 6 = 2 * 3 ord 7 (1) = 1 because 1 1 = 1 is not generator since 1 2 mod 7 1 ord 7 (2) = 3 because 2 3 = 1 is not generator since 2 2 mod 7 ! 1, but 2 3 mod 7 1 ord 7 (3) = 6 because 3 6 = 1 (3, 2, 6, 4, 5, 1) is a generator since 3 2 mod 7 ! 1, but 3 3 mod 7 ! 1 ord 7 (4) = 3 because 4 3 = 1 is not generator since 4 2 mod 7 ! 1, but 4 3 mod 7 1 ord 7 (5) = 6 because 5 6 = 1 is a generator since 5 2 mod 7 ! 1, but 5 3 mod 7 ! 1 ord 7 (6) = 2 because 6 2 = 1 is not generator since 6 2 mod 7 1, but 6 3 mod 7 ! 1 8 Generator (example) Find all generators of Z 17 *.
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lec8-4 - Admin Stuff E-mail Modern Cryptography Lecture 8...

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