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Unformatted text preview: Modern Cryptography Lecture 2 Yongdae Kim 2 Admin Stuff E-mail Subject should have  in front, e.g. “ Project proposal” CC TA and PostDoc: [email protected], [email protected] 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 1st assignment on-line (due: 02/02 9:00 AM) 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 d = gcd(a,b) d > 0 d | a and d | b e | a and e | b implies e | d lcm, relatively prime, existence of GCD Eucledean Algorithm d = gcd (a, b) ⇒ ∃ x, y such that d = a x + b y. 4 Few more useful stuff Let d = gcd (a, b) gcd (a/d, b/d) = ? a | bc and d = 1 ⇒ ? a | bc ⇒ (a/d) | c gcd (ma, mb) = md if m > 0 gcd (n, n+1) ? gcd (a, b) = gcd (a + kb, b) ? 5 Prime p ≥ 2 is prime if a | p ⇒ a = ± 1 or ± p Hereafter, p is prime [Euclid] p | ab ⇒ p | a or p | b [Euclid] There are infinite number of primes. Prime number theorem: let π (x) denote the number of prime numbers ≤ x, then lim x → Y ∞ π (x)/(x/ln x) = 1 Euler phi function : For n ≥ 1, let φ (n) denote the number of integers in [1, n] which are relatively prime to n. if p is a prime then φ (p)=p-1 if p is a prime, then φ (p r ) = p r-1 (p-1). φ is multiplicative. That is if gcd(m,n)=1 then φ (m*n) = φ (n) * φ (m) 6 Fundamental theorem of arithmetic Every positive integer greater than 1 can be uniquely written as a prime or as the product of two or more primes where the prime factors are written in order of non-decreasing size Examples 100 = 2 * 2 * 5 * 5 182 = 2 * 7 * 13 29820 = 2 * 2 * 3 * 5 * 7 * 71 7 Pairwise relative prime A set of integers a 1 , a 2 , … a n are pairwise relatively prime if, for all pairs of numbers, they are relatively prime Formally: The integers a 1 , a 2 , … a n are pairwise relatively prime if gcd( a i , a j ) = 1 whenever 1 ≤ i < j ≤ n . Example: are 10, 17, and 21 pairwise relatively prime? gcd(10,17) = 1, gcd (17, 21) = 1, and gcd (21, 10) = 1 Thus, they are pairwise relatively prime Example: are 10, 19, and 24 pairwise relatively prime? Since gcd(10,24) ≠ 1, they are not 8 Modular arithmetic If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a-b Notation: a 㲇 b (mod m ) Rephrased: m | a-b Rephrased: a mod m = b If they are not congruent: a 㲇 b (mod m ) Example: Is 17 congruent to 5 modulo 6?...
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- Spring '08