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1 Fingerprint

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2 Verifying set equality Verifying set equality v String Matching – Rabin-Karp Algorithm

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3 Verifying set equality Verifying set equality

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7 Fingerprinting Fingerprinting

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9 Fingerprinting Computation Fingerprinting Computation

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10 Fingerprinting Computation Fingerprinting Computation Horner’s Rule

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11 Protocol Protocol

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12 Prime Number q Prime Number q

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13 False Positive False Positive

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14 Prime Divisors Prime Divisors

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15 Density of Primes Density of Primes

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16 Density of Primes Density of Primes v (x) = número de primos menores ou iguais a x – (13) = 6 –Primos < = do que 13 = 2, 3, 5, 7, 11 e 13 v O valor de não muda até chegarmos ao próximo primo. – (13) = (14) = (15) = (16) –Ou seja, aumenta em salto de 1, mas o intervalo entre esses saltos é irregular

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17 Density of Primes Density of Primes Esses intervalos tornam-se cada vez maiores, isto é, a chance de um inteiro escolhido ao acaso ser primo diminui quando avançamos para os números maiores. PERGUNTA: O valor de não poderia ser aproximado por alguma função conhecida?

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18 Density of Primes Density of Primes Para um valor elevado de x, (x) ~ x/ ln x. Ou seja, lim (x) = 1 x x/ln x

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19 Sample Space Sample Space

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20 Probability of a bad prime Probability of a bad prime

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21 Final Protocol Properties Final Protocol Properties

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22 String Matching String Matching

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23 String Matching String Matching Many applications –While using editor/word processor/browser –Login name & password checking –Virus detection –Header analysis in data communications –DNA sequence analysis

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24 Naïve O(nm) algorithm Naïve O(nm) algorithm

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25 Rabin-Karp Algorithm Rabin-Karp Algorithm

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26 Fingerprinting Fingerprinting

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27 Fingerprinting function Fingerprinting function

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28 Fingerprinting computation Fingerprinting computation The only expensive operation

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29 False Positives? False Positives?

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30 Sample Space Sample Space

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31 False Positives False Positives

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32 Fingerprinting Fingerprinting

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33 Primality testing v A natural number n is prime iff the only natural numbers dividing n are 1 and n

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34 Primality testing v A natural number n is prime iff the only natural numbers dividing n are 1 and n v The following are prime: 2, 3, 5, 7, 11, 13,

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35 Primality testing v A natural number n is prime iff the only natural numbers dividing n are 1 and n v The following are prime: 2, 3, 5, 7, 11, 13, …and so are 1299709, 15485863, 22801763489, …

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36 Primality testing v A natural number n is prime iff the only natural numbers dividing n are 1 and n v The following are prime: 2, 3, 5, 7, 11, 13, …and so are 1299709, 15485863, 22801763489, … There is an infinite number of prime numbers

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37 Primality testing There is an infinite number of prime numbers Proof: Let us suppose the number of primes is Finite.

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38 Primality testing There is an infinite number of prime numbers Proof: Let us suppose the number of primes is Finite. Let p 1, p 2, … p k be all primes. Let n = p 1 p 2 … p k +1,

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39 Primality testing There is an infinite number of prime numbers Proof: Let us suppose the number of primes is Finite. Let p 1, p 2, … p k be all primes. Let n = p 1 p 2 … p k +1, n must be composite.

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40 Primality testing There is an infinite number of prime numbers Proof: Let us suppose the number of primes is Finite. Let p 1, p 2, … p k be all primes. Let n = p 1 p 2 … p k +1, n must be composite. there exists a prime p s.t. p | n (fund theo. arithmetic), and p cannot be any of the p 1, p 2, … p k

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41 Primality testing There is an infinite number of prime numbers Proof: Let us suppose the number of primes is Finite. Let p 1, p 2, … p k be all primes. Let n = p 1 p 2 … p k +1, n must be composite. there exists a prime p s.t. p | n (fund theo. arithmetic), and p cannot be any of the p 1, p 2, … p k Therefore, p 1, … p k were not all the prime numbers.

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42 Some questions? v Is 2 101 -1=2535301200456458802993406410751 prime? v How do we check whether a number is prime? v How do we generate huge prime numbers? v Why do we care?

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43 Some questions? v Is 2 101 -1=2535301200456458802993406410751 prime? v How do we check whether a number is prime? v How do we generate huge prime numbers? v Why do we care?

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44 Some questions? v Is 2 101 -1=2535301200456458802993406410751 prime? v How do we check whether a number is prime? v How do we generate huge prime numbers? v Why do we care?

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45 Some questions? v Is 2 101 -1=2535301200456458802993406410751 prime? v How do we check whether a number is prime? v How do we generate huge prime numbers? v Why do we care?

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46 Naïve solution: Finding the smallest divisor of n –For i=2,..., n do u Divide n by i until n mod i = 0 Check if i is a divisor of n for some i = 2,..., n

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47 An improvement Check if i is a divisor of n for some i = 2,..., n

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48 An improvement Check if i is a divisor of n for some i = 2,..., n Why can we do that?

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49 Theorem: Composit numbers have a divisor bellow their square root Theorem: Composit numbers have a divisor bellow their square root

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50 Theorem: Composit numbers have a divisor bellow their square root Theorem: Composit numbers have a divisor bellow their square root Proof Idea: n composite n = ab, 0 < a b < n

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51 Theorem: Composit numbers have a divisor bellow their square root Theorem: Composit numbers have a divisor bellow their square root Proof Idea: n composite n = ab, 0 < a b < n a sqrt(n)

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52 Theorem: Composit numbers have a divisor bellow their square root Theorem: Composit numbers have a divisor bellow their square root Proof Idea: n composite n = ab, 0 < a b < n a sqrt(n) Otherwise, we obtain ab > n (contradiction!!)

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53 Is there a more efficient way of checking primality? Is there a more efficient way of checking primality?

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54 Is there a more efficient way of checking primality? Is there a more efficient way of checking primality? Yes! At least if we are willing to accept a tiny probability of error.

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55 We can prove that a number is not prime without explicitly finding a divisor of it Is there a more efficient way of checking primality? Is there a more efficient way of checking primality? Yes! At least if we are willing to accept a tiny probability of error.

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56 We can prove that a number is not prime without explicitly finding a divisor of it Is there a more efficient way of checking primality? Is there a more efficient way of checking primality? Yes! At least if we are willing to accept a tiny probability of error. RANDOMNESS IS USEFUL IN COMPUTATION

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57 The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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58 Suppose that ra e sa have are the same modulo p, then we have r = s (mod p) Contradiction!! Aa, 2a, 3a,..., (p-1)a quando divididos por p possuem restos diferentes:1, 2,..., p-1 Proof: List the first p-1 positive multiple of a: a, 2a, 3a, 4a,..., (p-1) a The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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59 Suppose that ra and sa are the same modulo p, then we have r = s (mod p) Contradiction!! Aa, 2a, 3a,..., (p-1)a quando divididos por p possuem restos diferentes:1, 2,..., p-1 Proof: List the first p-1 positive multiple of a: a, 2a, 3a, 4a,..., (p-1) a The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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60 Suppose that ra and sa are the same modulo p, then we have r = s (mod p) Contradiction!! Aa, 2a, 3a,..., (p-1)a when divided by p have the different reminders:1, 2,..., p-1 Proof: List the first p-1 positive multiple of a: a, 2a, 3a, 4a,..., (p-1) a The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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61 a (p-1) (p-1)! = (p-1)! (mod p) Proof: a. 2a. 3a..... (p-1)a 1. 2. 3..... (p-1) (mod p) The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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62 a (p-1) (p-1)! = (p-1)! (mod p) Dividing by (p-1)! we get the result Proof: a. 2a. 3a..... (p-1)a 1. 2. 3..... (p-1) (mod p) The Fermat Primality Test Fermat’s little theorem: If p is a prime and p does not divide the integer a, then: a p-1 1(mod p)

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63 A Corollary: If p is a prime then, for any integer a: a p a (mod p)

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64 A Corollary: If p is a prime then, for any integer a: a p a (mod p) The result is trivial if p divides a: a(a p-1 – 1) 0 (mod p) If a does not divide a, then we need only multiply the congruence in Fermat´s little theorem by a to complete the proof

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65 A Corollary: If p is a prime then, for any integer a: a p a (mod p) The result is trivial if p divides a: a(a p-1 – 1) 0 (mod p) If p does not divide a, then we need only multiply the congruence in Fermat´s little theorem by a to complete the proof

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66 Corollary: Corollary: If a n ≠ a (mod n), for some a, then n is not a prime! Such an a is a witness to the compositeness of n. The Fermat Test: Do 100 times: Pick a random 1<a<n and compute a n (mod n). If a n a (mod n), then n is not a prime. If all 100 tests passed, declare n to be a prime.

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67 Fast computation of modular exponentiation (define (expmod a b m) (cond ((= b 0) 1) ((= b 0) 1) ((even? b) ((even? b) (remainder (expmod (remainder (expmod (remainder (* a a) m) (remainder (* a a) m) (/ b 2) (/ b 2) m) m)) m) m)) (else (else (remainder (* a (expmod a (- b 1) m)) (remainder (* a (expmod a (- b 1) m)) m)))) m))))

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68 Implementing Fermat test (define (test a n)(= (expmod a n n) a)) (define (rand-test n) (test (+ 1 (random (- n 1))) n)) (define (fermat-test n t); (cond ((= t 0) #t) ((rand-test n) (fermat-test n (- t 1))) (else #f))) Worst-case time complexity: (log n) Even if n is a 1000 digit number, it is still okay!

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69 Is the Fermat test correct? v If the Fermat test says that a number n is composite, then the number n is indeed a composite number. v If n is a prime number, the Fermat test will always say that n is prime. But, v Can the Fermat test say that a composite number is prime? v What is the probability that this will happen?

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70 Carmichael numbers A composite number n is a Carmichael number iff a n a (mod n) for every integer a. The first Carmichael numbers are: 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, … On Carmichael numbers, the Fermat test is always wrong! Carmichael numbers are fairly rare.

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71 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n.

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72 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Proof: Consider Z * n = {1, 2,..., n-1} Let B={x / x Z * n and x n-1 1 (mod n)} We are going to show that B is subgroup of Z * n For this: 1.1 B 2.x 1, x 2 B x 1. x 2 B 3.x B x -1 B

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73 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Proof: Consider Z * n = {1, 2,..., n-1} Let B={x / x Z * n and x n-1 1 (mod n)} We are going to show that B is subgroup of Z * n For this: 1.1 B : 1 n-1 1 (mod n) 2.x 1, x 2 B x 1. x 2 B 3.x B x -1 B

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74 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Proof: Consider Z * n = {1, 2,..., n-1} Let B={x / x Z * n and x n-1 1 (mod n)} We are going to show that B is subgroup of Z * n For this: 1.1 B : 1 n-1 1 (mod n) 2.x 1, x 2 B x 1. x 2 B 3.x B x -1 B (x 1 ) n-1 1 (mod n) (x 2 ) n-1 1 (mod n) (x 1.x 2 ) n-1 1 (mod n)

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75 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Proof: Consider Z * n = {1, 2,..., n-1} Let B={x / x Z * n and x n-1 1 (mod n)} We are going to show that B is subgroup of Z * n For this: 1.1 B : 1 n-1 1 (mod n) 2.x 1, x 2 B x 1. x 2 B 3.x B x -1 B (1) n-1 1 (mod n) (x.x -1 ) n-1 1 (mod n) (x -1 ) n-1 1 (mod n)

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76 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Proof: Pr(x n-1 1 (mod n)) = Pr(x B) It can be proved that 1 B and n-1 B and therefore, |B| 2 Since the order of a subgroup divides the subgroup we have that |B| |Z * n | / 2 Pr(x B) 1/2

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77 Theorem: (Rabin ’77) If n is a composite number that is not a Carmichael number, then at least half of the numbers between 1 and n are witnesses to the compositeness of n. Corollary: Let n be a composite number that is not a Carmichael number. If we pick a random number a, 1<a<n, then a is a witness with a probability of at least a 1/2 !

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78 “Correctness” of the Fermat test “Correctness” of the Fermat test v If n is prime, the Fermat test is always right. v If n is a Carmichael number, the Fermat test is always wrong! v If n is composite number that is not a Carmichael number, the Fermat test is wrong with a probability of at most 2 -100 2 -100 Is an error probability of 2 -100 acceptable? Yes!

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79 The Rabin-Miller test v A fairly simple modification of the Fermat test that is correct with a probability of at least 1-2 -100 also on Carmichael numbers. v Will not be covered in this course.

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80 A probabilistic algorithm An algorithm that uses random choices but outputs the correct result, with high probability, for every input! Randomness is a very useful algorithmic tool. Up to 2002, there were no efficient deterministic primality testing algorithms. In 2002, Agarwal, Kayal and Saxena found a fast deterministic primality testing algorithm.

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81 Finding large prime numbers Finding large prime numbers The prime number Theorem: The number of prime numbers smaller than n is asymptotically n / ln n. Thus, for every number n, there is “likely” to be a prime number between n and n + ln n. To find a prime number roughly the size of n, simply test n, n+2, n+4, … for primality.

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82 Primality testing versus Factoring v Fast primality testing algorithms determine that a number n is composite without finding any of its factors. v No efficient factoring algorithms are known. v Factoring a number is believed to be a much harder task. Primality testing - Easy Factoring - Hard But, factoring is not that hard on a quantum computer!

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