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Fingerprint. Verifying set equality. String Matching – Rabin-Karp Algorithm. Verifying set equality. Verifying set equality. Verifying set equality. Verifying set equality. Fingerprinting. Fingerprinting. Fingerprinting Computation.

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## Fingerprint

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**Verifying set equality**• String Matching – Rabin-Karp Algorithm**Fingerprinting Computation**Horner’s Rule**Density of Primes**• (x) = número de primos menores ou iguais a x • (13) = 6 • Primos < = do que 13 = 2, 3, 5 , 7, 11 e 13 • 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**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?**Density of Primes**Para um valor elevado de x, p(x) ~ x/ ln x . Ou seja, lim (x) = 1 x x/ln x**String Matching**• Many applications • While using editor/word processor/browser • Login name & password checking • Virus detection • Header analysis in data communications • DNA sequence analysis**Fingerprinting computation**The only expensive operation**Primality testing**• A natural number n is prime iff the only natural numbers dividing n are 1 and n**Primality testing**• A natural number n is prime iff the only natural numbers dividing n are 1 and n • The following are prime: 2, 3, 5, 7, 11, 13,**Primality testing**• A natural number n is prime iff the only natural numbers dividing n are 1 and n • The following are prime: 2, 3, 5, 7, 11, 13, …and so are 1299709,15485863, 22801763489, …**Primality testing**• A natural number n is prime iff the only natural numbers dividing n are 1 and n • The following are prime: 2, 3, 5, 7, 11, 13, …and so are 1299709,15485863, 22801763489, … There is an infinite number of prime numbers**Primality testing**There is an infinite number of prime numbers Proof:Let us suppose the number of primes is Finite.**Primality testing**There is an infinite number of prime numbers Proof:Let us suppose the number of primes is Finite. Let p1, p2, … pk beall primes. Let n = p1p2 … pk+1,**Primality testing**There is an infinite number of prime numbers Proof:Let us suppose the number of primes is Finite. Let p1, p2, … pk beall primes. Let n = p1p2 … pk+1, n must becomposite.**Primality testing**There is an infinite number of prime numbers Proof:Let us suppose the number of primes is Finite. Let p1, p2, … pk beall primes. Let n = p1p2 … pk+1, n must becomposite. there exists a prime p s.t. p | n (fundtheo. arithmetic), and p cannot be any of the p1,p2, … pk**Primality testing**There is an infinite number of prime numbers Proof:Let us suppose the number of primes is Finite. Let p1, p2, … pk beall primes. Let n = p1p2 … pk+1, n must becomposite. there exists a prime p s.t. p | n (fundtheo. arithmetic), and p cannot be any of the p1,p2, … pk Therefore, p1, … pk were not all the prime numbers.**Some questions?**• Is 2101-1=2535301200456458802993406410751 prime? • How do we check whether a number is prime? • How do we generate huge prime numbers? • Why do we care?**Some questions?**• Is 2101-1=2535301200456458802993406410751 prime? • How do we check whether a number is prime? • How do we generate huge prime numbers? • Why do we care?**Some questions?**• Is 2101-1=2535301200456458802993406410751 prime? • How do we check whether a number is prime? • How do we generate huge primenumbers? • Why do we care?**Some questions?**• Is 2101-1=2535301200456458802993406410751 prime? • How do we check whether a number is prime? • How do we generate huge primenumbers? • Why do we care?**Naïve solution: Finding the smallest divisor of n**• For i=2,..., n do • Divide n by i until n mod i = 0 Check if i is a divisor of n for some i = 2, ..., n**An improvement**• Check if i is a divisor of n for some i = 2, ..., n**An improvement**• Check if i is a divisor of n for some i = 2, ..., n Why can we do that?**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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