NumberTheorySrividyaGeorge

NumberTheorySrividyaGeorge - Number Theory Presented by...

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Number Theory Presented by Shrividya Shivkumar and George Frederick

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Contents Division Theorem Modular Exponential Prime Numbers Fermat’s Little Theorem Miller-Rabin Primes Is In P Relatively Prime numbers Euclid’s algorithm Extended Euclid algorithm Chinese Remainder Theorem RSA Pollard’s Rho
Division theorem For any integer a and a positive integer n there are unique integers q and r such that 0 ≤ r < n and a = qn + r or a = n + ( a mod n) If (a mod n) = (b mod n) then a is equivalent to b a b (mod n) Ex : 61 6 (mod 11)   a/n

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Properties of modular addition and multiplication: Let a a’ (mod n) b b’ (mod n) then a + b ( a’ + b’)( mod n) ab (a’b’) (mod n) Properties of common divisors: If d | a and d | b d | (a + b) If d | a and d | b d | ( a – b) If d | a and d | b d | (ax + by)
Modular Exponential Gives an efficient way to calculate n a b mod

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Modular Exponential
What are prime numbers? An integer having only trivial divisors ( 1 and itself) Ex : 2 , 3 , 5 , 7 , 11 …. What are relative Prime Numbers ? Numbers whose only common factor is 1 or the gcd(a,b) = 1. Ex: 6 and 35 are relatively prime (gcd = 1)

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Finding Prime numbers Trial division – testing for divisibility of each integer starting from 2 … sqrt(n) Even integers greater than 2 can be skipped. Worst case complexity : O (sqrt(n))
Fermat’s Little Theorem

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Fermat’s Little Theorem Disadvantages: Does not work with Carmichael numbers. Carmichael numbers - a Carmichael number is a composite positive integer n which satisfies the congruence for all integers b which are relatively prime to n . Ex : 561 = 11 * 3 * 17 ) (mod 1 1 n b n -
How to check if a number is prime? Use the Miller-Rabin test Uses several randomly chosen base values

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Miller-Rabin Test contd… Witness(a,n) 1. b(k),b(k-1)….b(0) . . Binary representation of n-1 2. D  1 3. For I  k to 0 Do x  d D  (d.d)mod(n) if d = 1 and (x not equal 1) and (x not equal n-1) return true if b(i) = 1 d  (d.a)mod n If ( d not equal 1)
PRIMES is in P Authored by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena Won the 2006 Gödel Prize Produced an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite Previous efforts were all conditional, randomized, or had exponential running times

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PRIMES is in P As with most primality tests, is based on Fermat’s Little Theorem (actually a generalization of) Fermat’s Little Theorem: For any integer : Generalization: Let and . Then is prime iff a ) (mod p a a p , 2 , , n N n Z a 1 ) , ( = n a n ) (mod ) ( n a X a X n n + = +
What is a greatest common divisor? The largest common divisor of a and b 1 < = gcd( a,b) <= min ( |a| , | b|)

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Euler’s Phi Function The number of positive integers less than equal to n that are relatively prime to n where, P  Number of primes dividing n.
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NumberTheorySrividyaGeorge - Number Theory Presented by...

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