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lecture11

# lecture11 - CS 70 Spring 2005 1 Primality Discrete...

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CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 11 1 Primality We are studying the complexity of two very fundamental, and intimately related, computational problems: P RIMALITY Given an integer x , is it a prime? F ACTORING Given an integer x , what are its prime factors? Obviously, P RIMALITY cannot be harder than F ACTORING , since, if we knew how to factor, we would definitely know how to test for primality. What is surprising and fundamental —and the basis of modern cryptography— is that P RIMALITY is easy while F ACTORING is hard! As we know, P RIMALITY can be trivially solved in O ( x ) time —in fact, we need only test factors up to x . But, of course, these are both exponential algorithms —exponential in the number n of bits of x , which is the more accurate and meaningful measure of the size of the problem (seen this way, the running times of the algorithms become O ( 2 n ) and O ( 2 n / 2 ) , respectively). In fact, pursuing this line (testing fewer and fewer factors) will get us nowhere: Since F ACTORING is hard, our only hope for finding a fast P RIMALITY algorithm is to look for an algorithm that decides whether n is prime without discovering a factor of n in case the answer is “no.” We describe such an algorithm next. This algorithm is based on the following fact about exponentiation modulo a prime: Theorem 11.1 : (Fermat’s Little Theorem.) If p is prime, then for all a 6 = 0 mod p we have a p - 1 = 1 mod p.

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lecture11 - CS 70 Spring 2005 1 Primality Discrete...

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