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Primality Testing Math 52 Jonathan Sands Work by: Megan Howley Yangxi Leng Juncai Liu

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Outline Introduction Fermat’s Little Theorem Proof of Miller’s Test Short homework question

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Definitions A prime number is an integer that has no integer factors other than 1 and itself. The opposite of a prime number is called a composite number. If p is prime and p |ab then p|a or p|b A primality test is an algorithm for determining whether an input number is prime.

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Primality Test for Applications safety cryptograph Electronic correspondence The security of this type of cryptograph primarily relies on difficulty involved in factoring very large number, a key one being the testing of numbers for primality. The scheme was used to encrypt plaintext into blocks in order to prevent third party to gain access to private message.

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Fermat's Little Theorem The little theorem is often used in number theory in the testing of large primes and simply states that: If n is a prime which does not divide a, then a (n-1) ≡ 1 (mod n).

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Example n=31, a=3

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Pseudoprimes Numbers which meet the conditions of Fermat's Little Theorem but are not prime are called pseudoprimes Example: 91 is a pseudoprime base 3

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Pseudoprimes

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The Miller Rabin Test The Miller Rabin primality test is essentially an extension of Fermat’s Little Theorem that utilizes factorization However, the Miller test allows one to test for primality with a much higher probability than Fermat’s Little Theorem.

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Miller Proposition Let n be an odd prime integer, and write n-1=2 t m where m is odd and m,t ∈ℤ. Then for all a ∈ℤ with gcd(a,n)=1: Either a m ≡ 1 (mod n), or a m ≡ -1 (mod n), or a 2m ≡ -1 (mod n) Or…

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Miller Test Proof We will first prove a factorization lemma by induction We will then apply this lemma to Fermat’s Little Theorem to prove the Miller’s Test

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Miller Proposition Let n be an odd prime integer, and write n-1=2 t m where m is odd and m,t ∈ℤ. Then for all a ∈ℤ with gcd(a,n)=1: Either a m ≡ 1 (mod n), or a m ≡ -1 (mod n), or a 2m ≡ -1 (mod n) Or…

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Example Use the Miller’s test to see if 29 is prime or composite.

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References Granville, Andrew. It is easy to determine whether a given integer is prime. Bulletin of the American Mathematical Society. Volume 42, Pages 3-38: McGregor-Dorsey, Zachary S. Methods of Primality Testing. MIT Undergraduate Journal of Mathematics. Boston: Rosen, Kenneth. Elementary number theory and its applications. Boston: Addison-Wesley, 2011.

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Our Mentor A special thank you to our mentor, John Voight!

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Homework Problem Use the Miller’s test to see if 11 is prime or composite.

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