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

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Presentation on theme: "Primality Testing Math 52 Jonathan Sands Work by: Megan Howley Yangxi Leng Juncai Liu."— Presentation transcript:

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

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

3 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.

4  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.

5 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).

6 Example n=31, a=3

7 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

8  Pseudoprimes

9 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.

10 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…

11 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

12 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…

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

14 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: 2004.  McGregor-Dorsey, Zachary S. Methods of Primality Testing. MIT Undergraduate Journal of Mathematics. Boston: 2010.  Rosen, Kenneth. Elementary number theory and its applications. Boston: Addison-Wesley, 2011.

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

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


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