Fibonacci Pseudoprimes and their Place in Primality Testing Carly Allen Faculty Mentor, Dr. Webster Butler University Department of Mathematics December 11, 2015 Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
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Fibonacci Pseudoprimes and their Place inPrimality Testing
Carly AllenFaculty Mentor, Dr. Webster
Butler UniversityDepartment of Mathematics
December 11, 2015
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Outline
1 Motivation
Primality TestingEncryption
2 Building Blocks of Fibonacci Pseudoprimes
3 The Fibonacci Primality Theorem
4 Theorem Becomes Primality Test
5 Pseudoprimes
6 The Test in a Larger Context
7 Future Work
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Motivation
Why study Fibonacci pseudoprimes?
1 Primality Testing
How do we determine if an extremely large integers are prime?
2 Encryption
How does primality testing and pseudoprimes play a role inreal-world applications? Does it at all?
Definition
Encryption: the process of converting data to an unrecognizable or”encrypted” form.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Building Blocks of Fibonacci Pseudoprimes
1 Prime Numbers
DefinitionExamples and Patterns
2 Trial Division3 Fermat Base-2 Primality Test
PseudoprimesPrevalence of Pseudoprimes using this Test
4 The Fibonacci Numbers
5 Legendre Symbol
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Prime Numbers and Testing for Primality
How do we determine if a large integer is prime or composite?
Definition
A prime number (or a prime) is a natural number greater than 1that has no positive divisors other than 1 and itself.
i.e. 2, 3, 5, 7, 11, 13, 17, 19, ...
Integer Factorization vs. Primality Testing
In testing for primality, we do not wish to break an integerdown into its’ prime factors, but rather we wish to statewhether the input is prime or not.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Trial Division
A simple, yet time-consuming method for primality testing.
To check if n, an integer, is prime, we may test up to√n integers.
What if n=589328201? There must be a better way!
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Fermat’s Little Theorem
Theorem
Fermat’s Little Theorem, as a special case, says for any oddprime p, 2p−1 ≡ 1 (mod p).
Theorem
Base-2 Fermat Test: For a given odd integer n > 1, if
2n−1 6≡ 1 (mod n),
then n is composite. If the result is 1, call n “probably prime.”
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Definition of a Pseudoprime
Using the Base-2 Fermat test, every prime number will return“probably prime.” The test, though is not perfect and somecomposite numbers will additionally return “probably prime.”When this happens, such a number is considered a pseudoprime.
Take n = 341.2341−1 (mod 341) ≡ 1 (mod 341).
=⇒ probably prime.BUT 341 = 11 · 31
Definition
A base-2 pseudoprime is is a composite number n that“passes”(or returns “probably prime”) the base-2 Fermat primality test.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Fibonacci Numbers
Definition
Fibonacci numbers are numbers in the Fibonacci sequence. Thesequence Fn of Fibonacci numbers is defined by the recurrencerelation:
F0 = 0 and F1 = 1
Fn = Fn−1 + Fn−2
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Legendre Symbol
Definition
Let p be an odd prime. For any integer m, the Legendre symbol( pm ) is defined as follows:
( pm
)=
+1 if m is quadratic residue modulo p.−1 if m is quadratic nonresidue modulo p.0 if p divides m.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Dominic W. Klyve and Daniel MonfreCarly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Fibonacci Primality Test in a Larger Context
As you can see, combining tests is POWERFUL. Let’s see anotherexample of a powerful combination of primality tests.
Definition
The BPSW (Baillie, Pomerance, Selfridge, Wagstaff) Test is a“probabilistic primality testing algorithm” that combines a base-2Fermat test with a Lucas Probable Prime Test.
No fpsp(2)’s yet found are congruent to 2 or 3 modulo 5.
The BPSW test finds the first D in the sequence 5, -7, 9, -11,13, -15, . . . for which the Jacobi symbol
(Dn
)is −1.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
BPSW
No composite up to 264 passes the BPSW Test.
The power of the BPSW test lies in the fact Fermatpseudoprimes and Lucas pseudoprimes share no knownnumbers.
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Fibonacci Primality Test in a Larger Context, Cont.
$620
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Future Work
What questions are still needing to be considered concerningFibonacci Pseudoprimes?
1 Proof or Counterexample of BPSW Primality Test
2 A more efficient algorithm at determining the number ofpseudoprimes. (Dr. Webster)
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing
Questions
Thank you! Questions?
Carly Allen Faculty Mentor, Dr. Webster Fibonacci Pseudoprimes and their Place in Primality Testing