Studying the efficiency of the Frobenius primality test

Abstract

In mathematics, especially number theory, prime numbers are essential concepts. Prime numbers are used in cryptography as one application.

Finding large prime numbers is crucial for cryptographic protocols; to do this, we must be able to tell whether a given number is prime or not. To test whether a number is a prime number, we require a computationally efficient primality testing algorithm. The primary objective of my research is to evaluate how well the tests work. Especially, in my research our main focus is on Grantham’s primality test. Grantham’s test is probabilistic and fast, but it comes with the risk of false positives. To determine how ‘good’ a test is, one must be aware of the possibility of false positives because in our development, deterministic tests are slower than false positive ones.

In this thesis, we will explain the definitions of ‘probable prime numbers’, such as ‘Frobenius pseudoprime’, as given by Jon Grantham. Our research goal is to find upper and lower bounds for the number of probable prime numbers by generalizing the work of Paul Erdös and Carl Pomerance on Fermat pseudoprimes, and Jon Grantham on Frobenius pseudoprimes.