Four primality testing algorithms

TL;DR

This paper reviews four primality testing algorithms, highlighting their efficiency, determinism, and practical use in identifying prime numbers with varying degrees of certainty and computational complexity.

Contribution

It provides an expository comparison of four primality tests, including their theoretical foundations and practical performance.

Findings

The first test is very efficient but probabilistic.

The second test is a deterministic polynomial-time algorithm.

The third and fourth tests are widely used in practice, exploiting cyclotomic fields and elliptic curves.

Abstract

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.