Explicit bounds for primality testing and related problems

Explicit Bounds for Primality Testing and Related Problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is 0(log m). This has been generalized by Lagañas. Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of n...

Primality testing

Primality Testing

Prime numbers are extremely useful, and are an essential input to many algorithms in large part due to the algebraic structure of arithmetic modulo a prime. In everyday life, perhaps the most frequent use for prime numbers is in RSA encryption, which requires quite large primes (typically≥ 128-bits long). Fortunately, there are lots of primes—for large n, the probability that a random integer l...

Algorithmic Primality Testing

This project consists of the implementation and analysis of a number of primality testing algorithms from Chapter 9 of 3]. The rst algorithm is added to provide a point of comparison and the section on Lucas-Lehmer testing was derived mainly from material found in 4]. Aside from simply implementing these tests in Maple and C, the goal of this paper is to provide a somewhat simpliied and more ge...

Four primality testing algorithms

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 t...