Primality testing with Gaussian periods

Primality Testing with Gaussian Periods

We exhibit a deterministic algorithm that, for some effectively computable real number c, decides whether a given integer n>1 is prime within time (logn)·(2+log logn). The same result, with 21/2 in the place of 6, was proved by Agrawal, Kayal, and Saxena. Our algorithm follows the same pattern as theirs, performing computations in an auxiliary ring extension of Z/nZ. We allow our rings to be ge...

Primality testing

Testing polynomial primality with pseudozeros

When polynomials have limited accuracy coefficients or are computed in finite precision, classical algebraic problems such that GCD, primality, divisibility have to be redefined. Such approximate algebraic problems are still challenging open questions in the symbolic computation community. In this paper, we show how a numerical tool, the pseudozero set, may provide solutions to some approximate...

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...