Primality testing and factoring

Factoring & Primality

In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount of research over the years. These problems started receiving attention in the mathematics community far before the appearance of computer science, however the emergence of the latter has given them additional importance. Gauss himself wrote in 1801 ...

Number-theoretic algorithms1 1 Factoring versus primality testing

One of the most fundamental dichotomies in the study of algorithms is between problems which can be solved efficiently and those which cannot. In the present chapter, we illustrate this divide with two number-theoretic tasks: factoring and deciding whether a number is prime. Despite the close connection between these two problems, they differ vastly in computational complexity. For factoring no...

Primality testing

A Brief History of Factoring and Primality Testing B. C. (Before Computers)

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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...