Primality and Factoring

Abstract

There are many situations where one wants to know if a large number n is prime. For example, in the RSA public key cryptosystem and in various cryptosystems based on the discrete log problem in finite fields, we need to find a large “random” prime. One interpretation of what this means is to choose a large odd integer $n_{0}$ using a generator of random digits and then test $n_{0}, n_{0}+2, \dotsc$ for primality until we obtain the first prime which is $\ge n_{0}$. A second type of use of primality testing is to determine whether an integer of a certain very special type is a prime. For example, for some large prime f we might want to know whether $2^{f}-1$ is a Mersenne prime. If we’re working in the field of $2^{f}$ elements, we saw that every element ≠ 0, 1 is a generator of ${\pmb{\text{F}}}^{*}_{2^{f}}$ if (and only if) $2^{f}-1$ is prime (see Exercise 13(a) of §11.1).