Generating large primes

1 A simple paradigm Generating a large prime is an important step in the RSA algorithm. We can generate a large prime by repeatedly selecting a large random integer and testing it for primality. We stop when a prime number is found. Therefore, assume that a large integer n is given, and choose a rational number α < 1 to perform the following algorithm: repeat generate a random integer p ∈ [n α , n] until p is prime There are two important aspects of the above algorithm. First, it is random-ized in the sense that our random choices determine its running time (through the number of iterations). Second, it relies heavily on a method for primality testing, which will dominate the running time of a single iteration (making a random choice takes O(log n) time since it requires generating that many bits). 2 Analysis of number of iterations Let us first determine the probability of choosing a prime in [n α , n]. We use a famous result in number theory: Prime number theorem Let π(n) be the number of primes ≤ n. Then π(n) ∼ n ln n This is an asymptotic characterization, i.e. lim n→∞ π(n) ln n n = 1. Since we are dealing with large values of n, making the approximation π(n) ≈