Generating REALLY big primes

Question

I'm playing around and trying to write an implementation of RSA. The problem is that I'm stuck on generating the massive prime numbers that are involved in generating a key pair. Could someone point me to a fast way to generate huge primes/probable primes?

Answer 1

You don't generate prime numbers exactly. You generate a large odd number randomly, then test if that number is prime, if not generate another one randomly. There are some laws of prime numbers that basically state that your odds of "hitting" a prime via random tries is (2/ln n)

For example, if you want a 512-bit random prime number, you will find one in 2/(512*ln(2)) So roughly 1 out of every 177 of the numbers you try will be prime.

There are multiple ways to test if a number is prime, one good one is the "Miller-Rabin test" as stated in another answer to this question.

Also, OpenSSL has a nice utility to test for primes:

$ openssl prime 119054759245460753
1A6F7AC39A53511 is not prime

Answer 2

Take a look at how TrueCrypt does it. Also, take a look at Rabin-Miller for testing large pseudoprimes.

Answer 3

You didn't mention what language you are using. Some have a method of doing this built in. For example, in java this is as easy as calling nextProbablePrime() on a BigInteger.

Answer 4

The previous answer is incorrect: 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031 = 509 * 59.

I think the poster is misremembering the (real) proof that there are are an uncountable number of prime numbers.

Answer 5

Mono has a BigInteger class that's open source as does java. You could take a look at those. They're probably portable :) g'luck

Answer 6

There is an algorithm due to U. Maurer that generates random provable (in contrast to statistically highly-probable) primes that are almost uniformly distributed over the set of all primes of a special size. I have a Python implementation of it that is fairly efficient at: http://s13.zetaboards.com/Crypto/topic/7234475/1/