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What is the currently industry-standard algorithm used to generate large prime numbers to be used in RSA encryption?

I'm aware that I can find any number of articles on the Internet that explain how the RSA algorithm works to encrypt and decrypt messages, but I can't seem to find any article that explains the algorithm used to generate the p and q large and distinct prime numbers that are used in that algorithm.

Paŭlo Ebermann
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Lukman
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6 Answers6

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The standard way to generate big prime numbers is to take a preselected random number of the desired length, apply a Fermat test (best with the base $2$ as it can be optimized for speed) and then to apply a certain number of Miller-Rabin tests (depending on the length and the allowed error rate like $2^{-100}$) to get a number which is very probably a prime number.

The preselection is done either by test divisions by small prime numbers (up to few hundreds) or by sieving out primes up to 10,000 - 1,000,000 considering many prime candidates of the form $b+2i$ ($b$ big, $i$ up to few thousands).

The deterministic prime number test by AKS is to my knowledge not yet used as it is slower and as the likeliness that an calculation error caused by the hardware is higher than $2^{-100}$.

Most smart cards offer a coprocessor for modular arithmetic with moduli from 1024 up to few thousand bits. The manufacturers often provide also libraries for RSA and RSA key generation using the coprocessor.

j.p.
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FIPS 186-3 tells you how they expect you to generate primes for cryptographic applications. It is essentially Miller-Rabin but it also specify what to do when you need extra properties from your primes.

ir01
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giuper
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The problem of generating prime numbers reduces to one of determining primality (rather than an algorithm specifically designed to generate primes) since primes are pretty common: π(n) ~ n/ln(n).

Probabilistic tests are used (e.g. in java.math.BigInteger.probablePrime()) rather than deterministic tests. See Miller-Rabin.

http://en.literateprograms.org/Miller-Rabin_primality_test_%28Java%29

As far as primes for RSA goes, there are some additional minor requirements, namely that (p-1) and (q-1) should not be easily factorable, and p and q should not be close together.

Jason S
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Instead of using probabilistic methods to generate large primes, as indicated in the other answers, one can (IMHO better) employ Maurer's algorithm of generating provable primes. I have a Python code implementing that algorithm, available at s13.zetaboards.com/Crypto/topic/7234475/1/

Mok-Kong Shen
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There are some tests to determines whether a given number is prime, like Miller–Rabin primality test. These algorithms determine whether a given number is prime with probability P.

ir01
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What you could do, is use a Mill's constant for generation. Then, a test for primarity would be good anyways...(in case that not exact enough constant is chosen, so you could end up with non prime)

Mill's constant is such a number, that if powered to 3, and then powered to any N, where N is an usigned integer, we get value V wich, rounded down, is a prime.

So, let C be constant

C = 1.3063778838630806904686144926 ....

Then let 

L = C^3

Then choose any N from <0, infinite> , N is whole

prime = round down( L^N )

It has several drawbacks, such as the constant must be very accurate, if you do not have accurate enough constant, you will end up with composed number. However, it is proven that this ,,algorithm" always generates prime numbers.

Next drawback I heard about is that as you increase the exponent, more computation power will be needed. However, all you need to do is generate two of them, so it could do...

For further reading, here is what I found as reference. https://en.wikipedia.org/wiki/Mills%27_constant

Jan Glaser
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