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7.5. Generating a Prime Number (Testing for Primality)

Problem

You need to generate a random prime number or test to see if a number is prime.

Solution

Use the routines provided by your arbitrary-precision math library, or generate a random odd number and use the Rabin-Miller primality test to see whether the number generated is actually prime.

Discussion

Good arbitrary-precision math libraries have functions that can automatically generate primes and determine to a near certainty whether a number is prime. In addition, these libraries should have functionality that produces "safe” primes (that is, a prime whose value minus 1 divided by 2 is also prime). You should also be able to ask for a prime that gives a particular remainder when you divide that prime by a particular number. The last two pieces of functionality are useful for generating parameters for Diffie-Hellman key exchange.

The OpenSSL functionality for generating and testing primes is discussed in Recipe 7.4.

The most common way primes are generated is by choosing a random odd number of the desired bit length from a secure pseudo-random source (we discuss pseudo-randomness in depth in Recipe 11.1). Generally, the output of the random number generator will have the first and last bits set. Setting the last bit ensures that the number is odd; no even numbers are primes. Setting the first bit ensures that the generated number really is of the desired bit length.

When generating RSA keys, people usually set the first two bits ...

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