Question

I’m about to implement the DSA algorithm, but there is a problem:

choose “p”, a prime number with L bits, where 512 <= L <= 1024 and L is a multiple of 64

How can I implement a random generator of that number? Int64 has “only” 63 bits length.

Answer 1

You can generate a random number with n bits using this code:

var rng = new RNGCryptoServiceProvider();
byte[] bytes = new byte[n / 8];
rng.GetBytes(bytes);

BigInteger p = new BigInteger(bytes);

The result is, of course, random and not necessarily a prime.

The BigInteger class was introduced in the .NET 4.0 Framework.

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For generating large prime numbers, Wikipedia says:

For the large primes used in cryptography, it is usual to use a modified form of sieving: a randomly-chosen range of odd numbers of the desired size is sieved against a number of relatively small odd primes (typically all primes less than 65,000). The remaining candidate primes are tested in random order with a standard primality test such as the Miller-Rabin primality test for probable primes.

So you could do something like this:

var p = Enumerable.Range(0, numberOfCandidates)
                  .Select(i => RandomOddNumber(bits))
                  .Where(x => !primesLessThan65000.Contains(x))
                  .Where(x => PrimalityTest(x))
                  .FirstOrDefault();