For the Diffie Helman prime and generator, what key length should I use?

Question

In the code below, from the Crypto++ wiki, is 128 the number I really should be using?

CryptoPP::AutoSeededRandomPool arngA;
CryptoPP::RandomNumberGenerator& rngA = *dynamic_cast<CryptoPP::RandomNumberGenerator *>(&arngA); 
CryptoPP::DH dhA(rngA, 128);
CryptoPP::Integer iPrime = dhA.GetGroupParameters().GetModulus();
CryptoPP::Integer iGenerator = dhA.GetGroupParameters().GetSubgroupGenerator();

Answer 1

2¹²⁸ is a ridiculously huge number*; so yes, that's more than enough.

* (It's about 340,000,000,000,000,000,000,000,000,000,000,000,000 [37 zeros]. A usual supercomputer can do about 2,000,000,000,000,000 [15 zeros] operations a second)

Answer 2

Integer factorization and discrete logarithm over Z/(pZ) are roughly equally difficult. Therefore the size of the modulus for Diffie-Hellman should be about the same size as you would choose for an RSA modulus. If you are comfortable with a 1024-bit RSA key then you can also be comfortable with a 1024-bit Diffie-Hellman key.

It is not easy to tell if key sizes in crypto++ are measured in bits or bytes. As Sebastian points out dhA(rngA, 128) may indeed generate a 128 bit Diffie-Hellman key, which would be too small. Going through the code it looks like this is indeed the case.

The size of the generator iGenerator does not affect the security of Diffie-Hellman. (I.e. iGenerator = 2 could be perfectly fine)