Let me add to this by talking about cyclic groups. All cyclic groups are equivalent to addition modulo an integer, but they can be different computationally: to

Let me add to this by talking about cyclic groups. All cyclic groups are equivalent to addition modulo an integer, but they can be different computationally: to go from an element of your cyclic group to the equivalent integer may be very difficult. For addition modulo n, the computational Diffie-Hellman problem is effectively: Given integers 1, a, and b, compute ab. This is obviously easy. The problem of translating from an arbitrary cyclic group to addition mod n is the discrete log problem, so perhaps that explains why it is important to cryptography that it be hard for your group.