In Discrete Logs, What Should g Be?

In reading about cryptography, have you ever come across the term of a cyclic group G of order p and with a generator g? This article will hopefully explain what this means.

The world of public key encryption is currently dominated by two things: discrete logarithms and elliptic curve methods. RSA is becoming a thing of the past for new applications, but it is only hanging on as it has such a monopoly in digital certificates. And so with discrete logarithms and the Diffie-Hellman method we end up with:

Y = gˣ mod p

where we have a generator value (g) and a prime number p. The challenge is that even though we know Y, g and p, it is extremely difficult to determine the x value if we use a large prime number.

So can we use any value of g, and should it be as large as possible? The answer to both of these questions is no. If select a prime number of 7, and then select g values of 2, 3, 4 …9, and then calculate the results we get [spreadsheet]:

Now look at g=2, we get an output of 2, 4, 1, 2, 4 … for the sequence values of 1, 2, … This means that we do not get a unique output for the values from 1 to 6(where the maximum…