I'm trying to understand how the safe primes numbers are used in Diffie–Hellman key exchange. According to wiki:
The order of G should have a large prime factor to prevent use of the Pohlig–Hellman algorithm to obtain a or b. For this reason, a Sophie Germain prime q is sometimes used to calculate p = 2q + 1, called a safe prime, since the order of G is then only divisible by 2 and q. g is then sometimes chosen to generate the order q subgroup of G, rather than G, so that the Legendre symbol of ga never reveals the low order bit of a. A protocol using such a choice is for example IKEv2
I'm trying to figure out the context of the paragraph above with small numbers.
q=11 is Sophie Germain prime -> safe prime
p=23. Than I need to find
g is then sometimes chosen to generate the order q subgroup of G.
Shall I find
g^11 (mod 23)will result in a number within the order-11 subgroup?
Or shall I abandon GF(23) and operate in GF(11)?
If you can provide a clear example with some small numbers that illustrate my misunderstanding, please, do it.