Proof that Paley Graphs are strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$

Question

A Paley graph is strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$. I need to prove that, and obtain the parameters too. Proving it is regular valency $\frac{p-1}{2}$ is easy but for the rest I am struggling. Also, proofs I found are very vague and only prove the third parameter and then use the equation $k(k-a-1)=b(v-k-1)$ to find the other. However, I dont think we can use that as we have not yet proved it is strongly regular, so we dont know if the equation holds.. is that right? Can someone please provide a proof of the statement?

http://en.wikipedia.org/wiki/Paley_graph

Answer 1

You can find the proof in these lecture notes.