  In this paper, we give a construction of strongly regular Cayley graphs and a
construction of skew Hadamard difference sets. Both constructions are based on
choosing cyclotomic classes in finite fields, and they generalize the
constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite
families of strongly regular graphs with new parameters are obtained. The main
tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
