A new characterization of balanced rotation symmetric (n, m)-functions is presented. Based on the characterization, the nonexistence of balanced rotation symmetric (pr, m)-functions is determined, where pis an odd prime and m= 2. And there exist balanced rotation symmetric (2r, m)-functions for 2 = m= 2r- r. With the help of these results, we also prove that there exist rotation symmetric resilient (2r, m)-functions for 2 = m= 2r- r- 1.