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