Computing permanents over fields of characteristic 3: where and why it becomes difficult

In this paper we consider the complexity of computing permanents over fields of characteristic 3. We present a polynomial time algorithm for computing per(A) for a matrix A such that the rank rg(AA^T-I)⩽1. On the other hand, we show that existence of a polynomial-time algorithm for computing per(A) for a matrix A such that rg(AA^T-I)⩾2 implies NP=R. As a byproduct we obtain that computing per(A) for a matrix A such that rg(AA^T-I)⩾2 is P(mod3) complete