Computing the permanent modulo a prime power

We show how to compute the permanent of an n x n integer matrix modulo p(k) in time n(k+ O(1)) if p = 2 and in time 2(n)/exp{Omega(gamma(2)n/p logp)} if p is an odd prime with kp < n, where gamma = 1-kp/n. Our algorithms are based on Ryser's formula, a randomized algorithm of Bax and Franklin, and exponential-space tabulation. Using the Chinese remainder theorem, we conclude that for each delta > 0 we can compute the permanent of an n x n integer matrix in time 2n/ exp{Omega(delta(2)n/beta(1/(1-delta)) log beta)}, provided there exists a real number beta such that vertical bar per A vertical bar <= beta(n) and beta <= (1/44 delta n)(1-delta) (C) 2017 Elsevier B.V. All rights reserved.