Many applications of wireless sensor networks (WSNs) require deploying sensors in hostile environments, where an adversary may eavesdrop communications. To secure communications in WSNs, the q-composite key predistribution scheme has been proposed in the literature. In this paper, we investigate secure k-connectivity in WSNs operating under the q-composite scheme, in consideration of the unreliability of wireless links. Secure k-connectivity ensures that any two sensors can find a path in between for secure communication, even when k - 1 sensors fail. We present conditions on how to set the network parameters such that the network has secure k-connectivity asymptotically almost surely. The result is given in the form of a sharp zero-one law.