Authentication is an important security mechanism for detecting forged messages in a sensor network. Each cluster head (CH) in dynamic key distribution schemes forwards a key dissemination message that contains encrypted authentication keys within its cluster to next-hop nodes for the purpose of authentication. The forwarding path of the key dissemination message strongly affects the number of nodes to which the authentication keys in the message are actually distributed. We propose a routing method for the key dissemination messages to increase the number of nodes that obtain the authentication keys. In the proposed method, each node selects next-hop nodes to which the key dissemination message will be forwarded based on secret key indexes, the distance to the sink node, and the energy consumption of its neighbor nodes. The experimental results show that the proposed method can increase by 50-70% the number of nodes to which authentication keys in each cluster are distributed compared to geographic and energy-aware routing (GEAR). In addition, the proposed method can detect false reports earlier by using the distributed authentication keys, and it consumes less energy than GEAR when the false traffic ratio (FTR) is ≥ 10%.