The data that flows through wireless sensor networks is often private. Although data encryption is required, the calculation ability of each sensor node is weak. Therefore, common key encryption systems are suitable for sensor networks. To encrypt data using a common key, a key exchange scheme is required. For large-scale mesh-type wireless sensor networks, Key Predistribution Schemes have been used. However, this scheme involves some nodes that cannot perform key exchange. Moreover, as the number of stolen nodes increases, the analysis of the common key becomes easier. In this paper, we propose a novel key exchange scheme that does not include nodes that cannot perform key exchange. Even if many nodes are stolen, the proposed scheme ensures that the analysis of keys in other nodes is not possible, in the case where the attacker cannot intercept communication between any of them.