Usually the main primitive in building a secure wireless authentication is a cryptographic algorithm, such as digital signature scheme. He et al. proposed a handover authentication protocol in (IEEE Trans. Wireless Commun., vol. 11, no. 1, 2011) and a distributed reprogramming protocol in (IEEE Trans. Ind. Electron., vol. 59, no. 11, 2012) for wireless networks. Both protocols are based on an identity-based signature scheme which is claimed to be secure yet efficient. Very recently, He et al. pointed out that such a signature scheme is vulnerable to the key compromised problem. They proposed a simple modification to fix this problem without losing the efficiency and security of the scheme in both (IEEE Commun. Lett., vol. 16, no. 8, 2012) and (IEEE Trans. Ind. Electron., to appear). In this letter, we show that the proposed modification remains vulnerable to the key compromised problem.