Authenticated broadcast, enabling a base station to send commands and requests to low-powered sensor nodes in an authentic manner, is one of the core challenges for securing wireless sensor networks. μTESLA and its multi-level variants based on delayed exposure of one-way chains are well known valuable broadcast authentication schemes, but concerns still remain for their practical application. To use these schemes on resource-limited sensor nodes, a 64-bit key chain is desirable for efficiency, but care must be taken. We will first show, by both theoretical analysis and rigorous experiments on real sensor nodes, that if μTESLA is implemented in a raw form with 64-bit key chains, some of the future keys can be discovered through time memory data tradeoff techniques. We will then present X-TESLA, as a new member of the TESLA family, to remedy the fact that previous schemes do not consider problems arising from sleep modes, network failures, idle sessions, as well as the time memory data tradeoff risk, and to reduce their high cost of countering DoS attacks. In X-TESLA, short key chains are used to continue indefinitely, and new interesting strategies and management methods are possible, significantly reducing unnecessary computation and buffer occupation, to be efficient solutions to the raised problems.