With the advancement and deployment of leading-edge telecommunication technologies for sensing and collecting traffic related information, the vehicular ad-hoc networks (VANETs) have emerged as a new application scenario that is envisioned to revolutionize the human driving experiences and traffic flow control systems. To avoid any possible malicious attack and resource abuse, employing lightweight cryptosystems is widely recognized as one of the most effective approaches for the VANETs to achieve confidentiality, integrity and authentication. As a typical substitution-permutation network lightweight cryptosystem, LED supports 64-bit and 128-bit secret keys, which are flexible to provide security for the RFID and other highly-constrained devices in the VANETs. Since its introduction, some research of fault analysis has been devoted to attacking the last three rounds of LED. It is an open problem to know whether provoking faults at a former round of LED allows recovering the secret key. In this paper, we give an answer to this problem by showing a novel impossible differential fault analysis on one round earlier of all LED keysize variants. Mathematical analysis and simulating experiments show that the attack could recover the 64-bit and 128-bit secret keys of LED by introducing 48 faults and 96 faults in average, respectively. The result in this study describes that LED is vulnerable to a half byte impossible differential fault analysis. It will be beneficial to the analysis of the same type of other iterated lightweight cryptosystems in the VANETs.