Jiang et al. proposed an authentication scheme for Vehicle-to-Infrastructure communications using a Binary Authentication Tree. The scheme can effectively eliminate the performance bottleneck when verifying a mass of signatures within a rigorously required interval and filtering bogus messages. In this paper, we show that Jiang et al.'s scheme is insecure against forgery attacks, replay attacks and Sybil attacks. To overcome the weaknesses, we reconstruct a conditional privacy-preserving authentication scheme, called sf CPP-BAT, for Vehicle-to-Infrastructure communications based on secure identity-based signature, aggregate signature schemes and the Binary Authentication Tree.