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In this paper, we propose a conditional privacy-preserving authentication scheme, called CPAS, using pseudo-identity-based signatures for secure vehicle-to-infrastructure communications in vehicular ad hoc networks. The scheme achieves conditional privacy preservation, in which each message launched by a vehicle is mapped to a distinct pseudo-identity, and a trust authority can always retrieve the real identity of a vehicle from any pseudo-identity. In the scheme, a roadside unit (RSU) can simultaneously verify multiple received signatures, thus considerably reducing the total verification time; an RSU can simultaneously verify 2540 signed-messages/s. The time for simultaneously verifying 800 signatures in our scheme can be reduced by 18%, compared with the previous scheme.