The concept of Designated Verifier Signatures (DVS) was introduced by Jakobsson, Sako and Impagliazzo at Eurocrypt '96. These signatures are intended to a specific verifier, who is the only one able to check their validity. In this context, we formalize the notion of privacy of signerrsquos identity which captures the strong designated verifier property investigated in their paper. We propose a variant of the pairing-based DVS scheme introduced at Asiacrypt '03 by Steinfeld, Bull, Wang and Pieprzyk. Contrary to their proposal, our new scheme can be used with any admissible bilinear map, especially with the low cost pairings and achieves the new anonymity property (in the random oracle model). Moreover, the unforgeability is tightly related to the Gap-Bilinear Diffie-Hellman assumption, in the random oracle model and the signature length is around 75% smaller than the original proposal.