Auctions are among the best-known market-based tools to solve the problem of dynamic spectrum redistribution. In recent years, a good number of strategy-proof auction mechanisms have been proposed to improve spectrum utilization and to prevent market manipulation. However, the issue of privacy preservation in spectrum auctions remains open. On the one hand, truthful bidding reveals bidders' private valuations of the spectrum. On the other hand, coverage/interference areas of the bidders may be revealed to determine conflicts. In this paper, we present PISA, which is a PrIvacy preserving and Strategy-proof Auction mechanism for spectrum allocation. PISA provides protection for both bid privacy and coverage/interference area privacy leveraging a privacy-preserving integer comparison protocol, which is well applicable in other contexts. We not only theoretically prove the privacy-preserving properties of PISA, but also extensively evaluate its performance. Evaluation results show that PISA achieves good spectrum allocation efficiency with light computation and communication overheads.