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We investigate how to obtain bid privacy in sealed-bid auctions. In particular, this paper focuses on unconditional full privacy, i.e., privacy that relies neither on trusted third parties (like auctioneers) or trusted fractions of bidders, nor on computational intractability assumptions (like the hardness of factoring). These constraints imply a scenario in which bidders exchange messages according to some predefined protocol in order to jointly determine the auction outcome without revealing any additional information. It turns out that the first-price sealed-bid auction can be emulated by an unconditionally fully private protocol. However, the protocolýs round complexity is exponential in the number of bits that represent a bid, and we show there is no more efficient protocol. On the other hand, we prove the impossibility of fully privately emulating the second-price sealed-bid (Vickrey) auction for more than two bidders. This impossibility holds even when relaxing various privacy constraints such as protecting just a single losing bid (while maintaining anonymity) or tolerating the revelation of complete information to a coalition of at least half of the bidders.