Privacy has become a factor of increasing importance in auction design. We propose general techniques for cryptographic first-price and (M+1)st-price auction protocols that only yield the winners' identities and the selling price. Moreover, if desired, losing bidders learn no information at all, except that they lost. Our security model is merely based on computational intractability. In particular, our approach does not rely on trusted third parties, e.g., auctioneers. We present an efficient implementation of the proposed techniques based on El Gamal encryption whose security only relies on the intractability of the decisional Diffie—Hellman problem. The resulting protocols require just three rounds of bidder broadcasting in the random oracle model. Communication complexity is linear in the number of possible bids.