First price sealed bid auction without auctioneers

We propose two protocol variants for a first price sealed-bid auction, without using intermediatory auctioneers. One version achieves full privacy for the bidders and their bids, the other provides a form of verifiability, at the cost of some privacy. Full privacy protects all bids. In particular the winner's identity and price are only known by the seller. Lesser privacy allows the winner to be known and verified publicly. Both versions provide non-repudiation. We demonstrate correctness and show how computational and communication costs vary with different privacy levels.