Quantum sealed-bid auction using a modified scheme for multiparty circular quantum key agreement

A feasible, secure and collusion-attack-free quantum sealed-bid auction protocol is proposed using a modified scheme for multi-party circular quantum key agreement. In the proposed protocol, the set of all ($n$) bidders is grouped in to $l$ subsets (sub-circles) in such a way that only the initiator (who prepares the quantum state to be distributed for a particular round of communication and acts as the receiver in that round) is a member of all the subsets (sub-circles) prepared for a particular round, while any other bidder is part of only a single subset. All $n$ bidders and auctioneer initiate one round of communication, and each of them prepares $l$ copies of a $\left(r-1\right)$-partite entangled state (one for each sub-circle), where $r=\frac{n}{l}+1$. The efficiency and security\textcolor{blue}{{} }of the proposed protocol are critically analyzed. It is shown that the proposed protocol is free from the collusion attacks that are possible on the existing schemes of quantum sealed-bid auction. Further, it is observed that the security against collusion attack increases with the increase in $l$, but that reduces the complexity (number of entangled qubits in each entangled state) of the entangled states to be used and that makes the scheme scalable and implementable with the available technologies. The additional security and scalability is shown to arise due to the use of a circular structure in place of a complete-graph or tree-type structure used earlier.