Verifiable M+1st-Price Auction without Manager

A.
$\mathrm{M}+1$
st-price auction is an auction method in which all bidders submit their bids simultaneously, and the
$M$
highest bidders purchase
$M$
identical goods at the
$M+$
1st bidding price. Previous researches use trusted third party (TTP), trusted mix server, and honest managers. We propose an efficient and secure auction protocol that guarantees financial fairness, robustness and correctness without any manager or TTP and is secure under malicious model for the first time. We also proposed a new mechanism to verify bids. This new mechanism can be used in all auction protocols which is based on encrypted bits.