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In an M + 1 st-price auction, all bidders submit their bids simultaneously, and the M highest bidders purchase M identical goods at the M + 1 st bidding price. Previous research is constructed based on trusted managers such as a trusted third party (TTP), trusted mix servers, and honest managers. All of the previous auctions are not fit for edge-assisted IoT since they need TTP. In this paper, we formalize a notion of commutative bi-homomorphic multiparty encryption and achieve no-TTP M + 1 -st auction based on blockchain with public verifiability. Our M + 1 st auction guarantees financial fairness, robustness, and correctness without TTP and is secure under a malicious model for the first time. Our M + 1 st auction can be executed over a distributed network and is thus fit for edge-assisted IoT. Furthermore, our formalized commutative bi-homomorphic multiparty encryption can be used in various applications for edge-assisted IoT, which needs to protect privacy and correctness.

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