Although secret sharing techniques have been applied to implement secure electronic sealed-bid auction for a long time, problems and attacks still exist in secret-sharing-based electronic sealed-bid auction schemes. In this paper, a new secret-sharing-based first-bid e-auction scheme is designed to achieve satisfactory properties and efficiency. Correctness and fairness of the new auction are based on hard computation problems and do not depend on any trust. Complete bid privacy based on a threshold trust is achieved in the new scheme. Attacks to existing secret-sharing-based sealed-bid e-auction schemes are prevented.