A new multi-stage secret sharing scheme using one-way function

He and Dawson proposed a multi-stage secret sharing scheme based on one-way function. In that scheme, many secrets are reconstructed stage-by-stage in the dealer's predetermined order, and only one secret shadow is kept by every participant. When all the secrets have been reconstructed, the dealer needs not redistribute fresh shadows to every participant. Later, Harn further improved the He-Dawson scheme to reduce the total number of public values. However, in this paper, we will show that both the He-Dawson scheme and Ham's scheme are one-time-use schemes and that many secrets cannot in fact be reconstructed stage-by-stage. At the same time, we shall also modify the He-Dawson scheme to improve the drawbacks above and show the improved scheme can be applied.