Abstract: He-Dawson one-way function based on multi-step secret sharing scheme is studied.The scheme is a time scheme and can not resist collusion attacks.Based on authentication，cryptographic multi-secret sharing scheme and the use of quadratic residue a digital signature scheme is proposed by means of a quadratic residue to construct a multi-secret sharing scheme.The scheme is a （t，n） threshold multi-secret sharing scheme.In the scheme，a secret is distributed by the distributor，but each participant can verify the distribution by the distributor of a secret，which can prevent the deception of secret distributor，and each participant can verify other partners’ cheating.In addition，for each participant，selected sub-secret can be re-used，and group secret can be reconstructed in any order，while the scheme is also able to resist collusion attacks.Safety of the scheme is based on Shamir threshold scheme，and RSA keys system.Under the assumption of difficulty in large integer factorization and the difficulty of discrete log decomposition，the proposal is proved to be safe.

Key words: quadratic residue, multi-secret sharing, decomposition of integers, decomposition of discrete logarithm

摘要： 研究了He-Dawson 所提出的基于单向函数的多步骤秘密共享方案，指出该方案是一次方案而且不能抵抗合谋攻击，结合基于身份验证的密码学多秘密共享方案和利用二次剩余构造的数字签名方案，提出了一种利用二次剩余构造一个多秘密共享方案，该方案功能是一种（t，n）门限的多秘密共享方案。该方案中，由秘密分发者分发秘密，但每个参与者可以验证由秘密分发者分发的秘密，可以防止秘密分发者的欺骗，并且每个参与者能够验证其他合作者的欺骗。另外，每个参与者选取的子秘密可以复用，组秘密可以以任意顺序重构，同时该方案还能够抵抗合谋攻击。其安全性是基于Shamir门限方案和RSA密钥体制。在大整数分解困难离散对数难分解等问题的假设下，证明了提出的方案是安全的。

关键词: 二次剩余, 多秘密共享, 大整数分解, 离散对数分解