基于Shamir秘密共享的密钥分发与恢复算法

在经典的Shamir秘密共享方案中，秘密分发者把秘密 分为 个影子秘密并分发给持有者；其中任意不少于t个影子秘密均能恢复秘密s，少于t个影子秘密则得不到秘密 的任何信息。现实的秘密恢复过程中可能存在超过t个参与者的情形，因此，在Shamir的秘密共享方案基础上讨论此种情形下秘密共享问题，通过引入影子秘密的线性组合——拉格朗日因子来恢复秘密，并进一步将其扩展为一个多秘密共享方案。理论分析与仿真实验表明：改进算法在同样复杂度条件下既保证影子秘密的安全，又能阻止欺骗者得到秘密，提高了整体安全性。

Key distribution and recovery algorithm based on Shamir's secret sharing

In Shamir's secret sharing scheme, the dealer divided the secret s into n shadows and distributed it to share-holders in such a way that any t or more than t shadows can recover this secret, while fewer than t shadows cannot obtain any information about the secret s. During the actual secret recovery process, there exist other cases with more than t par-ticipants. The case of secret sharing problem was discussed based on Shamir's secret sharing scheme and reconstructs the secret by introducing a linear combination of shadows—Lagrange factor. Then, the improved algorithm of key distribu-tion and recovery was proposed and extended to a multi-secret sharing scheme. Theoretical analysis and simulation show that the improved scheme improves its security under the same conditions of complexity.