Abstract:

In the reconstruction of[(t,n)]secret sharing, any more than[t]participants can recover a secret. But in a real application, it only needs any[ t ]participants to recover the secret when [kt≤k≤n]participants recover the secret, even if [k-t] participants do not provide shares. In a [(t,k,n)] tightly coupled secret sharing scheme, when [k ]participants seen as a group recover the secret, each one in the group needs to participate in the reconstruction of the secret. Any[k-1] participants can not get any information about the secret. In known tightly coupled secret sharing schemes, no matter what scheme based on Chinese Remainder Theorem（CRT） or based on Lagrange interpolating polynomial, they all have efficiency problems that information ratio does not equal 1. This paper generalizes CRT to a polynomial ring over finite fields and presents an ideal tightly coupled secret sharing scheme based on CRT over a polynomial ring.

Key words: secret sharing, Chinese Remainder Theorem（CRT）, tightly coupled secret sharing scheme, polynomial ring

摘要：

在[(t,n)]门限秘密共享恢复过程中，任意多于[t]个的参与者可以恢复得到秘密。但是在实际的应用过程中，当参与者人数为[k(t≤k≤n)]时，只需获得[t]个参与者的份额（share）即可恢复秘密，即使其中的[k-t]个参与者不提供子份额。[(t,k,n)]紧耦合秘密共享是指在[(t,n)]门限秘密共享中，当参与者人数为[k]时，[k]个参与者作为一个整体，其中的每个人均参与到秘密恢复中，任意的[k-1]个参与者无法获取秘密的任何信息。在现有的紧耦合秘密共享方案中，无论是基于中国剩余定理的紧耦合秘密共享方案或者是基于拉格朗日插值多项式的紧耦合秘密共享方案，均存在信息率不为1，导致效率低下的缺陷。将中国剩余定理推广到有限域上的多项式环上，利用多项式环上的中国剩余定理构造出理想型[(t,k,n)]紧耦合秘密共享方案。

关键词: 秘密共享, 中国剩余定理, 紧耦合秘密共享, 多项式环