一类理想的存取结构的构造

摘要

The construction of ideal access structure has an important role for designing secret sharing scheme with high information rate. The access structures corresponding to Shamir(k,n)’s type scheme( different from Shamir’s threshold type scheme) are ideal, but how to get these access structures which are not mutually isomorphic is a problem needed to be solved. First of all, the deifnition that the tracks are mutually equivalent is proposed, and then the problem for judging whether two minimal access structures are isomorphic in Shamir(k,n)’s type scheme is converted into the problem for judging whether their corresponding tracks are equivalent. This paper designs an algorithm which can be used to calculate all minimal privileged arrays that exist in the Shamir(k,n)’s type scheme and can be used to calculate all the tracks existing in the Shamir(k,n)’s type scheme that are not mutually equivalent. So this paper perfectly solves the problem that how to construct all ideal access structures which are not mutually isomorphic in the Shamir(k,n)’s type scheme. Particularly, this paper gives all the minimal privileged arrays with 7 participants in ifnite ifeldF13 and obtains all the tracks that are not mutually equivalent, and thus gives all the ideal minimal access structures with 7 participants that are not mutually isomorphic by the above judgment.%构造理想的存取结构对于设计信息率高的秘密共享方案具有重要作用。Shamir(k,n)型方案（区别于Shamir门限方案）对应的存取结构是理想的，但如何求出这类方案对应的互不同构的存取结构是一个需要解决的问题。文章首先提出Shamir(k,n)型方案中两组迹等价的概念，然后将Shamir(k,n)型方案中极小存取结构的同构的判定转化为对应的两组迹的等价问题。文章进而给出了Shamir(k,n)型方案中求极小特权数组的一个算法，利用这个算法可以求出Shamir(k,n)型方案中所有互不等价的迹，从而在理论上完满地解决了Shamir(k,n)型方案中互不同构的理想的存取结构的构造问题。特别地，文章给出有限域F13中当有7个参与者时的所有极小特权数组，并得到了互不等价的迹，进而利用文中的判定方法给出了当有7个参与者时， Shamir(k,n)型方案的所有互不同构的理想的极小存取结构。