Efficient integer span program for hierarchical threshold access structure
Introduction
Integer span program (ISP), i.e., monotone span program (MSP) over , is a very useful primitive, since it can be used to construct black-box secret sharing scheme (BBSSS) [12], [11], [5], [7] and linear integer secret sharing scheme (LISSS) [8]. The relations between ISP and BBSSS were given by Cramer and Fehr [5]. There are many applications of BBSSS, such as threshold RSA [12], [11], black-box ring multiparty computation [6], and zero-knowledge [4]. The relations between ISP and LISSS were given by Damgård and Thorbek [8]. LISSS can be used to build secure distributed protocols for exponentiation in any group [8], [10] and zero-knowledge [9].
ISP was introduced by Cramer and Fehr [5] to construct BBSSS. Cramer and Fehr argued that there exists an ISP for every access structure. But the existing methods for constructing corresponding ISPs are inefficient. The efficiency of ISPs and secret sharing schemes is usually measured by the ratio between the number of the rows of the monotone span matrix and the number of participants. Efficient construction of ISPs, which have the ratios being polynomial on the number of participants is of interested to this work. The efficiency of ISP is very important because only efficient ISPs have the real applications in practice. Nowadays, only efficient ISP realizing threshold access structure was constructed by Desmedt and Frankel [11], and Cramer and Fehr [5], especially, Cramer and Fehrʼs construction is optimal. But efficient ISPs with the realization of other access structures are yet to be proposed. Efficient MSPs realizing non-threshold access structures are of great importance and have attracted a great amount of research interest. There are several reports of research on efficient MSPs over finite fields, including [2], [3], [13], [14], [15], [16], [18], [19], [20], [21]. But the construction of efficient ISPs for non-threshold access structures is still missing.
In this paper, an efficient ISP realizing hierarchical threshold access structure is constructed, which can be used to construct efficient BBSSS and efficient LISSS for hierarchical threshold access structure. We extend the approach that was proposed by Cramer and Fehr [5] to introduce the concept of weak MSPs, which can be used to construct ISP for hierarchical threshold access structure. One of the main contributions of this work is that a method of constructing efficient ISPs by using MSPs over finite fields [20] and rational field is proposed. The proposed method may be applied to construct efficient ISPs for other access structures.
The paper is organized as follows. Section 2 gives an introduction on related notations and definitions about ISP, as well as the theory of Birkhoff interpolation. Section 3 introduces the definitions about weak MSPs, and identifies the relations among these definitions. The important result is presented in Section 4, an efficient ISP realizing hierarchical threshold is constructed by using MSPs over finite fields [20] and rational fields.
Section snippets
The notions about integer span program
The following notation convention is adopted in this work: Vectors are denoted by boldface letters, while their components are denoted with the corresponding italic-type indexed letter. We start by defining an access structure which is a collection of authorized sets that can reconstruct the secret.
Definition 1 Access structure Let be the set of participants. A collection is monotone if and imply that . An access structure is a monotone collection of nonempty subsets of . Sets in Γ are
Weak monotone span program
This section presents an introduction on some new definitions, which extend the definition of MSP introduced by Cramer and Fehr [5] followed by the relations among these definitions.
Definition 6 is a I-weak MSP for Γ, if there exist such that for all the following holds. If , then . If , then there exists with .I-weak MSP
If , is called a I-weak ISP for Γ.
Definition 7 is a II-weak MSP for Γ, if there exists such that for all theII-weak MSP
Integer span program for hierarchical threshold access structures
This section presents a way of constructing efficient ISP that realizes . From the above section, it is known that if it is possible to construct coprime II-weak MSPs for over R, then it is possible to construct an ISP for . We first give an approach to construct a weak MSP for over .
Conclusions
This work proposed an approach to construct efficient ISPs for hierarchical threshold access structures. The proposed approach can also be used to construct efficient ISPs for other access structures. However, the size of our construction of ISP is not minimal, the issue of constructing ISPs for hierarchical threshold structure with minimal sizes remains to be investigated.
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Research supported in part by the Guangzhou Research Infrastructure Development Fund (No. 2012224-12), the Guangdong Nature Science Fund (No. S2012030006242), the Guangdong-CAS Cooperation Fund (No. 2011A090100003), the Guangdong Emerging Industry Key Technology R&D Fund (No. 2011A010801007), the Guangzhou Zhujiang Science and Technology Future Fellow Fund (No. 2011J2200089), and the MOE-China Mobile Research Fund (No. MCM20121051).