(k, n)-Consecutive access structures

IF 1.2 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-05-24 DOI:10.1007/s10623-025-01651-7

Javier Herranz, Germán Sáez

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引用次数: 0

Abstract

We consider access structures over a set of n participants, defined by a parameter k with \(1 \le k \le n\) in the following way: a subset is authorized if it contains at least k consecutive participants. Depending on whether we consider the participants placed in a line (that is, participant 1 is not next to participant n) or in a circle, we obtain two different families, that we call (k, n)-line-consecutive and (k, n)-circle-consecutive access structures, respectively. Such access structures can appear in real-life situations involving distributed cryptography, which makes it more interesting to look for the best secret sharing schemes that can realize them. For both families, we characterize which are the configurations (k, n) that admit ideal secret sharing schemes. For the non-ideal (k, n)-consecutive access structures, we give both upper and lower bounds on the information ratio of the best secret sharing schemes that can realize them. Some of these bounds are obtained after proving relations between the information ratios of access structures in the two considered families.

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（k, n）-连续存取结构

我们考虑由参数k用\(1 \le k \le n\)定义的n个参与者集合上的访问结构：如果一个子集包含至少k个连续的参与者，则该子集被授权。根据我们考虑的参与者是放在一条直线上（即参与者1不在参与者n旁边）还是放在一个圆上，我们得到了两个不同的族，我们分别称之为(k, n)-线连续访问结构和(k, n)-圆连续访问结构。这种访问结构可能出现在涉及分布式加密的实际情况中，这使得寻找能够实现它们的最佳秘密共享方案变得更加有趣。对于这两个族，我们描述了哪些构型（k, n）允许理想的秘密共享方案。对于非理想（k, n）连续访问结构，给出了实现它们的最佳秘密共享方案的信息比的上界和下界。在证明了两个考虑族的访问结构的信息比率之间的关系之后，得到了其中的一些界限。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.