基于自平行同构的拉丁方临界集
Critical sets of Latin squares based on autoparatopisms
- Universidad de Sevilla（塞维利亚大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文针对秘密共享中临界集信息持有者缺失问题，提出利用拉丁方自平行同构群轨道计算临界集，并据此设计新秘密共享方案。
中文摘要 AI 辅助
在密码学中，拉丁方的临界集已被特别用于设计秘密共享方案。这些密码协议中的一个主要问题源于不同临界集所共有的信息片段的持有者缺失，因为这些持有者对于恢复秘密变得不可或缺。本文通过利用所考虑拉丁方的自平行同构群所描述的条目轨道来解决这一问题。为此，我们引入了更一般的问题，即计算在其自平行同构群中具有给定平行同构的拉丁方的临界集。这些临界集仅依赖于自平行同构的共轭类以及所考虑拉丁方的主类。基于这一事实，作为一个说明性示例，我们确定了阶数不超过六的拉丁方的自平行同构所关联的临界集的最小和最大尺寸。我们将此方法应用于设计一种新的秘密共享方案。
英文摘要
In cryptography, critical sets of Latin squares have particularly been implemented to design secret sharing schemes. A main problem in these cryptographic protocols arises from absent holders of pieces of information that are common to different critical sets, because they become indispensable to recover the secret. This paper solves this problem by making use of the orbits of entries described by the autoparatopism group of the Latin square under consideration. To this end, we introduce the more general problem of computing critical sets of Latin squares having a given paratopism in their autoparatopism group. These critical sets depend only on the conjugacy class of the autoparatopism and the main class of the Latin square under consideration. Based on this fact, as an illustrative example, we determine the smallest and largest sizes of critical sets associated with autoparatopisms of Latin squares of order up to six. We implement this approach in the design of a new secret sharing scheme.
