Secret sharing scheme using Hypercubes

Reference No.	2025a044
Type/Category	Grant for Supporting the Advancement of Female Researchers- Short-term Visiting Researcher
Title of Research Project	Secret sharing scheme using Hypercubes
Principal Investigator	Tomoko Adachi（Shizuoka Institute of Science and Technology・Professor）
Research Period	September 8,2025. - September 12,2025. February 24,2026. - February 27,2026.
Keyword(s) of Research Fields	Secret sharing scheme, Hypercubes
Abstract for Research Report	A secret sharing scheme is one of the cryptography, and (k, n) threshold scheme by Shamir in 1979 is the most famous in secret sharing schemes. Dawson et al. (1993) proposed a secret sharing scheme using orthogonal arrays. However, as it is, it is hard to use as cryptography and it is complicated to construct orthogonal arrays. MOLS is a set of the mutually orthogonal Latin squares. MOLS is a kind of orthogonal array and it is simple among orthogonal arrays. Nuida and Adachi (2024) researched the characteristics of a special form of MOLS that is easy to handle as a secret sharing scheme and gave an upper bound on the parameters. This work was supported by Institute of Mathematics for Industry, Joint Usage/Research Center in Kyushu University (FY2023 Short-term Visiting Researcher “Cryptography using Latin squares” (2023a016)). Latin squares are two-dimensional, and the higher-dimensional version of Latin squares are hypercubes. In this study, we deal with a set of the mutually orthogonal hypercubes, and it is also a kind of orthogonal array. In hypercubes, there is an indicator called type. A k-dimensional hypercube has types j (j=1, 2, …, k-1), and is called a (k,j)-cube. Lu and Adachi (2020) constructed a set of the mutually orthogonal (3, 2)-cubes with the order q=p^2 (p: prime). Adachi and Nuida have constructed a set of the mutually orthogonal (k, k-1)-cubes with the order q=p^h (p: prime), and talk at JSIAM in March 2025. This work was supported by Institute of Mathematics for Industry, Joint Usage/Research Center in Kyushu University (FY2024 Short-term Visiting Researcher “Secret sharing scheme using orthogonal arrays” (2024a041)). That is the set of hypercubes of the same type. In this study, we consider a set of hypercubes of different types. As a research idea, based on the MSD code construction of Rawat et al. (2018), I will construct Vandermond-liked matrices. Combining those matrices with the results of Either (2012), I will construct a set of hypercubes of different types. As a promising outcome, a secret sharing scheme with perfect security can be constructed by hypercubes.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Tomoko Adachi（Shizuoka Institute of Science and Technology・Professor） Yujie Gu（Faculty of Information Science and Electrical Engineering Department of Informatics, Kyushu University・Associate Professor）