A Latin square autotopism secret sharing scheme

Stones, Rebecca J.; Su, Ming; Liu, Xiaoguang; Wang, Gang; Lin, Sheng Hsien

doi:10.1007/s10623-015-0123-1

Cited by 15 publications

(12 citation statements)

References 15 publications

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“…Diagonally cyclic equi-n-squares are equivalent to equi-n-squares that admit an n-cycle automorphism. Latin squares that admit automorphisms are used in secret sharing [32], and the Latin square could easily be replaced by an equi-n-square, and interpreted as a graph decomposition [7] (see [30] for a broad and detailed treatment). In abstract algebra, diagonally cyclic equi-n-squares correspond to n-element magmas (sometimes called groupoids) that admit n-cycle automorphisms.…”

Section: Discussionmentioning

confidence: 99%

Balanced Equi-$n$-Squares

Akbari¹,

Marbach²,

Stones³

et al. 2020

Electron. J. Combin.

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We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \times n$ matrix containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \times (n/d)$ subrectangles. In graph theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n/d,n/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \in \mathbb{Z}_n$, which correspond to cyclic such decompositions of $K_{n,n}$ (and thus $\alpha$-labellings). We identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \times (n/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \in \{1,2,3\}$. Along the way, we identify a bijection between $\alpha$-labellings of $d$-regular bipartite graphs and what we call $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\alpha$-labellings.

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Section: Discussionmentioning

confidence: 99%

Balanced Equi-$n$-Squares

Akbari¹,

Marbach²,

Stones³

et al. 2020

Electron. J. Combin.

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“…An early Latin-square secret-sharing scheme was given by Cooper, Donovan, and Seberry [23] (see also [24]) based on critical sets in 1994, but it was subsequently harshly criticized [25]- [27]. Afterward, changing the secret from the Latin square to one of its autotopisms was suggested in [28] and developed in [29]. Compared with critical sets, autotopisms of Latin squares are better understood [30]- [33] and are more practical to work with (e.g., determining if a partial Latin square has a completion as required by the original Latin square secret-sharing scheme is NP-complete [34]).…”

Section: Secret Sharing Schemesmentioning

confidence: 99%

“…This paper applies the autotopism-based scheme LASS [29] to the proposed dispersal scheme, for a high-level of key protection. We introduce the basic notions and properties of LASS, as well as why we use it in Section III.…”

Section: Secret Sharing Schemesmentioning

confidence: 99%

“…With the corresponding autotopism θ , a contour C constructs a Latin square L; whereas, with a faulty autotopism θ , a contour C is unlikely to construct a Latin square L. Thus, during reconstruction we can verify that whether the autotopism θ reconstructed from downloaded secret shares is correct or not. Verifiability is one of the primary benefits of LASS over both SSSS and Blakey's secret sharing scheme [29].…”

Section: ) Verifiabilitymentioning

confidence: 99%

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Gecko: A Resilient Dispersal Scheme for Multi-Cloud Storage

et al. 2019

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We have entered an era where copious amounts of sensitive data are being stored in the cloud. To meet the rising privacy, reliability, and verifiability needs, we propose Gecko, a multi-cloud dispersal scheme where: (a) the key used to encrypt the data file is the secret in a Latin-square-autotopism secretsharing scheme, (b) data files and encryption keys are dispersed separately to multiple clouds, and (c) a blockchain-based integrity-check protocol is devised to pinpoint faulty data. Gecko enables fast and thorough key renewal: when a portion of the key (the secret) is leaked, we replace all shares of the partially-leaked secret without replacing the secret itself; this immediately resists targeted attack to certain file without reencrypting the data file itself. Key renewal is further accelerated by the blockchain-based integrity check. We evaluate Gecko theoretically and experimentally against the traditional AONT-RS dispersal scheme, drawing two conclusions: 1) Gecko admits powerful key renewal and identification of damaged data, with a minor transfer overhead; and 2) Gecko performs key renewal three to five times faster than AONT-RS hybrid-slice renewal (the closest thing AONT-RS has to key renewal). INDEX TERMS Blockchain, data recovery, dispersal scheme, integrity check, Latin square, multi-cloud.

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“…Recent advances about the sets of autotopisms, automorphisms and autoparatopisms of Latin squares are exposed, for instance, in [145][146][147][148]. We note in particular on the implementation of autotopisms of Latin squares into the design of authentication schemes [149], secret sharing schemes [150,151] and cryptographic transformations [152] in Cryptography. In addition, an implementation of autotopisms and paratopisms of Latin squares into the design of new graph colouring games [153,154] is being currently developed.…”

Section: Quasigroups Latin Squares and Related Structuresmentioning

confidence: 99%

A Historical Perspective of the Theory of Isotopisms

2018

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In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

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A Latin square autotopism secret sharing scheme

Cited by 15 publications

References 15 publications

Balanced Equi-$n$-Squares

Balanced Equi-$n$-Squares

Gecko: A Resilient Dispersal Scheme for Multi-Cloud Storage

A Historical Perspective of the Theory of Isotopisms

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