Critical sets of Latin squares based on autoparatopisms
Abstract
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.
Related
- Symmetric Quantum Latin Squares with Maximal Cardinality
- Orthogonal Quantum Latin Squares with Maximal Cardinality from Full-Rank Factorizations
- Resolvable quantum Latin squares with maximal cardinality
- Latin Squares with Few Transversals
- Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems
BibTeX
@misc{gonzalezregadera2026critical,
title = {{Critical sets of Latin squares based on autoparatopisms}},
author = {Manuel González-Regadera and Raúl M. Falcón and María Dolores Frau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21532},
eprint = {2609.21532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical sets of Latin squares based on autoparatopisms
AU - Manuel González-Regadera
AU - Raúl M. Falcón
AU - María Dolores Frau
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21532
UR - https://arxiv.org/abs/2609.21532
AB - 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.
ER -