A Discussion of a Cryptographical Scheme Based in F-Critical Sets of a Latin Square

Laura M Johnson ,Stephanie Perkins

doi:10.3390/math9030285

Abstract

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.

Highlights

Introduction and PreliminariesA Latin square of order n is an n×n array comprising of n distinct elements, such that each element occurs exactly once in each row and column [1,2]
A partial Latin square is an n×n array with all entries of the array belonging to the set {0, 1, . . ., n − 1}
The triples of Ent(T) and autotopisms in F associated with K are distributed to the k participants in the secret sharing scheme in such a way that when a group of t participants come together, the union of whose shares form an F-critical set of K

Summary

Introduction and Preliminaries

A Latin square of order n is an n×n array comprising of n distinct elements, such that each element occurs exactly once in each row and column [1,2]. The triples of Ent(T) and autotopisms in F associated with K are distributed to the k participants in the secret sharing scheme in such a way that when a group of t participants come together, the union of whose shares form an F-critical set of K They are able to combine their shares in order to find the key K. Notice that the removal of any F-orbit of Ent(Q1) will generate a partial Latin square q1=Ent(Q1) that is not uniquely completable to K This demonstrates that F-orbits of T\{T1, T11} under the autotopism θ3 form an F-critical set. Since order 2 Latin squares have a critical set of size 1, if a partial Latin square does not contain at least one entry in each intercalate I(i,j) ∈ K, P will not be uniquely completable to K.

Discussion of Example

Discussion

Summary of Findings

Conclusions