Latin squares associated to principal autotopisms of long cycles. Application in Cryptography

Abstract

Fixed a principal isotopism $\Theta=(\alpha,\beta,\epsilon)\in S_n^3$, where $S_n$ is the symmetric group of the set $N=\{0,1,...,n-1\}$, we are going to study in this paper the number $\Delta(\Theta)$ of Latin squares having $\Theta$ as a principal autotopism. As an application in Cryptography, we use it in the construction of secret sharing schemes based in $\mathfrak{F}$-critical sets of Latin squares.