On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography

This paper introduced a condition called $\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic loops, satisfying the and possessing trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for loops. Isotopy-isomorphy loops was generally characterized. With $\mathcal{R}$-condition, it that positive integers $r$, $s$ $t$, if there is quasigroup order $3k$ with an inverse-cycle length $gcd(k,r+s+t)>1$, then exists $gcd\big(k(r+s+t), (r+s+t)^2\big)$. The procedure application such cryptography described explained, while feasibility illustrated sample values $k,r,s$ $t$.