On quasigroups with few associative triples
The paper focuses on the idea of employing quasigroups in designs of hash functions. It turns out that quasigroups with a relatively small number of associative triples are more suitable as building blocks in these designs. Given a quasigroup \(Q\) of order \(n\), one is interested in the total number \(a(Q)\) of associative triples in \(Q\). The main result of the paper is a lower bound for the number of associative triples: \(a(Q)\geq 2n-i(Q)\), where \(i(Q)\) is the cardinality of the set of all idempotent elements of the quasigroup \(Q\). An upper bound for \(a(Q)\) is investigated as well and constructions of quasigroup sequences \(\{Q_n\}_{n=1}^\infty\) are described for which \(a(Q_n)=|Q_n|^t\) for \(t=\frac{\ln 15}{\ln 5},\frac{\ln 19}{\ln 6}\) and \(\frac{\ln 19}{\ln 7}\).
scientific article; zbMATH DE number 6037532