Four-generated direct powers of partition lattices and authentication

For an integer $n\geq 5$, H. Strietz (1975) and L. Zádori (1986) proved that the lattice $\operatorname{Part}n$ of all partitions of $\{1,2,\dots,n\}$ is four-generated. Developing L. Zádori's particularly elegant construction further, we prove that even the $k$-th direct power $\operatorname{Part}n^k$ of $\operatorname{Part}n$ is four-generated for many but only finitely many exponents $k$. E.g., $\operatorname{Part}{100}^k$ is four-generated for every $k\leq 3\cdot 10^{89}$, and it has a four-element generating set that is not an antichain for every $k\leq 1.4\cdot 10^{34}$. In connection with these results, we outline a protocol how to use these lattices in authentication and secret key cryptography.