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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 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 Part$(n)^k$ of Part$(n)$ is four-generated for many but only finitely many exponents $k$. E.g., Part$(n)^k$ 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.