  For an integer $n\geq 5$, H. Strietz (1975) and L. Z\'adori (1986) proved
that the lattice Part$(n)$ of all partitions of $\{1,2,\dots,n\}$ is
four-generated. Developing L. Z\'adori'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.
