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Let Sp($k$) denote the number of the $\lfloor k/2\rfloor$-element subsets ofa finite $k$-element set. We prove that the least size of a generating subsetof the Boolean lattice with $n$ atoms (or, equivalently, the powerset latticeof an $n$-element set) is the least number $k$ such that $n\leq$ Sp($k$). Basedon this fact and our 2021 protocol based on equivalence lattices, we outline acryptographic protocol for exchanging session keys, that is, frequentlychanging secondary keys. In the present paper, which belongs mainly to latticetheory, we do not elaborate and prove those details of this protocol thatmodern cryptology would require to guarantee security; the security of theprotocol relies on heuristic considerations. However, as a first step, we provethat if an eavesdropper could break every instance of an easier protocol inpolynomial time, then P would equal NP. As a byproduct, it turns out that ineach nontrivial finite lattice that has a prime filter, in particular, in eachnontrivial finite Boolean lattice, the solvability of systems of equations withconstant-free left sides but constant right sides is an NP-complete problem.

Generating Boolean lattices by few elements and exchanging session keys