Generating Boolean lattices by few elements and exchanging session keys

Abstract

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