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Two functions Δ and Δ_{$b$}, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ($d, n$) is the maximum diameter of convex polyhedra of dimension$d$with$n$faces of dimension$d$−1; similarly, Δ_{$b$}($d,n$) is the maximum diameter of bounded polyhedra of dimension$d$with$n$faces of dimension$d$−1. The diameter of a polyhedron$P$is the smallest integer$l$such that any two vertices of$P$can be joined by a path of$l$or fewer edges of$P$. It is shown that the bounded$d$-step conjecture, i.e. Δ_{$b$}$(d,2d)=d$, is true for$d$≤5. It is also shown that the general$d$-step conjecture, i.e. Δ($d, 2d$)≤$d$, of significance in linear programming, is false for$d$≥4. A number of other specific values and bounds for Δ and Δ_{$b$}are presented.

Anonymous veto networks (AV-nets), originally proposed by Hao and Zielinski (2006), are particularly lightweight protocols for evaluating a veto function in a peer-to-peer network such that anonymity of all protocol participants is preserved. Prior to this work, anonymity in all AV-nets from the literature relied on the decisional Diffie-Hellman (DDH) assumption and can thus be broken by (scalable) quantum computers. In order to defend against this threat, we propose two practical and completely lattice-based AV-nets. The first one is secure against passive and the second one is secure against active adversaries. We prove that anonymity of our AV-nets reduces to the ring learning with errors (RLWE) assumption. As such, our AV-nets are the first ones with post-quantum anonymity. We also provide performance benchmarks to demonstrate their practicality.

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