We propose the signature scheme Hawk, a concrete instantiation of proposals to use Lattice Isomorphism Problem (LIP) as foundation for cryptography that focuses on simplicity. This simplicity stems from LIP, which allows lattices such $$\mathbb Z^n$$ , leading algorithms with no floats, rejection sampling, and compact precomputed distributions. Such design features are desirable constrained dev...

We prove the following result: Theorem. Every algebraic distributive lattice D with at most א1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the א1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. F...

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system ` is algebraized by a variety K, then ` has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to b...

Let G be a locally compact topological group. A lattice in G is a discrete subgroup Γ such that Γ\G carries a finite G–invariant measure, and Γ is uniform or cocompact if Γ\G is compact. Lattices in Lie groups have been well-studied. See, for example, Raghunathan [48], and for open problems the section on “Lattices in Lie groups” in this wiki. Much less is known about lattices in other locally ...

We prove the following result: Theorem. Every algebraic distributive lattice D with at most א1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the א1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. F...