Lattices whose ideal lattice is Stone

Ideal of Lattice homomorphisms corresponding to the products of two arbitrary lattices and the lattice [2]

Abstract. Let L and M be two finite lattices. The ideal J(L,M) is a monomial ideal in a specific polynomial ring and whose minimal monomial generators correspond to lattice homomorphisms ϕ: L→M. This ideal is called the ideal of lattice homomorphism. In this paper, we study J(L,M) in the case that L is the product of two lattices L_1 and L_2 and M is the chain [2]. We first characterize the set...

Ideal Lattices of Lattices

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducib...

Density of Ideal Lattices

The security of many efficient cryptographic constructions, e.g. collision-resistant hash functions, digital signatures, identification schemes, and more recently public-key encryption has been proven assuming the hardness of worst-case computational problems in ideal lattices. These lattices correspond to ideals in the ring Z[ζ], where ζ is some fixed algebraic integer. Under the assumption th...

Ideal Lattices and NTRU

Identifying Ideal Lattices

Micciancio defined a generalization of cyclic lattices, called ideal lattices. These lattices can be used in cryptosystems to decrease the number of parameters necessary to describe a lattice by a square root, making them more efficient. He proves that the computational intractability of classic lattice problems for these lattices gives rise to provably secure one-way and collision-resistant ha...