Isochronous Gaussian Sampling and its applying to the Falcon Signature Scheme

This paper describes a block framework for generating discrete Gaussian components with arbitrary center and standard deviation. Simplicity makes it easy to implement secure, portable, effective, and time-resistant. This sampler is a good candidate for any sample analysis. Most importantly, it was recently introduced into the Falcon Signature Scheme. Gaussian sampling over integers is a central element of algebraic lattice cryptography, which is difficult to perform efficiently and safely. Given the vast number of uses of sampling processes, it is important to select Gaussian samplers that are effective provably secure, resistant to transient attacks, and generally easy to implement. Sampling with other distributions, other than Gaussian, is yet an open question.

Alpha Ideals and the Space of Prime Alpha Ideals in Universal Algebras

The purpose of this paper is to study α -ideals in a more general context, in universal algebras having a constant 0 . Several characterizations are obtained for an ideal I of an algebra A to be an α -ideal. It is shown that the class of all α -ideals of an algebra A forms an algebraic lattice. Prime α -ideals and several related properties are investigated. Some properties of the spectral space of prime α -ideals equipped with the hull-kernel topology are derived.

THE CONVERGENCE ON ALGEBRAIC LATTICE NORMED SPACES

The multiplicative convergence on Riesz algebras introduced and investigated with respect to norm and order convergences. If X is a Riesz space and E is a Riesz algebra then the vector norm μ:X→E_+ can be considered. Then (X,μ,E) is called algebraic lattice normed spaces. A net (x_α )_(α∈A) in an (X,μ,E) is said to be multiplicative μ-convergent to x∈X if μ(x_α-x)∙u□(→┴o ) 0 holds for all u∈E_+. In this paper, the general properties of this convergence are studied.

m-Algebraic lattices in formal concept analysis

The notion of an m-algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m-approximable concept on each context. The m-approximable concept lattice derived from the notion is an m-algebraic lattice, and conversely, every m-algebraic lattice is isomorphic to an m-approximable concept lattice of some context. Morphisms on m-algebraic lattices and those on contexts are provided, called m-continuous functions and m-approximable morphisms, respectively. We establish a categorical equivalence between LATm, the category of m-algebraic lattices and m-continuous functions, and CXTm, the category of contexts and mapproximable morphisms.We prove that LATm is cartesian closed whenevermis regular and m > 2. By the equivalence of LATm and CXTm, we obtain that CXTm is also cartesian closed under same circumstances. The notions of a concept, an approximable concept, and a weak approximable concept are showed to be special cases of that of an m-approximable concept.

REPRESENTING REGULAR PSEUDOCOMPLEMENTED KLEENE ALGEBRAS BY TOLERANCE-BASED ROUGH SETS

We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.

A new view of relationship between atomic posets and complete (algebraic) lattices

In the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.

Irredundant Decomposition of Algebras into One-Dimensional Factors

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.