A Factorization Algorithm for G-Algebras and Applications

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is any $G$-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gr\"obner basis algorithm for $G$-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from $\mathcal{G}$. Additionally, it is possible to include inequality constraints for ideals in the input

VDOO: A Short, Fast, Post-Quantum Multivariate Digital Signature Scheme

Hard lattice problems are predominant in constructing post-quantum cryptosystems. However, we need to continue developing post-quantum cryptosystems based on other quantum hard problems to prevent a complete collapse of post-quantum cryptography due to a sudden breakthrough in solving hard lattice problems. Solving large multivariate quadratic systems is one such quantum hard problem. Unbalanced Oil-Vinegar is a signature scheme based on the hardness of solving multivariate equations. In this work, we present a post-quantum digital signature algorithm VDOO (Vinegar-Diagonal-Oil-Oil) based on solving multivariate equations. We introduce a new layer called the diagonal layer over the oil-vinegar-based signature scheme Rainbow. This layer helps to improve the security of our scheme without increasing the parameters considerably. Due to this modification, the complexity of the main computational bottleneck of multivariate quadratic systems i.e. the Gaussian elimination reduces significantly. Thus making our scheme one of the fastest multivariate quadratic signature schemes. Further, we show that our carefully chosen parameters can resist all existing state-of-the-art attacks. The signature sizes of our scheme for the National Institute of Standards and Technology's security level of I, III, and V are 96, 226, and 316 bytes, respectively. This is the smallest signature size among all known post-quantum signature schemes of similar security

The representation theory of Hc(Sn~Z/lZ)

In this thesis we examine various aspects of the representation theory of the restricted rational Cherednik algebra Hc(Sn~Z/lZ). We prove several multiplicity results for graded modules, in particular for modules over an algebra that admits a triangular decomposition. This includes the algebra Hc(Sn~Z/lZ). Furthermore, if the projective covers admit a radical preserving filtration, we show that we can calculate the multiplicities of the simple modules inside the radical layers of the projective covers. We give an explicit presentation of the centre of the restricted rational Cherednik algebra Hc(Sn~Z/lZ) for suitably generic c. This is done by first deriving a presentation of the centre of Hc(Sn) using Schubert cells, then extending this to the more general wreath product group using the action of the cyclic group Z/lZ. This presentation is given for each block of the centre. Using the bijection between irreducible representations of Sn~Z/lZ and l-multipartitions of n, we prove that this explicit presentation can be read directly from the l-multipartition of n

An introduction to constructive algebraic analysis and its applications

This text is an extension of lectures notes I prepared for les Journées Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal analysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts. We are quickly led to Björk's books which, at first glance, may look difficult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty of the existing references than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. To a very small extent, these lectures notes were planned to fill this gap, at least for the basic ideas of algebraic analysis. Since, we can only teach well what we have clearly understood, I have chosen to focus on my work on the constructive aspects of algebraic analysis and its applications.Ce texte est une extension des notes de cours que j'ai préparés pour les les Journées Nationales de Calcul Formel qui ont eu lieu au CIRM, Luminy (France) du 3 au 7 Mai 2010. Le but principal de ce cours était d'introduire la communauté française du calcul formel à l'analyse algébrique constructive, et particulièrement à la théorie des D-modules algébriques, à ses applications à la théorie mathématique des systèmes et à ses implantations dans des logiciels de calcul formel tels que Maple ou GAP4. Parce que l'analyse algébrique est une théorie mathématique qui utilise différentes techniques venant de la théorie des modules, de l'algèbre homologique, de la théorie des faisceaux, de la géométrie algébrique et de l'analyse microlocale, il peut être difficile d'entrer dans ce domaine, nouveau et fascinant, des mathématiques. En effet, il existe peu de textes introductifs. Nous sommes rapidement conduits aux livres de Björk qui, à première vue, peuvent sembler difficiles aux membres de la communauté de calcul formel et aux mathématiciens appliqués. Je pense que le problème vient moins de la difficulté technique de la littérature existante que du manque d'introductions pédagogiques qui donnent une idée globale du domaine, montrent quels types de résultats et d'applications on peut attendre et qui développent les différents calculs à mener sur des exemples explicites. A leur humble niveau, ces notes de cours ont pour but de combler ce manque, tout du moins en ce qui concerne les idées de base de l'analyse algébrique. Puisque l'on ne peut enseigner bien que les choses que l'on a bien comprises, j'ai choisi de restreindre cette introduction à mes travaux sur les aspects constructifs de l'analyse algébrique et sur ses applications

Polygraphs: From Rewriting to Higher Categories

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half

VDOO: A Short, Fast, Post-Quantum Multivariate Digital Signature Scheme

Hard lattice problems are predominant in constructing post-quantum cryptosystems. However, we need to continue developing post-quantum cryptosystems based on other quantum hard problems to prevent a complete collapse of post-quantum cryptography due to a sudden breakthrough in solving hard lattice problems. Solving large multivariate quadratic systems is one such quantum hard problem. Unbalanced Oil-Vinegar is a signature scheme based on the hardness of solving multivariate equations. In this work, we present a post-quantum digital signature algorithm VDOO (Vinegar-Diagonal-Oil-Oil) based on solving multivariate equations. We introduce a new layer called the diagonal layer over the oil-vinegar-based signature scheme Rainbow. This layer helps to improve the security of our scheme without increasing the parameters considerably. Due to this modification, the complexity of the main computational bottleneck of multivariate quadratic systems i.e. the Gaussian elimination reduces significantly. Thus making our scheme one of the fastest multivariate quadratic signature schemes. Further, we show that our carefully chosen parameters can resist all existing state-of-the-art attacks. The signature sizes of our scheme for the National Institute of Standards and Technology\u27s security level of I, III, and V are 96, 226, and 316 bytes, respectively. This is the smallest signature size among all known post-quantum signature schemes of similar security

Mathematical Logic and Its Applications 2020

The issue "Mathematical Logic and Its Applications 2020" contains articles related to the following three directions: Descriptive Set Theory (3 articles). Solutions for long-standing problems, including those of A. Tarski and H. Friedman, are presented. Exact combinatorial optimization algorithms, in which the complexity relative to the source data is characterized by a low, or even first degree, polynomial (1 article). III. Applications of mathematical logic and the theory of algorithms (2 articles). The first article deals with the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability; for these purposes, non-standard analysis is used. The second article provides a quantitative description of the balance and adaptive resource of a human. Submissions are invited for the next issue "Mathematical Logic and Its Applications 2021