Efficient Lattice-Based Cryptosystems with Key Dependent Message Security

Key-dependent message (KDM) security is of great research significance, to better analyse and solve the potential security problems in complex application scenarios. Most of the current KDM security schemes are based on traditional hard mathematical problems, where the public key and ciphertext are not compact enough, and make the ciphertext size grow linearly with the degree of the challenge functions. To solve the above problems and the inefficient ciphertext operation, the authors propose a compact lattice-based cryptosystem with a variant of the RLWE problem, which applies an invertible technique to obtain the RLWE* problem. It remains hard after the modification from the RLWE problem. Compared with the ACPS scheme, our scheme further expands the set of challenge functions based on the affine function of the secret key, and the size of public key and ciphertext is O˜(n), which is independent of the challenge functions. In addition, this scheme enjoys a high level of efficiency, the cost of encryption and decryption is only ploylog(n) bit operations per message symbol, and we also prove that our scheme is KDM-CPA secure under the RLWE* assumption.

A Note on Confusion between Linear and Affine Functions and the Generalized Forms of Gradient

Background: Arguably the most frequently used term in science, particularly in mathematics and statistics, is linear. However, confusion arises from the various meanings of linearity instructed in different levels of mathematical courses. The definition of linearity taught in high school is less correct than the one learned in a linear algebra class. The correlation coefficient of two quantitative variables is a numerical measure of the affinity, not only linearity, of two variables. However, every statistics book loosely says it is a measure of linear relationship. This clearly show that there is some confusion between use of the terms the linear function and affine function. Objective: This article aims at clarifying the confusion between use of the terms linear function and affine function. It also provides more generalized forms of the gradient in different branches of mathematics and show their equivalency. Materials and Methods: We have used the pure analytical deductive methods to proof the statements. Results: We have clearly presented that gradient is the measure of affinity, not just linearity. It becomes a special case of the derivative in calculus, of the least-squares estimate of the regression coefficient in statistics and matrix theory. The gradient can ­­­­be seen in terms of the inverse of the informative matrix in the most general setting of the linear model estimation. Conclusion: The article has been clearly written to show the distinction between the linear and affine functions in a concise and unambiguous manner. We hope that readers will clearly see various generalizations of the gradient and article itself would be a simple exposition, enlightening, and fun to read.

Modelagem Matemática e o tema Pecuária: investigando os custos para cercar uma propriedade

Resumo Neste artigo apresenta-se um recorte dos resultados de uma prática pedagógica, aplicada, durante os meses de julho e agosto, a um grupo voluntário de oito alunos do Ensino Médio de uma escola estadual do Município de Novo Mundo no estado do Mato Grosso. Desenvolvida à luz da Modelagem Matemática, a pesquisa teve como objetivo principal analisar as implicações do uso desta metodologia, no processo de estudo de diferentes funções com alunos do Ensino Médio, partindo do tema Pecuária. A pesquisa, de cunho qualitativo com análise descritiva, utilizou-se dos seguintes instrumentos de coleta de dados: gravação dos encontros em áudio e vídeo, caderno de campo da professora/pesquisadora, fórum interativo pelo Google sala de aula e entrevista. No decorrer da prática pedagógica, evidenciaram-se a motivação e a dedicação dos alunos em estudarem conceitos matemáticos a partir do contexto em que se encontram inseridos. Destaca-se também a capacidade de pesquisa, tanto no processo de coleta de dados como na matematização, dos alunos envolvidos. Durante o desenvolvimento da prática, construíram-se conceitos iniciais de função afim e função quadrática, bem como foram utilizados conceitos de porcentagem, regra de três, perímetro e área. Por fim, salienta-se que o tema Pecuária proporcionou discussões econômicas, sociais e culturais acerca dos diferentes tipos dessa atividade desenvolvida na comunidade, tornando os estudantes mais críticos e atentos ao contexto em que vivem. Palavras-chave: Modelagem Matemática. Função Afim. Função Quadrática. Pecuária. Ensino Médio. AbstractThis article presents the outcomes of a teaching practice applied from July to August 2020 to a volunteer group of eight Secondary School students in a state school in the municipality of Novo Mundo in Mato Grosso, Brazil. Based on Mathematical Modelling, the main purpose of this research was analyzing the implications of using this methodology in the process of studying distinct functions with Secondary School students, based on the theme of the Livestock Industry. This research was qualitative, used descriptive analysis, and collected data through audio recordings and video shootings of the meetings, teacher/researcher’s field journal, interactive forum through Google Classroom and interview. During this practice, students’ motivation and dedication became evident in studying mathematical concepts related to their environment. Furthermore, their researching capacity stood out, both in collecting the data and in mathematization. In the practice, initial concepts for affine and quadratic functions were developed, as well as concepts of percentage, rule of three, perimeter, and area. Finally, the theme of Livestock Industry promoted economic, social, and cultural discussions about the distinct types of this activity developed within the community, making students more aware and critical of the environment where they live. Keywords: Mathematical Modelling. Affine Function. Quadratic Function. Livestock Industry. Secondary School.

Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

The paper is concerned with the finite-time stabilization of a coupled PDE-ODE system describing the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is described by the wave equation with a variable coefficient which is an affine function of the curvilinear abscissa along the cable. Using several changes of variables, a backstepping transformation, and a finite-time stable second-order ODE for the dynamics of a conveniently chosen variable, we prove that a global finite-time stabilization occurs for the full system constituted of the platform and the cable. The kernel equations and the finite-time stable ODE are numerically solved in order to compute the nonlinear feedback law, and numerical simulations validating our finite-time stabilization approach are presented.

DISCRETE-TIME OPTIMAL EXECUTION UNDER A GENERALIZED PRICE IMPACT MODEL WITH MARKOVIAN EXOGENOUS ORDERS

This paper examines a discrete-time optimal execution problem with generalized price impact. Our main objective is to investigate the effect of price impact caused by aggregate random trade orders posed by small traders on the optimal execution strategy when orders of the small traders have a Markovian dependence. Our problem is formulated as a Markov decision process with state variables which include the last small traders’ aggregate orders. Over a finite horizon, a large trader with Constant Absolute Risk Aversion (CARA) von Neumann–Morgenstern (vN-M) utility function maximizes the expected utility from the final wealth. By applying the backward induction method of dynamic programming, we characterize the optimal execution strategy and optimal value function and conclude that the optimal execution strategy is a time-dependent affine function of three state variables. Moreover, numerical analysis prevails that the optimal execution strategy admits a “statistical arbitrage” via a round-trip trading, although our model considers a linear permanent price impact. The result differs from the previous prevailing one that a linear permanent price impact model precludes any price manipulation or arbitrage. Thus, considering a price impact caused by small traders’ orders with a Markovian dependence is significant.

Box-constrained optimization for minimax supervised learning

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

Agency Contracts under Maximum-Entropy

This article proposes the application of the maximum-entropy principle (MEP) to agency contracting (where a principal hires an agent to make decisions on their behalf) in situations where the principal and agent only have partial knowledge on the probability distribution of the output conditioned on the agent’s actions. The paper characterizes the second-best agency contract from a maximum entropy distribution (MED) obtained from applying the MEP to the agency situation consistently with the information available. We show that, with the minimum shared information about the output distribution for the agency relationship to take place, the second-best compensation contract is (a monotone transformation of) an increasing affine function of output. With additional information on the output distribution, the second-best optimal contracts can be more complex. The second-best contracts obtained theoretically from the MEP cover many compensation schemes observed in real agency relationships.

Calmness of Linear Constraint Systems under Structured Perturbations with an Application to the Path-Following Scheme

We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter. Such structured perturbations provide a unified framework for different parametric models in the literature, as block, directional and/or partial perturbations of both inequalities and equalities. We extend some recent results about calmness of the feasible set mapping and provide an application to the convergence of a certain path-following algorithmic scheme. We underline the fact that our formula for the calmness modulus depends only on the nominal data, which makes it computable in practice.

Sandwich Type Results for m-Convex Real Functions

We establish necessary and sufficient conditions allowing separation of pair of real functions by an m-convex and by an m-affine function. Some examples and a geometric interpretation of m-convexity of a function is exhibited, as well as a Jensen’s inequality for this kind of function.

Flexible Continuous-Time Modeling for Multi-Objective Day-Ahead Scheduling of CHP Units

Increasing applications of CHP units have turned the problem of finding the best optimization model into a significant subject for scholars. In this respect, this paper is aimed at driving a novel formulation to the multi-objective day-ahead scheduling of CHP units using Bernstein polynomials, which more optimally schedules power and heat generations as well as ramping trajectories. This procedure includes yielding an affine function that closely approximates real-time net-load and generation trajectories, which is demonstrated to have a superior performance to the conventional hourly day-ahead scheduling of CHP units based on discrete-time approximation. The problem of how to handle various objective functions by function space method is also addressed. The simulations conducted on the sample test systems, which consist of CHP systems, thermal and heat-only units, as well as thermal and electrical loads, show that the suggested multi-objective model can perfectly cover the total heat and electrical loads in terms of economic and environmental criteria. More importantly, the results indicate that the accuracy of the proposed approach renders cost saving of 1.67% and emission saving of 1.46% in comparison with the conventional hourly-based model, apart from leading to fewer ramping scarcities in real-time operations.