MATRIX POLYNOMIAL AND LAMBDA MATRIX

A matrix x is a solvent of the matrix polynomials. f(x)=A o xm +…..+Am if f(x) = 0.Where A o, x, f are square matrices. In this paper we develop the linear algebra of matrix polynomials and solvents. we define division and interpolation the properties of matrices and define and study the existence of a complete set of solvents. we study the relation between the matrix polynomial problems and Theorems and lambda matrix problem, Which is to find scalar λ such that A o λm+A1 λm-1+ ……. +Am is singular. In a future paper we extend Cayley Hamilton Theorem for calculating of scalar polynomials to matrix polynomials. And lambda matrix for column and row Hermite calculating of lambda matrix

A generalization of the Mignotte's scheme over Euclidean domains and applications to secret image sharing

Secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. In 1979, the pioneer construction on this area was given by Shamir and Blakley independently. After these initial studies, Asmuth-Bloom and Mignotte have proposed a different $(k,n)$ threshold modular secret sharing scheme by using the Chinese remainder theorem. In this study, we explore the generalization of Mignotte's scheme to Euclidean domains for which we obtain some promising results. Next, we propose new algorithms to construct threshold secret image sharing schemes by using Mignotte's scheme over polynomial rings. Finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security

On the security of HMFEv

In this short report, we study the security of the new multivariate signature scheme HMFEv proposed at PQCrypto 2017

Interpolation-based condensation of algebraic semi-discrete models with frequency response application

AbstractCondensation model reduction theory, a method of degree-of-freedom-elimination for semi-discrete system models with response-prediction fidelity in the retained degrees-of-freedom (DOF), is applied to algebraic semi-discrete models. The condensation process makes use of an interpolation over a user-chosen subset, denoted as a “window,” of the set of continuous-independent-variable values. The window's “size” and “location,” as well as the accuracy of the method within the window, are hence controllable by the user. (There is a computational-cost versus accuracy/windowsize tradeoff for a given DOF reduction, as would be expected.) One target of this capability is the DOF reduction of spatially-discrete, continuous-time-transformed (Fourier, Laplace, etc.) linear system models, for which the resulting semi-discrete model has frequency as the continuous independent variable. The window would then correspond to a selected frequency range, (a region of the complex frequency plane in the most general case). Another target of this capability is the DOF reduction of nonlinear, path-independent static or quasistatic models, for which the window corresponds to a region of the reduced-DOF-model solution space itself. As a demonstration, the method is applied to the frequency response of a non-periodic linear elastic laminate over a rectangular window in the complex frequency plane. It is seen that the frequency-response predicted by the reduced-DOF model at each of various values within the window, as well as the eigenvalues predicted by the reduced-DOF model within the window, agree well with the corresponding predictions of the original, full-DOF model

PIE: pp-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption

A large part of current research in homomorphic encryption (HE) aims towards making HE practical for real-world applications. In any practical HE, an important issue is to convert the application data (type) to the data type suitable for the HE. The main purpose of this work is to investigate an efficient HE-compatible encoding method that is generic, and can be easily adapted to apply to the HE schemes over integers or polynomials. $p$-adic number theory provides a way to transform rationals to integers, which makes it a natural candidate for encoding rationals. Although one may use naive number-theoretic techniques to perform rational-to-integer transformations without reference to $p$-adic numbers, we contend that the theory of $p$-adic numbers is the proper lens to view such transformations. In this work we identify mathematical techniques (supported by $p$-adic number theory) as appropriate tools to construct a generic rational encoder which is compatible with HE. Based on these techniques, we propose a new encoding scheme PIE, that can be easily combined with both AGCD-based and RLWE-based HE to perform high precision arithmetic. After presenting an abstract version of PIE, we show how it can be attached to two well-known HE schemes: the AGCD-based IDGHV scheme and the RLWE-based (modified) Fan-Vercauteren scheme. We also discuss the advantages of our encoding scheme in comparison with previous works

Leveled Fully Homomorphic Encryption Schemes with Hensel Codes

We propose the use of Hensel codes (a mathematical tool lifted from the theory of $p$-adic numbers) as an alternative way to construct fully homomorphic encryption (FHE) schemes that rely on the hardness of some instance of the approximate common divisor (AGCD) problem. We provide a self-contained introduction to Hensel codes which covers all the properties of interest for this work. Two constructions are presented: a private-key leveled FHE scheme and a public-key leveled FHE scheme. The public-key scheme is obtained via minor modifications to the private-key scheme in which we explore asymmetric properties of Hensel codes. The efficiency and security (under an AGCD variant) of the public-key scheme are discussed in detail. Our constructions take messages from large specialized subsets of the rational numbers that admit fractional numerical inputs and associated computations for virtually any real-world application. Further, our results can be seen as a natural unification of error-free computation (computation free of rounding errors over rational numbers) and homomorphic encryption. Experimental results indicate the scheme is practical for a large variety of applications

Establishment of a high accuracy geoid correction model and geodata edge match

This research has developed a theoretical and practical methodology for efficiently and accurately determining sub-decimeter level regional geoids and centimeter level local geoids to meet regional surveying and local engineering requirements. This research also provides a highly accurate static DGPS network data pre-processing, post-processing and adjustment method and a procedure for a large GPS network like the state level HRAN project. The research also developed an efficient and accurate methodology to join soil coverages in GIS ARE/INFO.;A total of 181 GPS stations has been pre-processed and post-processed to obtain an absolute accuracy better than 1.5cm at 95% of the stations, and at all stations having a 0.5 ppm average relative accuracy. A total of 167 GPS stations in Iowa and around Iowa have been included in the adjustment.;After evaluating GEOID96 and GEOID99, a more accurate and suitable geoid model has been established in Iowa. This new Iowa regional geoid model improved the accuracy from a sub-decimeter 10~20 centimeter to 5~10 centimeter. The local kinematic geoid model, developed using Kalman filtering, gives results better than third order leveling accuracy requirement with 1.5 cm standard deviation

NONLINEAR OPERATORS FOR IMAGE PROCESSING: DESIGN, IMPLEMENTATION AND MODELING TECHNIQUES FOR POWER ESTIMATION

1998/1999Negli ultimi anni passati le applicazioni multimediali hanno visto uno sviluppo notevole, trovando applicazione in un gran numero di campi. Applicazioni come video conferenze, diagnostica medica, telefonia mobile e applicazioni militari necessitano il trattamento di una gran mole di dati ad alta velocità. Pertanto, l'elaborazione di immagini e di dati vocali è molto importante ed è stata oggetto di numerosi sforzi, nel tentativo di trovare algoritmi sempre più veloci ed efficaci. Tra gli algoritmi proposti, noi crediamo che gli operatori razionali svolgano un ruolo molto importante, grazie alla loro versatilità ed efficacia nell'elaborazione di dati. Negli ultimi anni sono stati proposti diversi algoritmi, dimostrando che questi operatori possono essere molto vantaggiosi in diverse applicazioni, producendo buoni risultati. Lo scopo di questo lavoro è di realizzare alcuni di questi algoritmi e, quindi, dimostrare che i filtri razionali, in particolare, possono essere realizzati senza ricorrere a sistemi di grandi dimensioni e possono raggiungere frequenze operative molto alte. Una volta che il blocco fondamentale di un sistema basato su operatori razionali sia stato realizzato, esso pu6 essere riusato con successo in molte altre applicazioni. Dal punto di vista del progettista, è importante avere uno schema generale di studio, che lo renda capace di studiare le varie configurazioni del sistema da realizzare e di analizzare i compromessi tra le variabili di progetto. In particolare, per soddisfare l'esigenza di metodi versatili per la stima della potenza, abbiamo sviluppato una tecnica di macro modellizazione che permette al progettista di stimare velocemente ed accuratamente la potenza dissipata da un circuito. La tesi è organizzata come segue: Nel Capitolo 1 alcuni sono presentati alcuni algoritmi studiati per la realizzazione. Ne viene data solo una veloce descrizione, lasciando comunque al lettore interessato dei riferimenti bibliografici. Nel Capitolo 2 vengono discusse le architetture fondamentali usate per la realizzazione. Principalmente sono state usate architetture a pipeline, ma viene data anche una descrizione degli approcci oggigiorno disponibili per l'ottimizzazione delle temporizzazioni. Nel Capitolo 3 sono presentate le realizzazioni di due sistemi studiati per questa tesi. Gli approcci seguiti si basano su ASIC e FPGA. Richiedono tecniche e soluzioni diverse per il progetto del sistema, per cui é interessante vedere cosa pu6 essere fatto nei due casi. Infine, nel Capitolo 4, descriviamo la nostra tecnica di macro modellizazione per la stima di potenza, dando una breve visione delle tecniche finora proposte e facendo vedere quali sono i vantaggi che il nostro metodo comporta per il progetto.In the past few years, multimedia application have been growing very fast, being applied to a large variety of fields. Applications like video conference, medical diagnostic, mobile phones, military applications require to handle large amount of data at high rate. Images as well as voice data processing are therefore very important and they have been subjected to a lot of efforts in order to find always faster and effective algorithms. Among image processing algorithms, we believe that rational operators assume an important role, due to their versatility and effectiveness in data processing. In the last years, several algorithms have been proposed, demonstrating that these operators can be very suitable in different applications with very good results. The aim of this work is to implement some of these algorithm and, therefore, demonstrate that rational filters, in particular, can be implemented without requiring large sized systems and they can operate at very high frequencies. Once the basic building block of a rational based system has been implemented, it can be successfully reused in many other applications. From the designer point of view, it is important to have a general framework, which makes it able to study various configurations of the system to be implemented and analyse the trade-off among the design variables. In particular, to meet the need far versatile tools far power estimation, we developed a new macro modelling technique, which allows the designer to estimate the power dissipated by a circuit quickly and accurately. The thesis is organized as follows: In chapter 1 we present some of the algorithms which have been studied for implementation. Only a brief overview is given, leaving to the interested reader some references in literature. In chapter 2 we discuss the basic architectures used for the implementations. Pipelined structures have been mainly used for this thesis, but an overview of the nowaday available approaches for timing optimization is presented. In chapter 3 we present two of the implementation designed for this thesis. The approaches followed are ASIC driven and FPGA drive. They require different techniques and different solution for the design of the system, therefore it is interesting to see what can be done in both the cases. Finally, in chapter 4, we describe our macro modelling techniques for power estimation, giving a brief overview of the up to now proposed techniques and showing the advantages our method brings to the design.XII Ciclo1969Versione digitalizzata della tesi di dottorato cartacea