Pell Equation - Theory and applications to cryptography

The Pell equation $x^2 − d y^2 = 1$ is a classical topic in number theory. There are well known methods for solving this equation, but there are still several important issues. One of the most interesting from the point of view of cryptographic applications is the study of its solutions over a generic field, in which case new interesting open problems arise. This work focuses on studying the theoretical and practical potential of the Pell equation in this context. Firstly, the required theoretical results from the state of the art are collected using a new unique and simple notation. This allows to obtain easily and elegantly new properties also for the generalization of the Pell equation in the cubic case. Then, all the theoretical results are adopted to formulate new public–key encryption and digital signature schemes with security based on the integer factorization problem or on the discrete logarithm problem, namely new RSA–like and ElGamal cryptosystems, and new Digital Signature Algorithms. The obtained cryptosystems are compared in terms of security, data–size and performance with the classical alternatives, and the results are very interesting especially in the case of the quadratic Pell equation. Finally, the properties of the Pell equation are exploited for defining new powerful probabilistic primality tests, related to the Lucas test included in the widely used Baillie–PSW test. In particular, the new primality tests are equipped with adaptations of the Selfridge method for choosing the parameters, resulting in very powerful tests

On the cubic Pell equation over finite fields

The classical Pell equation can be extended to the cubic case considering the elements of norm one in Z[ √3 r], which satisfy x 3 + ry 3 + r 2 z 3 − 3rxyz = 1. The solution of the cubic Pell equation is harder than the classical case, indeed a method for solving it as Diophantine equation is still missing [3]. In this paper, we study the cubic Pell equation over finite fields, extending the results that hold for the classical one. In particular, we provide a novel method for counting the number of solutions in all possible cases depending on the value of r. Moreover, we are also able to provide a method for generating all the solutions

A symbiosis between cellular automata and genetic algorithms

Cellular automata are systems which use a rule to describe the evolution of a population in a discrete lattice, while genetic algorithms are procedures designed to find solutions to optimization problems inspired by the process of natural selection. In this paper, we introduce an original implementation of a cellular automaton whose rules use a fitness function to select for each cell the best mate to reproduce and a crossover operator to determine the resulting offspring. This new system, with a proper definition, can be both a cellular automaton and a genetic algorithm. We show that in our system the Conway’s Game of Life can be easily implemented and, consequently, it is capable of universal computing. Moreover two generalizations of the Game of Life are created and also implemented with it. Finally, we use our system for studying and implementing the prisoner’s dilemma and rock-paper-scissors games, showing very interesting behaviors and configurations (e.g., gliders) inside these games

Exploring Deep Learning for In-Field Fault Detection in Microprocessors

Nowadays, due to technology enhancement, faults are increasingly compromising all kinds of computing machines, from servers to embedded systems. Recent advances in ma- chine learning are opening new opportunities to achieve fault detection exploiting hardware metrics inspection, thus avoiding the use of heavy software techniques or product-specific errors reporting mechanisms. This paper investigates the capability of different deep learning models trained on data collected through simulation-based fault injection to generalize over different software applications

Primality tests, linear recurrent sequences and the Pell equation

We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce a new specific primality test, which we will call generalized Pell test. We perform some numerical computations on the new primality tests and, for the generalized Pell test, we do not any pseudoprime up to $10^{10}$

Toward a Post-Quantum Zero-Knowledge Verifiable Credential System for Self-Sovereign Identity

The advent of quantum computers brought a large interest in post-quantum cryptography and in the migration to quantum-resistant systems. Protocols for Self-Sovereign Identity (SSI) are among the fundamental scenarios touched by this need. The core concept of SSI is to move the control of digital identity from third-party identity providers directly to individuals. This is achieved through Verificable Credentials (VCs) supporting anonymity and selective disclosure. In turn, the implementation of VCs requires cryptographic signature schemes compatible with a proper Zero-Knowledge Proof (ZKP) framework. We describe the two main ZKP VCs schemes based on classical cryptographic assumptions, that is, the signature scheme with efficient protocols of Camenisch and Lysyanskaya, which is based on the strong RSA assumption, and the BBS+ scheme of Boneh, Boyen and Shacham, which is based on the strong Diffie-Hellman assumption. Since these schemes are not quantum-resistant, we select as one of the possible post-quantum alternatives a lattice-based scheme proposed by Jeudy, Roux-Langlois, and Sander, and we try to identify the open problems for achieving VCs suitable for selective disclosure, non-interactive renewal mechanisms, and efficient revocation

Supporting CPS Modeling Through a New Method for Solving Complex Non-holomorphic Equations

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Disease-Modifying Therapies and Coronavirus Disease 2019 Severity in Multiple Sclerosis

Objective: This study was undertaken to assess the impact of immunosuppressive and immunomodulatory therapies on the severity of coronavirus disease 2019 (COVID-19) in people with multiple sclerosis (PwMS). Methods: We retrospectively collected data of PwMS with suspected or confirmed COVID-19. All the patients had complete follow-up to death or recovery. Severe COVID-19 was defined by a 3-level variable: mild disease not requiring hospitalization versus pneumonia or hospitalization versus intensive care unit (ICU) admission or death. We evaluated baseline characteristics and MS therapies associated with severe COVID-19 by multivariate and propensity score (PS)-weighted ordinal logistic models. Sensitivity analyses were run to confirm the results. Results: Of 844 PwMS with suspected (n = 565) or confirmed (n = 279) COVID-19, 13 (1.54%) died; 11 of them were in a progressive MS phase, and 8 were without any therapy. Thirty-eight (4.5%) were admitted to an ICU; 99 (11.7%) had radiologically documented pneumonia; 96 (11.4%) were hospitalized. After adjusting for region, age, sex, progressive MS course, Expanded Disability Status Scale, disease duration, body mass index, comorbidities, and recent methylprednisolone use, therapy with an anti-CD20 agent (ocrelizumab or rituximab) was significantly associated (odds ratio [OR] = 2.37, 95% confidence interval [CI] = 1.18-4.74, p = 0.015) with increased risk of severe COVID-19. Recent use (<1 month) of methylprednisolone was also associated with a worse outcome (OR = 5.24, 95% CI = 2.20-12.53, p = 0.001). Results were confirmed by the PS-weighted analysis and by all the sensitivity analyses. Interpretation: This study showed an acceptable level of safety of therapies with a broad array of mechanisms of action. However, some specific elements of risk emerged. These will need to be considered while the COVID-19 pandemic persists

COVID-19 Severity in Multiple Sclerosis: Putting Data Into Context

Background and objectives: It is unclear how multiple sclerosis (MS) affects the severity of COVID-19. The aim of this study is to compare COVID-19-related outcomes collected in an Italian cohort of patients with MS with the outcomes expected in the age- and sex-matched Italian population. Methods: Hospitalization, intensive care unit (ICU) admission, and death after COVID-19 diagnosis of 1,362 patients with MS were compared with the age- and sex-matched Italian population in a retrospective observational case-cohort study with population-based control. The observed vs the expected events were compared in the whole MS cohort and in different subgroups (higher risk: Expanded Disability Status Scale [EDSS] score > 3 or at least 1 comorbidity, lower risk: EDSS score ≤ 3 and no comorbidities) by the χ2 test, and the risk excess was quantified by risk ratios (RRs). Results: The risk of severe events was about twice the risk in the age- and sex-matched Italian population: RR = 2.12 for hospitalization (p < 0.001), RR = 2.19 for ICU admission (p < 0.001), and RR = 2.43 for death (p < 0.001). The excess of risk was confined to the higher-risk group (n = 553). In lower-risk patients (n = 809), the rate of events was close to that of the Italian age- and sex-matched population (RR = 1.12 for hospitalization, RR = 1.52 for ICU admission, and RR = 1.19 for death). In the lower-risk group, an increased hospitalization risk was detected in patients on anti-CD20 (RR = 3.03, p = 0.005), whereas a decrease was detected in patients on interferon (0 observed vs 4 expected events, p = 0.04). Discussion: Overall, the MS cohort had a risk of severe events that is twice the risk than the age- and sex-matched Italian population. This excess of risk is mainly explained by the EDSS score and comorbidities, whereas a residual increase of hospitalization risk was observed in patients on anti-CD20 therapies and a decrease in people on interferon