Class Numbers of Real Cyclotomic Fields of Conductor pq

The class numbers h+ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and that is why other methods have been developed, which approach the problem from different angles. In this thesis we extend a method of Schoof that was designed for real cyclotomic fields of prime conductor to real cyclotomic fields of conductor equal to the product of two distinct odd primes. Our method calculates the index of a specific group of cyclotomic units in the full group of units of the field. This index has h+ as a factor. We then remove from the index the extra factor that does not come from h+ and so we have the order of h+. We apply our method to real cyclotomic fields of conductor < 2000 and we test the divisibility of h+ by all primes < 10000. Finally, we calculate the full order of the l-part of h+ for all odd primes l < 10000.</italic

On the Selmer group and rank of a family of elliptic curves

For arbitrary $m,n \in \mathbb{Z}$ with $gcd(2m , 3n) = 1$ we denote by $D = 4m^3 - 27n^2 \neq -3, -4$ the discriminants which are squarefree, and we define the family of elliptic curves $E_{D}: y^2 = x^3 + 16D$. These curves admit a rational 3-isogeny $\lambda$. In this paper we show that the rank of the $\lambda$-Selmer group of $E_{D}$ and of the $\lambda$-isogenous curves $E_{D'} = E_{-27D}$ have specific values related to the 3-rank $r_{3}(D)$ of the ideal class group of the quadratic field $K_{D} = \mathbb{Q}(\sqrt{D})$. Employing a known result on the parity of \Sha[E_{D}], we obtain that the rank of these curves is bounded below by $1$ for $D > 0$ and by $2$ for $D < -4$. Finally, we give an explicit, infinite subfamily of curves $E_{D}$ with $D = -p$, where $p$ are primes of a specific form.Comment: Submitted 26/10/2021 - Under review, 13 page

On homomorphic encryption using abelian groups: Classical security analysis

In [15], Leonardi and Ruiz-Lopez propose an additively homomorphic public key encryption scheme whose security is expected to depend on the hardness of the $\textit{learning homomorphism with noise problem}$ (LHN). Choosing parameters for their primitive requires choosing three groups $G$, $H$, and $K$. In their paper, Leonardi and Ruiz-Lopez claim that, when $G$, $H$, and $K$ are abelian, then their public-key cryptosystem is not quantum secure. In this paper, we study security for finite abelian groups $G$, $H$, and $K$ in the classical case. Moreover, we study quantum attacks on instantiations with solvable groups

Nationwide Survey in Greece about Knowledge, Risk Perceptions, and Preventive Behaviors for COVID-19 during the General Lockdown in April 2020

Background: The aim of this study was to investigate the knowledge, attitudes, and practices of the Greek general population toward coronavirus disease 2019 (COVID-19) during the lockdown period in April 2020, to examine factors associated with misperceptions and to determine behavioral patterns that may require interventions. Methods: A cross-sectional study of the general Greek population (N = 1858) was conducted. A geographically stratified cluster sampling was implemented. A questionnaire was composed consisting of 35 questions. Data collection took place from 15 April to 2 May 2020. A random-digit dialing survey was conducted by 29 interviewers. Results: The majority of respondents (62.7%) answered &ge;12/17 questions correctly. Participants aged 18&ndash;44 years, male gender, specific occupations (freelancer, unemployed, housewife, retiree) and those who sought information about COVID-19 from less than two sources received lower aggregated scores on knowledge questions. Regarding attitudes toward future vaccination, 18.9% declared that were against it, while 81.1% that they may consider or will be vaccinated. About 40% were not using a face mask and only 42% washed their hands appropriately. Conclusion: Adjusting information campaigns targeting especially people below 45 years of age can help to sensitize them and realise their role to control the spread. Further targeted surveys are needed to adjust/design prevention campaigns