19 papers · ranked by Valyu relevance
Thomas Hader, Alex Ozdemir
In the last few years there have been rapid developments in SMT solving for finite fields. These include new decision procedures, new implementations of SMT theory solvers, and new software verifiers that rely on SMT solving for finite fields. To support interoperability in this emerging ecosystem, we propose the…
Matias Relyea
Cubic and biquadratic reciprocity have long since been referred to as "the forgotten reciprocity laws", largely since they provide special conditions that are widely considered to be unnecessary in the study of number theory. However, this paper aims to approach reciprocity with ample detail to motivate its existence.…
Brent Everitt
| 0. | What is Galois Theory? | 3 | | --- | --- | --- | | 1. Rings I: Polynomials | | 8 | | 2. | Roots and Irreducibility | 14 | | 3. | Fields I: Basics, Extensions and Concrete Examples | 22 | | 4. Rings II: Quotients | | 28 | | 5. | Fields II: Constructions and More Examples | 33 | | 6. | Ruler and Compass…
Lucian M. Ionescu, Mina M. Zarrin
| 1. Introduction | | 1 | | --- | --- | --- | | 2. | Finite Fields: the "Abstract Way" | 3 | | 3. What are | Number Fields? | 4 | | 3.1. Gaussian Integers | | 5 | | 3.2. | Eisenstein Integers | 5 | | 3.3. | From Number Fields to Finite Fields | 5 | | 4. | Lattice Models: the "Geometric Way" | 5 | | 5. | Applications to…
Anastasia Chavez, Christopher O’Neill
A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois theory to prove it. Yet the combinatorial and algebraic coding theory applications of finite fields can show up early on for students in STEM.…
Ryan Grady, Mark Poston
In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based course and some linear algebra; the appendix utilizes abstract algebra. A more…
Yizhou Zhao, Hua Sun, Cyril Branciard
We consider the secure computation problem in a minimal model, where Alice and Bob each holds an input and wish to securely compute a function of their inputs at Carol without revealing any additional information about the inputs. For this minimal secure computation problem, we propose a novel coding scheme built from…
Akhat Bakirov, Dinara Matrassulova, Yelizaveta Vitulyova, Dina Shaltykova + 1 more
'Dina Shaltykova' 'Ibragim Suleimenov'] An algorithm of digital logarithm calculation for the Galois field $GF257$ is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it…
Ibragim E. Suleimenov, Yelizaveta S. Vitulyova, Dinara K. Matrassulova, Pierluigi Vellucci
'Dinara K. Matrassulova' 'Pierluigi Vellucci'] An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1. For such numbers, it…
Khodakhast Bibak, Bruce M. Kapron, Venkatesh Srinivasan
Authentication plays a critical role in the security of quantum key distribution (QKD) protocols. We propose using Polynomial Hash and its variants for authentication of variable length messages in QKD protocols. Since universal hashing is used not only for authentication in QKD but also in other steps in QKD like…
Qichen Huang, Haoyang Guo
Cellular automata and graph reaction–diffusion systems encode local spatial interactions in different mathematical forms. We develop a cochain-operator calculus for these two settings. Over a finite field F_q_, every local rule on a finite neighborhood has a unique reduced polynomial representative. On an oriented…
G. Richard Christie, Poul M.F. Nielsen, Shane A. Blackett, Chris P. Bradley + 1 more
'Chris P. Bradley' 'Peter J. Hunter'] The field modelling language FieldML is being developed as a standard for modelling and interchanging field descriptions in software, suitable for a wide range of computation techniques. It comprises a rich set of operators for defining generalized fields as functions of other…
Yue Mei, Jiahao Liu, Xu Guo, Brandon Zimmerman + 2 more
This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving…
Authors not listed
NMR chemical shifts depend on the applied magnetic flux density, and this becomes more and more important as stronger and stronger magnetic fields are becoming available. Herein, we develop a theory of the field dependence of NMR shifts of paramagnetic molecules in solution. Our derivation leads to two distinct…
Robersy Sanchez, Jesús Barreto
Experimental studies reveal that genome architecture splits into natural domains suggesting a well-structured genomic architecture, where, for each species, genome populations are integrated by individual mutational variants. Herein, we show that the architecture of population genomes from the same or closed related…
Jheyne N. Ortiz, Robson R. de Araujo, Diego F. Aranha, Sueli I. R. Costa + 3 more
'Sueli I. R. Costa' 'Ricardo Dahab' 'Amin Sakzad' 'Khoa Nguyen'] Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring…
Mehri Baniasadi, Daniele Proverbio, Jorge Gonçalves, Frank Hertel + 1 more
Deep brain stimulation (DBS) is a surgical therapy to alleviate symptoms of certain brain disorders by electrically modulating neural tissues. Computational models predicting electric fields and volumes of tissue activated are key for efficient parameter tuning and network analysis. Currently, we lack efficient and…
Fabio Ferri, Henri Johnston
Let L/K be a Galois extension of number fields and let $G=\textrm{Gal}L/K$. We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt’s conjecture at p for certain proper intermediate fields of L/K implies Leopoldt’s conjecture at p for L. We also obtain relations between the Leopoldt defects of…
Ian Holmes
We describe a strategy for constructing codes for DNA-based information storage by serial composition of weighted finite-state transducers. The resulting state machines can integrate correction of substitution errors; synchronization by interleaving watermark and periodic marker signals; conversion from binary to…