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CHARACTERIZATION OF PRODUCT FORMULA FIELDS
Title
A CHARACTERIZATION OF PRODUCT FORMULA FIELDS.
Creator
HELLMAN, ALLEN PAUL, Florida Atlantic University
Abstract/Description
In this thesis we present a characterization of fields which admit a product formula. We prove that a field which admits a product formula consisting of admissible prime spots is a global field. This result was originally proved by Artin and Whaples in 1945. By limiting the admissible prime spots to those that are archimedean or discrete with finite residue class field, we are able to obtain a more elementary proof than that given by Artin and Whaples. The proof given here is, to our...
Show moreIn this thesis we present a characterization of fields which admit a product formula. We prove that a field which admits a product formula consisting of admissible prime spots is a global field. This result was originally proved by Artin and Whaples in 1945. By limiting the admissible prime spots to those that are archimedean or discrete with finite residue class field, we are able to obtain a more elementary proof than that given by Artin and Whaples. The proof given here is, to our knowledge, The render should notice that Artin and Whaples obtain, as a part of their result, that only the two types of prime spots mentioned above can occur in a product formula.
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Date Issued
1973
PURL
http://purl.flvc.org/fcla/dt/13582
Subject Headings
Algebraic fields, Algebraic number theory
Format
Document (PDF)
existence of minimal logarithmic signatures for classical groups
Title
The existence of minimal logarithmic signatures for classical groups.
Creator
Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
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Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172943
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)
the minimal logarithmic signature conjecture
Title
On the minimal logarithmic signature conjecture.
Creator
Singhi, Nidhi., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups,...
Show moreThe minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BN-pair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS.
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Date Issued
2011
PURL
http://purl.flvc.org/FAU/3172946
Subject Headings
Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
Format
Document (PDF)
New Geometric Large Sets
Title
New Geometric Large Sets.
Creator
Hurley, Michael Robert, Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Let V be an n-dimensional vector space over the field of q elements. By a geometric t-[q^n, k, λ] design we mean a collection D of k-dimensional subspaces of V, called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t-[q^n, k, λ] designs that partitions [V K], the collection of k-dimensional subspaces of V. In this work we construct non-isomorphic large sets using...
Show moreLet V be an n-dimensional vector space over the field of q elements. By a geometric t-[q^n, k, λ] design we mean a collection D of k-dimensional subspaces of V, called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t-[q^n, k, λ] designs that partitions [V K], the collection of k-dimensional subspaces of V. In this work we construct non-isomorphic large sets using methods based on incidence structures known as the Kramer-Mesner matrices. These structures are induced by particular group actions on the collection of subspaces of the vector space V. Subsequently, we discuss and use computational techniques for solving certain linear problems of the form AX = B, where A is a large integral matrix and X is a {0,1} solution. These techniques involve (i) lattice basis-reduction, including variants of the LLL algorithm, and (ii) linear programming. Inspiration came from the 2013 work of Braun, Kohnert, Ostergard, and Wassermann, [17], who produced the first nontrivial large set of geometric designs with t ≥ 2. Bal Khadka and Michael Epstein provided the know-how for using the LLL and linear programming algorithms that we implemented to construct the large sets.
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Date Issued
2016
PURL
http://purl.flvc.org/fau/fd/FA00004732, http://purl.flvc.org/fau/fd/FA00004732
Subject Headings
Group theory., Finite groups., Factorial experiment designs., Irregularities of distribution (Number theory), Combinatorial analysis.
Format
Document (PDF)
Elliptic curves: identity-based signing and quantum arithmetic
Title
Elliptic curves: identity-based signing and quantum arithmetic.
Creator
Budhathoki, Parshuram, Steinwandt, Rainer, Eisenbarth, Thomas, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Pairing-friendly curves and elliptic curves with a trapdoor for the discrete logarithm problem are versatile tools in the design of cryptographic protocols. We show that curves having both properties enable a deterministic identity-based signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identity-based signature scheme along with a provable security reduction. We propose a modification of their scheme with several performance benefits. In...
Show morePairing-friendly curves and elliptic curves with a trapdoor for the discrete logarithm problem are versatile tools in the design of cryptographic protocols. We show that curves having both properties enable a deterministic identity-based signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identity-based signature scheme along with a provable security reduction. We propose a modification of their scheme with several performance benefits. In addition to faster signing, for batch signing the signature size can be reduced, and if multiple signatures for the same identity need to be verified, the verification can be accelerated. Neither the signing nor the verification algorithm rely on the availability of a (pseudo)random generator, and we give a provable security reduction in the random oracle model to the (`-)Strong Diffie-Hellman problem. Implementing the group arithmetic is a cost-critical task when designing quantum circuits for Shor’s algorithm to solve the discrete logarithm problem. We introduce a tool for the automatic generation of addition circuits for ordinary binary elliptic curves, a prominent platform group for digital signatures. Our Python software generates circuit descriptions that, without increasing the number of qubits or T-depth, involve less than 39% of the number of T-gates in the best previous construction. The software also optimizes the (CNOT) depth for F2-linear operations by means of suitable graph colorings.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004182, http://purl.flvc.org/fau/fd/FA00004182
Subject Headings
Coding theory, Computer network protocols, Computer networks -- Security measures, Data encryption (Computer science), Mathematical physics, Number theory -- Data processing
Format
Document (PDF)
study of divisors and algebras on a double cover of the affine plane
Title
A study of divisors and algebras on a double cover of the affine plane.
Creator
Bulj, Djordje., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x - 1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.
Date Issued
2012
PURL
http://purl.flvc.org/FAU/3355618
Subject Headings
Algebraic number theory, Geometry, Data processing, Noncommutative differential geometry, Mathematical physics, Curves, Algebraic, Commutative rings
Format
Document (PDF)
Unique decomposition of direct sums of ideals
Title
Unique decomposition of direct sums of ideals.
Creator
Ay, Basak., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R.
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Date Issued
2010
PURL
http://purl.flvc.org/FAU/2683133
Subject Headings
Algebraic number theory, Modules (Algebra), Noetherian rings, Commutative rings, Algebra, Abstract
Format
Document (PDF)
Minimal zero-dimensional extensions
Title
Minimal zero-dimensional extensions.
Creator
Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
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Date Issued
2009
PURL
http://purl.flvc.org/FAU/210447
Subject Headings
Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format
Document (PDF)
Fermat's assertion about primes of the form x² + 3y²
Title
Fermat's assertion about primes of the form x² + 3y².
Creator
Lukacs, Olimpia., Harriet L. Wilkes Honors College
Abstract/Description
Fermat's assertion that "every prime number which surpasses by one a multiple of three [sic] is composed of a square and the triple of another square" raises further questions about primes and other quadratic forms. The interest over primes does not stop at x² + 3y², although this form by itself is a complex way of analyzing simple primes. We present an algorithm for solving p= x² + 3y², first discovered by Lagrange, and show that this algorithm does not work for other representations of...
Show moreFermat's assertion that "every prime number which surpasses by one a multiple of three [sic] is composed of a square and the triple of another square" raises further questions about primes and other quadratic forms. The interest over primes does not stop at x² + 3y², although this form by itself is a complex way of analyzing simple primes. We present an algorithm for solving p= x² + 3y², first discovered by Lagrange, and show that this algorithm does not work for other representations of numbers by quadratic forms.
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Date Issued
2007
PURL
http://purl.flvc.org/FAU/11614
Subject Headings
Number theory, Fermat numbers
Format
Document (PDF)
CFD Study of Pectoral Fins of Larval Zebrafish: Effect of Reynolds Number, Swimming Kinematics and Fin Bending on Fluid Structures and Transport.
Title
CFD Study of Pectoral Fins of Larval Zebrafish: Effect of Reynolds Number, Swimming Kinematics and Fin Bending on Fluid Structures and Transport.
Creator
Islam, Toukir, Curet, Oscar M., Florida Atlantic University, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
Abstract/Description
Flow Structure and fluid transport via advection around pectoral fin of larval ZebraFish are studied numerically using Immersed Boundary Method, Lagrangian Coherent Structure, passive particle tracing, vortex core evolution and four statistically defined mixing numbers. Experimental fish kinematics for nominal swimming case are obtained from previous researchers and numerically manipulated to analyze the role of different body motion kinematics, Reynolds number and fin morphology on flow...
Show moreFlow Structure and fluid transport via advection around pectoral fin of larval ZebraFish are studied numerically using Immersed Boundary Method, Lagrangian Coherent Structure, passive particle tracing, vortex core evolution and four statistically defined mixing numbers. Experimental fish kinematics for nominal swimming case are obtained from previous researchers and numerically manipulated to analyze the role of different body motion kinematics, Reynolds number and fin morphology on flow structure and transport. Hyperbolic strain field and vortex cores are found to be effective particle transporter and their relative strength are driving force of varying flow structure and fluid transport. Translation and lateral undulation of fish; as a combination or individual entity, has coherent advantages and drawbacks significant enough to alter the nature of fluid advection. Reynolds number increase enhances overall fluid transport and mixing in varying order for different kinematics and nominal bending position of fin has average transport capability of other artificially induced fin morphology.
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Date Issued
2016
PURL
http://purl.flvc.org/fau/fd/FA00004606, http://purl.flvc.org/fau/fd/FA00004606
Subject Headings
Reynolds number., Aquatic animals (Physiology), Transport theory., Computational fluid dynamics., Dynamical systems., Continuum physics., Turbulence--Mathematical models.
Format
Document (PDF)
A Study on Partially Homomorphic Encryption Schemes
Title
A Study on Partially Homomorphic Encryption Schemes.
Creator
Mithila, Shifat P., Karabina, Koray, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
High processing time and implementation complexity of the fully homomorphic encryption schemes intrigued cryptographers to extend partially homomorphic encryption schemes to allow homomorphic computation for larger classes of polynomials. In this thesis, we study several public key and partially homomorphic schemes and discuss a recent technique for boosting linearly homomorphic encryption schemes. Further, we implement this boosting technique on CGS linearly homomorphic encryption scheme to...
Show moreHigh processing time and implementation complexity of the fully homomorphic encryption schemes intrigued cryptographers to extend partially homomorphic encryption schemes to allow homomorphic computation for larger classes of polynomials. In this thesis, we study several public key and partially homomorphic schemes and discuss a recent technique for boosting linearly homomorphic encryption schemes. Further, we implement this boosting technique on CGS linearly homomorphic encryption scheme to allow one single multiplication as well as arbitrary number of additions on encrypted plaintexts. We provide MAGMA source codes for the implementation of the CGS scheme along with the boosted CGS scheme.
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Date Issued
2017
PURL
http://purl.flvc.org/fau/fd/FA00004840, http://purl.flvc.org/fau/fd/FA00004840
Subject Headings
Computer networks--Security measures., Computer security., Computers--Access control--Code words., Cyberinfrastructure., Computer network architectures., Cryptography., Number theory--Data processing.
Format
Document (PDF)
Subjecting the CHIMERA supernova code to two hydrodynamic test problems, (i) Riemann problem and (ii) Point blast explosion
Title
Subjecting the CHIMERA supernova code to two hydrodynamic test problems, (i) Riemann problem and (ii) Point blast explosion.
Creator
Ahsan, Abu Salah M., Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
A Shock wave as represented by the Riemann problem and a Point-blast explosion are two key phenomena involved in a supernova explosion. Any hydrocode used to simulate supernovae should be subjected to tests consisting of the Riemann problem and the Point-blast explosion. L. I. Sedov's solution of Point-blast explosion and Gary A. Sod's solution of a Riemann problem have been re-derived here from one dimensional fluid dynamics equations . Both these problems have been solved by using the idea...
Show moreA Shock wave as represented by the Riemann problem and a Point-blast explosion are two key phenomena involved in a supernova explosion. Any hydrocode used to simulate supernovae should be subjected to tests consisting of the Riemann problem and the Point-blast explosion. L. I. Sedov's solution of Point-blast explosion and Gary A. Sod's solution of a Riemann problem have been re-derived here from one dimensional fluid dynamics equations . Both these problems have been solved by using the idea of Self-similarity and Dimensional analysis. The main focus of my research was to subject the CHIMERA supernova code to these two hydrodynamic tests. Results of CHIMERA code for both the blast wave and Riemann problem have then been tested by comparing with the results of the analytic solution.
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Date Issued
2008
PURL
http://purl.flvc.org/FAU/172665
Subject Headings
Mathematical physics, Continuum mechanics, Number theory, Supernovae, Data processing, Shock waves, Fluid dynamics
Format
Document (PDF)