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 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.
Show less  Date Issued
 1973
 PURL
 http://purl.flvc.org/fcla/dt/13582
 Subject Headings
 Algebraic fields, Algebraic number theory
 Format
 Document (PDF)
 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.
Show less  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)
 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 BNpair, 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.
Show less  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)
 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 ndimensional vector space over the field of q elements. By a geometric t[q^n, k, λ] design we mean a collection D of kdimensional subspaces of V, called blocks, such that every tdimensional 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 kdimensional subspaces of V. In this work we construct nonisomorphic large sets using...
Show moreLet V be an ndimensional vector space over the field of q elements. By a geometric t[q^n, k, λ] design we mean a collection D of kdimensional subspaces of V, called blocks, such that every tdimensional 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 kdimensional subspaces of V. In this work we construct nonisomorphic large sets using methods based on incidence structures known as the KramerMesner 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 basisreduction, 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 knowhow for using the LLL and linear programming algorithms that we implemented to construct the large sets.
Show less  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)
 Title
 Elliptic curves: identitybased 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

Pairingfriendly 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 identitybased signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identitybased signature scheme along with a provable security reduction. We propose a modification of their scheme with several performance benefits. In...
Show morePairingfriendly 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 identitybased signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identitybased 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 DiffieHellman problem. Implementing the group arithmetic is a costcritical 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 Tdepth, involve less than 39% of the number of Tgates in the best previous construction. The software also optimizes the (CNOT) depth for F2linear operations by means of suitable graph colorings.
Show less  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)
 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)
 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 Rmodule 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 13 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 Rmodule 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 13 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of onedimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the KrullSchmidt property for direct sums of torsionfree rank one modules for a reduced local commutative Noetherian onedimensional ring R.
Show less  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)
 Title
 Minimal zerodimensional extensions.
 Creator
 Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The structure of minimal zerodimensional 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 onetoone correspondence between isomorphism classes of minimal zerodimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zerodimensional 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 onetoone correspondence between isomorphism classes of minimal zerodimensional 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 zerodimensional extensions of general ZPIrings.
Show less  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)
 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.
Show less  Date Issued
 2007
 PURL
 http://purl.flvc.org/FAU/11614
 Subject Headings
 Number theory, Fermat numbers
 Format
 Document (PDF)
 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.
Show less  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., TurbulenceMathematical models.
 Format
 Document (PDF)
 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.
Show less  Date Issued
 2017
 PURL
 http://purl.flvc.org/fau/fd/FA00004840, http://purl.flvc.org/fau/fd/FA00004840
 Subject Headings
 Computer networksSecurity measures., Computer security., ComputersAccess controlCode words., Cyberinfrastructure., Computer network architectures., Cryptography., Number theoryData processing.
 Format
 Document (PDF)
 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 Pointblast 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 Pointblast explosion. L. I. Sedov's solution of Pointblast explosion and Gary A. Sod's solution of a Riemann problem have been rederived 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 Pointblast 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 Pointblast explosion. L. I. Sedov's solution of Pointblast explosion and Gary A. Sod's solution of a Riemann problem have been rederived here from one dimensional fluid dynamics equations . Both these problems have been solved by using the idea of Selfsimilarity 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.
Show less  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)