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Cryptanalysis of small private key RSA
Title
Cryptanalysis of small private key RSA.
Creator
Guild, Jeffrey Kirk, Florida Atlantic University, Klingler, Lee
Abstract/Description
RSA cryptosystems with decryption exponent d less than N 0.292, for a given RSA modulus N, show themselves to be vulnerable to an attack which utilizes modular polynomials and the LLL Basis Reduction Algorithm. This result, presented by Dan Boneh and Glenn Durfee in 1999, is an improvement on the bound of N0.25 established by Wiener in 1990. This thesis examines in detail the LLL Basis Reduction Algorithm and the attack on RSA as presented by Boneh and Durfee.
Date Issued
1999
PURL
http://purl.flvc.org/fcla/dt/15730
Subject Headings
Cryptography, Algorithms, Data encryption (Computer science)
Format
Document (PDF)
On the Loewy structure of the projective indecomposable representations of some stabilizer subgroups of A(8) in characteristic 2
Title
On the Loewy structure of the projective indecomposable representations of some stabilizer subgroups of A(8) in characteristic 2.
Creator
Hindman, Peter Blake, Florida Atlantic University, Klingler, Lee
Abstract/Description
Given a module over a ring for which the Jordan-Holder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of n-tuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with...
Show moreGiven a module over a ring for which the Jordan-Holder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of n-tuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with special attention toward obtaining the Loewy structure of their projective indecomposable representations.
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Date Issued
1993
PURL
http://purl.flvc.org/fcla/dt/14971
Subject Headings
Representations of groups, Projective modules (Algebra), Indecomposable modules
Format
Document (PDF)
Pruefer domains, the strong 2-generator property, and integer-valued polynomials
Title
Pruefer domains, the strong 2-generator property, and integer-valued polynomials.
Creator
Roth, Heather., Florida Atlantic University, Klingler, Lee
Abstract/Description
We present several results involving three concepts: Prufer domains, the strong 2-generator property, and integer-valued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2-generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2-generator property . We note that, if D has the strong 2-generator property, then D is a...
Show moreWe present several results involving three concepts: Prufer domains, the strong 2-generator property, and integer-valued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2-generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2-generator property . We note that, if D has the strong 2-generator property, then D is a Prufer domain. If Q is the field of fractions of D, and E is a finite nonempty subset of D; we define Int(E, D ) = {f(X) ∈ Q[ X] ∣ f(a) ∈ D for every a ∈ E} to be the ring of integer-valued polynomials on D with respect to the subset E. We show that D is a Prufer domain if and only if Int(E, D) is a Prufer domain. Our main theorem is that Int(E, D) has the strong 2-generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).
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Date Issued
2004
PURL
http://purl.flvc.org/fcla/dt/13151
Subject Headings
Prüfer rings, Rings of integers, Polynomials, Ideals (Algebra), Mathematical analysis
Format
Document (PDF)
Integer-valued polynomials and pullbacks of arithmetical rings
Title
Integer-valued polynomials and pullbacks of arithmetical rings.
Creator
Boynton, Jason, Florida Atlantic University, Klingler, Lee
Abstract/Description
Let D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the n-generator property for D is equivalent to the n-generator property for Int(E, D), which is equivalent to strong (n + 1)-generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient...
Show moreLet D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the n-generator property for D is equivalent to the n-generator property for Int(E, D), which is equivalent to strong (n + 1)-generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Prufer domains R between D[X] and K[X]such that the conductor C of K[X] into R is non-zero. As an application, we show that for n > 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
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Date Issued
2006
PURL
http://purl.flvc.org/fcla/dt/12221
Subject Headings
Polynomials, Ideals (Algebra), Rings of integers, Categories (Mathematics), Arithmetical algebraic geometry
Format
Document (PDF)
prime spectrum of a ring: A survey
Title
The prime spectrum of a ring: A survey.
Creator
Fernandez, James Stephen, Florida Atlantic University, Klingler, Lee
Abstract/Description
This thesis has as its motivation the exploration, on an informal level, of a correspondence between Algebra and Topology. Specifically, it considers the prime spectrum of a ring, that is, the set of prime ideals, endowed with the Zariski topology. Questions posed by M. Atiyah and I. MacDonald in their book, "Introduction to Commutative Algebra", serve as a guideline through most of this work. The final section, however, follows R. Heitmann's paper, "Generating Non-Noetherian Modules...
Show moreThis thesis has as its motivation the exploration, on an informal level, of a correspondence between Algebra and Topology. Specifically, it considers the prime spectrum of a ring, that is, the set of prime ideals, endowed with the Zariski topology. Questions posed by M. Atiyah and I. MacDonald in their book, "Introduction to Commutative Algebra", serve as a guideline through most of this work. The final section, however, follows R. Heitmann's paper, "Generating Non-Noetherian Modules Efficiently". This section examines the patch topology on the prime spectrum of a ring where the patch topology has as a closed subbasis the Zariski closed and Zariski quasi-compact open sets. It is proven that the prime spectrum of a ring with the patch topology is a compact Hausdorff space, and several relationships between the patch and Zariski topologies are established. The final section concludes with a technical theorem having a number of interesting corollaries, among which are a stable range theorem and a theorem of Kronecker, both generalized to the non-Noetherian setting.
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Date Issued
1991
PURL
http://purl.flvc.org/fcla/dt/14763
Subject Headings
Rings (Algebra)
Format
Document (PDF)
H-LOCAL RINGS
Title
H-LOCAL RINGS.
Creator
Omairi, Akeel, Klingler, Lee, Florida Atlantic University, 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 _nite 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 and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian...
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 _nite 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 and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian rings in general.
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Date Issued
2019
PURL
http://purl.flvc.org/fau/fd/FA00013336
Subject Headings
Noetherian rings, Prüfer rings, Local rings
Format
Document (PDF)
ANNIHILATORS AND A + B RINGS
Title
ANNIHILATORS AND A + B RINGS.
Creator
Epstein, Alexandra Nicole, Klingler, Lee, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
A + B rings are constructed from a ring A and nonempty set of prime ideals of A. Initially, these rings were created to provide examples of reduced rings which satisfy certain annihilator conditions. We describe precisely when A + B rings have these properties, based on the ring A and set of prime ideals of A. We continue by giving necessary and su cient conditions for A + B rings to have various other properties. We also consider annihilators in the context of frames of ideals of reduced rings.
Date Issued
2020
PURL
http://purl.flvc.org/fau/fd/FA00013588
Subject Headings
Rings (Algebra)
Format
Document (PDF)
Maximally Prüfer rings
Title
Maximally Prüfer rings.
Creator
Sharma, Madhav, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong...
Show moreIn this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004465, http://purl.flvc.org/fau/fd/FA00004465
Subject Headings
Approximation theory, Commutative algebra, Commutative rings, Geometry, Algebraic, Ideals (Algebra), Mathematical analysis, Prüfer rings, Rings (Algebra), Rings of integers
Format
Document (PDF)