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- Title
- The discrete logarithm problem in non-abelian groups.
- Creator
- Iliâc, Ivana., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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This dissertation contains results of the candidate's research on the generalized discrete logarithm problem (GDLP) and its applications to cryptology, in non-abelian groups. The projective special linear groups PSL(2; p), where p is a prime, represented by matrices over the eld of order p, are investigated as potential candidates for implementation of the GDLP. Our results show that the GDLP with respect to specic pairs of PSL(2; p) generators is weak. In such cases the groups PSL(2; p) are...
Show moreThis dissertation contains results of the candidate's research on the generalized discrete logarithm problem (GDLP) and its applications to cryptology, in non-abelian groups. The projective special linear groups PSL(2; p), where p is a prime, represented by matrices over the eld of order p, are investigated as potential candidates for implementation of the GDLP. Our results show that the GDLP with respect to specic pairs of PSL(2; p) generators is weak. In such cases the groups PSL(2; p) are not good candidates for cryptographic applications which rely on the hardness of the GDLP. Results are presented on generalizing existing cryptographic primitives and protocols based on the hardness of the GDLP in non-abelian groups. A special instance of a cryptographic primitive dened over the groups SL(2; 2n), the Tillich-Zemor hash function, has been cryptanalyzed. In particular, an algorithm for constructing collisions of short length for any input parameter is presented. A series of mathematical results are developed to support the algorithm and to prove existence of short collisions.
Show less - Date Issued
- 2010
- PURL
- http://purl.flvc.org/FAU/3356783
- Subject Headings
- Data encryption (Computer science), Computer security, Cryptography, Combinatorial group theory, Data processing, Mapping (Mathematics)
- 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
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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
- On the Laplacian and fractional Laplacian in exterior domains, and applications to the dissipative quasi-geostrophic equation.
- Creator
- Kosloff, Leonardo., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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In this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this...
Show moreIn this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this allowed us to obtain a maximum principle for solutions of the dissipative two-dimensional quasi-geostrophic equation the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/FAU/3355570
- Subject Headings
- Fluid dynamics, Data processing, Laplacian matrices, Attractors (Mathematics), Differential equations, Partial
- Format
- Document (PDF)
- Title
- A min/max algorithm for cubic splines over k-partitions.
- Creator
- Golinko, Eric David, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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The focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of interest since the trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a minimum/maximum algorithm that will identify potential knots. Then we employ least squares estimation to find potential regression...
Show moreThe focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of interest since the trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a minimum/maximum algorithm that will identify potential knots. Then we employ least squares estimation to find potential regression coefficients based upon the cubic spline model and the knots chosen by the minimum/maximum algorithm. We then utilize the best subsets regression method to aid in model selection in which we find the minimum value of the Bayesian Information Criteria. Finally, we preent the R2adj as a measure of overall goodness of fit of our selected model. We have found among the fifty states and Washington D.C., 42 out of 51 showed an R2adj value that was greater than 90%. We also present an overall model of the United States. Also, we show additional applications our algorithm for data which show a non linear association. It is hoped that our method can serve as a unified model for violent crime rate over future years.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/FAU/3342107
- Subject Headings
- Spline theory, Data processing, Bayesian statistical decision theory, Data processing, Neural networks (Computer science), Mathematical statistics, Uncertainty (Information theory), Probabilities, Regression analysis
- Format
- Document (PDF)
- Title
- Shamir's secret sharing scheme using floating point arithmetic.
- Creator
- Finamore, Timothy., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
Implementing Shamir's secret sharing scheme using floating point arithmetic would provide a faster and more efficient secret sharing scheme due to the speed in which GPUs perform floating point arithmetic. However, with the loss of a finite field, properties of a perfect secret sharing scheme are not immediately attainable. The goal is to analyze the plausibility of Shamir's secret sharing scheme using floating point arithmetic achieving the properties of a perfect secret sharing scheme and...
Show moreImplementing Shamir's secret sharing scheme using floating point arithmetic would provide a faster and more efficient secret sharing scheme due to the speed in which GPUs perform floating point arithmetic. However, with the loss of a finite field, properties of a perfect secret sharing scheme are not immediately attainable. The goal is to analyze the plausibility of Shamir's secret sharing scheme using floating point arithmetic achieving the properties of a perfect secret sharing scheme and propose improvements to attain these properties. Experiments indicate that property 2 of a perfect secret sharing scheme, "Any k-1 or fewer participants obtain no information regarding the shared secret", is compromised when Shamir's secret sharing scheme is implemented with floating point arithmetic. These experimental results also provide information regarding possible solutions and adjustments. One of which being, selecting randomly generated points from a smaller interval in one of the proposed schemes of this thesis. Further experimental results indicate improvement using the scheme outlined. Possible attacks are run to test the desirable properties of the different schemes and reinforce the improvements observed in prior experiments.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/FAU/3342048
- Subject Headings
- Signal processing, Digital techniques, Mathematics, Data encryption (Computer science), Computer file sharing, Security measures, Computer algorithms, Numerical analysis, Data processing
- Format
- Document (PDF)
- Title
- Cryptography in the presence of key-dependent messages.
- Creator
- Gonzalez, Madeline., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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The aim of this work is to investigate a security model in which we allow an adversary to have access to functions of the secret key. In recent years, significant progress has been made in understanding the security of encryption schemes in the presence of key-dependent plaintexts or messages (known as KDM). Here, we motivate and explore the security of a setting, where an adversary against a message authentication code (MAC) or signature scheme can access signatures on key-dependent messages...
Show moreThe aim of this work is to investigate a security model in which we allow an adversary to have access to functions of the secret key. In recent years, significant progress has been made in understanding the security of encryption schemes in the presence of key-dependent plaintexts or messages (known as KDM). Here, we motivate and explore the security of a setting, where an adversary against a message authentication code (MAC) or signature scheme can access signatures on key-dependent messages. We propose a way to formalize the security of message authentication schemes in the presence of key-dependent MACs (KD-EUF) and of signature schemes in the presence of key-dependent signatures (KDS). An attack on a message recognition protocol involving a MAC is presented. It turns out that the situation is quite different from key-dependent encryption: To achieve KD-EUF-security or KDS-security under non-adaptive chosen message attacks, the use of a stateful signing algorithm is inevitable even in the random oracle model. After discussing the connection between key-dependent signing and forward security, we describe a compiler which lifts any EUF-CMA secure one-time signature scheme to a forward secure signature scheme offering KDS-CMA security. Then, we discuss how aggregate signatures can be used to combine the signatures in the certificate chain used in the compiler. A natural question arises about how to combine the security definitions of KDM and KDS to come up with a signcryption scheme that is secure. We also offer a connection with Leakage-Resilient Signatures, which take into account side-channel attacks. Lastly, we present some open problems for future research.
Show less - Date Issued
- 2009
- PURL
- http://purl.flvc.org/FAU/2182087
- Subject Headings
- Cryptography, Data processing, Digital signatures, Computer security, Data encryption (Computer science), Software protection
- Format
- Document (PDF)