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On the Hilbert Characteristic Polynomial
Title
On the Hilbert Characteristic Polynomial.
Creator
Wilson, G. Peter, Bastida, Julio R., Florida Atlantic University
Abstract/Description
The purpose of this thesis is to provide complete proofs for several results on integral-valued polynomials, which are used in Serre's proof of Hilbert's Theorem found in the theory of characteristic polynomials. These results, however elementary, are not found in the literature. The proof of Hilbert's Theorem is also given.
Date Issued
1970
PURL
http://purl.flvc.org/fau/fd/FA00000855
Subject Headings
Hilbert space, Polynomials
Format
Document (PDF)
ANALYSIS OF CRYPTOGRAPHIC EFFICIENCY: ELLIPTIC CURVE SCALAR MULTIPLICATION AND CONSTANT-TIME POLYNOMIAL INVERSION IN POST-QUANTUM CRYPTOGRAPHY
Title
ANALYSIS OF CRYPTOGRAPHIC EFFICIENCY: ELLIPTIC CURVE SCALAR MULTIPLICATION AND CONSTANT-TIME POLYNOMIAL INVERSION IN POST-QUANTUM CRYPTOGRAPHY.
Creator
Dutta, Abhraneel, Persichetti, Edoardo, Karabina, Koray, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
An efficient scalar multiplication algorithm is vital for elliptic curve cryptosystems. The first part of this dissertation focuses on a scalar multiplication algorithm based on scalar recodings resistant to timing attacks. The algorithm utilizes two recoding methods: Recode, which generalizes the non-zero signed all-bit set recoding, and Align, which generalizes the sign aligned columns recoding. For an ℓ-bit scalar split into d subscalars, our algorithm has a computational cost of ⌈⌈ℓ logk...
Show moreAn efficient scalar multiplication algorithm is vital for elliptic curve cryptosystems. The first part of this dissertation focuses on a scalar multiplication algorithm based on scalar recodings resistant to timing attacks. The algorithm utilizes two recoding methods: Recode, which generalizes the non-zero signed all-bit set recoding, and Align, which generalizes the sign aligned columns recoding. For an ℓ-bit scalar split into d subscalars, our algorithm has a computational cost of ⌈⌈ℓ logk(2)⌉/d⌉ point additions and k-scalar multiplications and a storage cost of kd−1(k − 1) – 1 points on E. The “split and comb” method further optimizes computational and storage complexity. We find the best setting to be with a fixed base point on a Twisted Edwards curve using a mix of projective and extended coordinates, with k = 2 generally offering the best performance. However, k = 3 may be better in certain applications. The second part of this dissertation is dedicated to constant-time polynomial inversion algorithms in Post-Quantum Cryptography (PQC). The computation of the inverse of a polynomial over a quotient ring or finite field is crucial for key generation in post-quantum cryptosystems like NTRU, BIKE, and LEDACrypt. Efficient algorithms must run in constant time to prevent side-channel attacks. We examine constant-time algorithms based on Fermat’s Little Theorem and the Extended GCD Algorithm, providing detailed time complexity analysis. We find that the constant-time Extended GCD inversion algorithm is more efficient, performing fewer field multiplications. Additionally, we explore other exponentiation algorithms similar to the Itoh-Tsuji inversion method, which optimizes polynomial multiplications in the BIKE/LEDACrypt setup. Recent results on hardware implementations are also discussed.
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Date Issued
2024
PURL
http://purl.flvc.org/fau/fd/FA00014492
Subject Headings
Cryptography, Curves, Elliptic, Polynomials
Format
Document (PDF)
Random Harmonic Polynomials
Title
Random Harmonic Polynomials.
Creator
Thomack, Andrew, Lundberg, Erik, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization....
Show moreThe study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization. Three prevalent models of signi cant study are the Kostlan model, the Weyl model, and the naive model in which the coe cients of the polynomial are standard Gaussian random variables. A harmonic polynomial is a complex function of the form h(z) = p(z) + q(z) where p and q are complex analytic polynomials. Li and Wei adapted the Kac integral formula for the expected number of zeros to study random harmonic polynomials and take particular interest in their interpretation of the Kostlan model. In this thesis we nd asymptotic results for the number of zeros of random harmonic polynomials under both the Weyl model and the naive model as the degree of the harmonic polynomial increases. We compare the ndings to the Kostlan model as well as to the analytic analogs of each model. We end by establishing results which lead to open questions and conjectures about random harmonic polynomials. We ask and partially answer the question, \When does the number and behavior of the zeros of a random harmonic polynomial asymptotically emulate the same model of random complex analytic polynomial as the degree increases?" We also inspect the variance of the number of zeros of random harmonic polynomials, motivating the work by the question of whether the distribution of the number of zeros concentrates near its as the degree of the harmonic polynomial increases.
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Date Issued
2017
PURL
http://purl.flvc.org/fau/fd/FA00004986
Subject Headings
Dissertations, Academic -- Florida Atlantic University, Random polynomials., Functions., Polynomials.
Format
Document (PDF)
GENERALIZED PADE APPROXIMATION TECHNIQUES AND MULTIDIMENSIONAL SYSTEMS
Title
GENERALIZED PADE APPROXIMATION TECHNIQUES AND MULTIDIMENSIONAL SYSTEMS.
Creator
MESSITER, MARK A., Florida Atlantic University, Shamash, Yacov A., College of Engineering and Computer Science, Department of Computer and Electrical Engineering and Computer Science
Abstract/Description
Two algorithms for greatest common factor (GCF) extraction from two multivariable polynomials, based on generalized Pade approximation, are presented. The reduced transfer matrices for two-dimensional (20) systems are derived from two 20 state-space models. Tests for product and sum separabilities of multivariable functions are also given.
Date Issued
1983
PURL
http://purl.flvc.org/fcla/dt/14175
Subject Headings
Multivariate analysis, Padé approximant, Polynomials
Format
Document (PDF)
Polynomials that are integer valued on the image of an integer-valued polynomial
Title
Polynomials that are integer valued on the image of an integer-valued polynomial.
Creator
Marshall, Mario V., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions.
Date Issued
2009
PURL
http://purl.flvc.org/FAU/216411
Subject Headings
Polynomials, Ring of integers, Ideals (Algebra)
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)
Internal waves on a continental shelf
Title
Internal waves on a continental shelf.
Creator
Jagannathan, Arjun., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
Abstract/Description
In this thesis, a 2D CHebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collation method in both spatial directions. The numerical model solves the...
Show moreIn this thesis, a 2D CHebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collation method in both spatial directions. The numerical model solves the inviscid, incomprehensible, fully non-linear, non-hydrostatic Boussinesq equations in the vorticity-streamfunction formulation. A number of important features of internal waves over topography are captured with the present model, including the onset of wave-breaking at sub-critical Froude numbers, up to the point of overturning of the pycnoclines. Density contours and wave spectra are presented for different combinations of Froude numbers, stratifications and topographic slope.
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Date Issued
2012
PURL
http://purl.flvc.org/FAU/3358549
Subject Headings
Engineering geology, Mathematical models, Chebyshev polynomials, Fluid dynamics, Continuum mechanics, Spectral theory (Mathematics)
Format
Document (PDF)
Rings of integer-valued polynomials and derivatives
Title
Rings of integer-valued polynomials and derivatives.
Creator
Villanueva, Yuri., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c...
Show moreFor D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field.
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Date Issued
2012
PURL
http://purl.flvc.org/FAU/3356899
Subject Headings
Rings of integers, Ideals (Algebra), Polynomials, Arithmetic algebraic geometry, Categories (Mathematics), Commutative algebra
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)