Master of Arts, The Ohio State University, 1921, Graduate School

Committee: Not Provided (Other) Subjects:

Master of Science, University of Akron, 2007, Mathematics

Let p be a prime number and consider the vector space consisting of all p-by-p-by-pmatrices with entries taken from the field with p elements. We wish to construct, list, and describe all those subspaces that are simultaneously invariant under three particular linear transformations on this vector space. Even for small primes p, this is an extensive and difficult computational problem. Using an elaborate overall strategy based on concepts from linear algebra, we completely solve this problem for the prime p=2, and we have completed several cases of this problem for the prime p=3. This problem has connections with classification problems for certain subgroups of wreath product finite groups of prime-power order.

Committee: Jeffrey Riedl (Advisor) Subjects: Mathematics

Master of Science in Cyber Security (M.S.C.S.), Wright State University, 2019, Computer Science

Elliptic Curve Cryptography (ECC) is a public-key cryptography system. Elliptic Curve Cryptography (ECC) can achieve the same level of security as the public-key cryptography system, RSA, with a much smaller key size. It is a promising public key cryptography system with regard to time efficiency and resource utilization. This thesis focuses on the software implementations of ECC over finite field GF(p) with two distinct implementations of the Big Integer classes using character arrays, and bit sets in C++ programming language. Our implementation works on the ECC curves of the form y^2 = x^3 + ax + b (mod p). The point addition operation and the scalar multiplication are implemented on a real SEC (Standards for Efficient Cryptography) ECC curve over a prime field with two different implementations. The Elliptic Curve Diffie-Hellman key exchange, the ElGamal encryption/decryption system, and the Elliptic Curve Digital Signature Algorithm (ECDSA) on a real SEC ECC curve with two different implementations of the big integer classes are tested, and validated. The performances of the two different implementations are compared and analyzed.

Committee: Meilin Liu Ph.D. (Advisor); Junjie Zhang Ph.D. (Committee Member); Keke Chen Ph.D. (Committee Member) Subjects: Computer Science; Information Technology

BA, Oberlin College, 2018, Mathematics

At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic curves.

Committee: Benjamin Linowitz (Advisor) Subjects: Computer Science; Mathematics

BS, Kent State University, 2018, College of Arts and Sciences / Department of Mathematical Sciences

Our goal will be to find a weak solution to the Beltrami flow. A Beltrami flow in three-dimensional space is an incompressible (divergence free) vector field that is everywhere parallel to its curl. That is, curl(B) = λ B for some function. These flows arise naturally in many physical problems. In astrophysics and in plasma fusion Beltrami fields are known as force-free fields. They describe the equilibrium of perfectly conducting pressure-less plasma in the presence of a strong magnetic field. In fluid mechanics, Beltrami flows arise as steady states of the 3D Euler equations. Numerical evidence suggests that in certain regimes turbulent flows organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows. Given the importance of Beltrami fields, there are several approaches to proving existence of solutions, for instance use the calculus of variations, and use fixed point arguments. In this thesis we instead use a Hilbert space approach through the Lax-Milgram lemma.

Committee: Benjamin Jaye (Advisor); Andrew Tonge (Committee Member); Dexheimer Veronica (Committee Member); Jeremy Williams (Committee Member) Subjects: Applied Mathematics; Astrophysics

Doctor of Philosophy, The Ohio State University, 1984, Graduate School

Committee: Not Provided (Other) Subjects: Mathematics

Master of Science in Engineering, University of Akron, 2013, Civil Engineering

Site response analysis plays an important role in the design of seismic resistant structures. This is traditionally performed by assigning average soil properties to the site without considering the effects of uncertain spatial variability of soils and other sources of soil uncertainties e.g., measurement uncertainty and transformation relation uncertainty. With increased focus on safety and reliability of infrastructures, there is a need to capture the effects of uncertainties in soil properties in our designs and/or simulations. This thesis uses Monte Carlo technique together with deterministic finite element method to quantify the effects of uncertainties in soil properties on nonlinear site response analysis at a site in northern California. The soil deposit at the site is mostly comprised of stiff clay and is modeled as a stack of eight-noded brick elements using open source finite element code, OpenSees. Probabilistic elastic-perfectly plastic von Mises model, which works reasonably well for clay and whose parameters are easy to obtain from in-situ tests, is used to describe the constitutive behavior of the soil. The uncertainties in soil properties are captured by modeling the soil properties as random fields. The parameters of the random fields are estimated by analyzing cone penetration test (CPT) soundings available through the United States Geological Survey (USGS). The 1989 Loma Prieta motion recorded at station Bran by the University of California, Santa Cruz (UCSC) is applied to the base of the finite element model and the probabilistic propagation of the waves is simulated using Monte Carlo technique. The surface displacement (displacement response at the top node) is presented in terms of its mean and standard deviation. Constitutive response of the soil is also presented in terms of mean and standard deviation of stress-strain hysteresis loop.

Committee: Kallol Sett Dr. (Advisor); Robert Liang Prof. (Committee Member); David Roke Dr. (Committee Member) Subjects: Civil Engineering