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Full-Text Articles in Algebra
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
All Dissertations
Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …
The Hfd Property In Orders Of A Number Field, Grant Moles
The Hfd Property In Orders Of A Number Field, Grant Moles
All Theses
We will examine orders R in a number field K. In particular, we will look at how the generalized class number of R relates to the class number of its integral closure R. We will then apply this to the case when K is a quadratic field to produce a more specific relation. After this, we will focus on orders R which are half-factorial domains (HFDs), in which the irreducible factorization of any element α∈R has fixed length. We will determine two cases in which R is an HFD if and only if its ring of …
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
All Theses
In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …