Date of Award

May 2020

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

Committee Member

Shuhong Gao

Committee Member

Colin Gallagher

Committee Member

Kevin James

Committee Member

Felice Manganiello


The vast amount of personal data being collected and analyzed through internet connected devices is vulnerable to theft and misuse. Modern cryptography presents several powerful techniques that can help to solve the puzzle of how to harness data for use while at the same time protecting it---one such technique is homomorphic encryption that allows computations to be done on data while it is still encrypted. The question of security for homomorphic encryption relates to the broader field of lattice cryptography. Lattice cryptography is one of the main areas of cryptography that promises to be secure even against quantum computing.

In this dissertation, we will touch on several aspects of homomorphic encryption and its security based on lattice cryptography. Our main contributions are:

1. proving some heuristics that are used in major results in the literature for controlling the error size in bootstrapping for fully homomorphic encryption,

2. presenting a new fully homomorphic encryption scheme that supports k-bit arbitrary operations and achieves an asymptotic ciphertext expansion of one,

3. thoroughly studying certain attacks against the Ring Learning with Errors problem,

4. precisely characterizing the performance of an algorithm for solving the Approximate Common Divisor problem.



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