Homomorphic encryption in algebraic settings

Abstract

Cryptography methods have been around for a long time to protect sensitive data. With data sets becoming increasingly large we wish to not only store sensitive data in public clouds but in fact, analyse and compute there too. The idea behind homomorphic encryption is that encryption preserves the structure and allows us to perform the same operations on ciphertext as we would on the plaintext. A lot of the work so far restricts the operations that can be performed correctly on ciphertexts. The goal of this thesis is to explore methods for encryption which should greatly increase the amount of analysis and computation that can be performed on ciphertexts. First of all, we will consider the implications of quantum computers on cryptography. There has already been research conducted into quantum-resistant encryption methods. The particular method we will be interested in is still classical. We are assuming these schemes are going to be used in a post-quantum world anyway, we look at how we can use the quantum properties to improve the cryptosystem. More speci cally, we aim to remove a restriction that naturally comes with the scheme restricting how many operations we can perform on ciphertexts. Secondly, we propose a key exchange protocol that works in a polynomial ideal setting. We do this so that the key can be used for a homomorphic cryptography protocol. The advantage of using key exchange over a public key system is that a large proportion of the process needs to be carried out only once instead of needing a more complicated encryption function to use for each piece of data. Polynomial rings are an appropriate choice of structure for this particular type of scheme as they allow us to do everything we need. We will examine how we can perform computation correctly on ciphertexts and address some of the potential weaknesses of such a process. Finally after establishing a fully homomorphic encryption system we will take a more in-depth look at complexity. Measuring the complexity of mathematical problems is, of course, crucial in cryptography, but the choice of measure is something we need to consider seriously. In the nal chapter we will look at generic complexity as its gives us a good feel for how di cult the typical instances of a problem are to solve.