Homomorphic Encryption and Cryptanalysis of Lattice Cryptography

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

Applying Fully Homomorphic Encryption: Practices and Problems

Fully homomorphic encryption (FHE) has been regarded as the "holy grail" of cryptography for its versatility as a cryptographic primitive and wide range of potential applications. Since Gentry published the first theoretically feasible FHE design in 2008, there has been a lot of new discoveries and inventions in this particular field. New schemes significantly reduce the computational cost of FHE and make practical deployment within reach. As a result, FHE schemes have come off the paper and been explored and tested extensively in practice. However, FHE is made possible with many new problems and assumptions that are not yet well studied. In this thesis we present a comprehensive and intuitive overview of the current applied FHE landscape, from design to implementation, and draw attention to potential vulnerabilities both in theory and in practice. In more detail, we show how to use currently available FHE libraries for aggregation and select parameters to avoid weak FHE instances