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Homomorphic Post-Quantum Cryptography - Evaluation of Module Learning with Error in Homomorphic Cryptography

This thesis investigates the conversion of Ring-LWE (R-LWE)-based homomorphic encryption schemes to Module-LWE (M-LWE) and analyses the resulting perfor- mance differences. The advantage of M-LWE is that a fixed-sized polynomial de- gree can be utilized and the security of the system can be changed by increas- ing the vector/matrix dimension. This is the same concept that is utilized in the CRYSTALS-Kyber encryption scheme. The feasibility of transferring R-LWE to M- LWE is demonstrated based on the BFV homomorphic encryption scheme, showing that a functioning homomorphic encryption can be maintained. While the addition is straightforward, the multiplication necessitates the generation of multiple relin- earization (evaluation) keys. It is demonstrated that the practical performance is only slightly inferior to that of R-LWE, with the advantage of smaller ciphertext sizes. However, there is still considerable scope for improvement in theoretical as- pects, such as the study of security benefits and in practice, in enhancing the general performance.