Modern Fully Homomorphic Encryption (FHE)

The concept of Fully Homomorphic Encryption (FHE) was believed to be the ultimate goal of cryptography until 2009 when Gentry presented the first FHE scheme. FHE allows multiple operations on plaintext without the need for decryption, hence enable arbitrary complex computation on encrypted data. It addresses privacy concerns in different fields, such as machine learning, cloud computing and fog computing. In order to achieve this goal both addition and multiplication must be executed homomorphically since any Boolean (arithmetic) circuit can be represented by XOR (addition) and AND (multiplication). This thesis work includes the explanation of FHE from theoretical definitions and security properties to the possible use cases in which it can be applied. It then presents the still existing and most widely used libraries, with emphasis on Zama’s concrete and openFHE, which are the most popular and comprehensive, showing their respective strengths and weaknesses, so as to allow justification for the choice of library to carry out testing. At the end, some tests made with concrete are presented which aim to verify the performance of the library in each step that has to be done to enable and compile homomorphically and to calculate the execution times. Starting with some general exhaustive tests in which the times and sizes of the circuits created are analyzed, then it is investigated the execution times of the main mathematical operations, which are: addition, subtraction, multiplication, division, power and square root. These times are compared with those of the operations without FHE enabled in order to calculate how much it costs in terms of performance to decide to apply full homomorphism. In addition, some tests were conducted on the machine learning algorithms implemented in the concrete-ml library by comparing them using the respective metrics with those without using FHE implemented in Python's scikitlearn library.