Optimization Algorithms for the computation of mathematical functions in CKKS cryptosystems

The widespread adoption of technology has made data processing a central aspect of our daily lives. Cloud computing provides an efficient infrastructure for managing huge amounts of information, but transferring sensitive data to remote servers raises significant security and privacy concerns. Traditional encryption ensures data security but has a major limitation: performing operations on encrypted data requires decryption before any kind of computation, exposing it to potential vulnerabilities and compromising privacy. To overcome this limitation, homomorphic encryption allows computations to be performed directly on encrypted data, preserving confidentiality even in untrusted environments. OpenFHE is an open-source library that efficiently implements this technology but supports only a limited set of mathematical functions. This thesis explores new methodologies for implementing mathematical functions in a homomorphic setting, focusing on division and square root operations. To this end, three optimization algorithms are considered—namely, Gradient Descent, Hill Climbing, and Simulated Annealing—and a comparative evaluation is conducted to determine the most efficient one in terms of accuracy and execution time. Results have shown that the Hill Climbing algorithm performs outperforms other methods in terms of both accuracy and execution time, providing a powerful approach for computing division and square root functions in a homomorphic environment. This novel solution can enable the widespread adoption of homomorphic encryption by expanding the set of mathematical functions that can be used in this context.