HERFA: A Homomorphic Encryption-Based Root-Finding Algorithm

Edge-cloud computing architectures are exposed to significant security challenges. Although general encryption methods can mitigate some of these concerns, they require decryption to perform operations on data, exposing the data and secret keys to potential attacks. Homomorphic encryption (HE), which allows operations on encrypted data without decryption, provides an effective solution to this issue. Applying HE schemes to root-finding algorithms can expand the use of HE to a wider range of real-world applications that involve solving equations. This letter presents an adaptation of the well-known Newton’s method for use in the HE domain. Specifically, it employs a division-free approach to remove the division operation, which is not a basic HE operation. In addition, the proposed method is extended to handle a polynomial multiplicity greater than one for faster convergence. Compared to an alternative implementation that uses a numerical method for division, the proposed HE-based root-finding algorithm (HERFA) significantly reduces the number of sequential multiplications, which is a key factor limiting the feasibility of applications in the HE domain. This reduction allows HERFA to achieve faster execution speeds or higher accuracy.