As eye tracking becomes pervasive with screen-based devices and head-mounted displays, privacy concerns regarding eye-tracking data have escalated. While state-of-the-art approaches for privacy-preserving eye tracking mostly involve differential privacy and empirical data manipulations, previous research has not focused on methods for scanpaths. We introduce a novel privacy-preserving scanpath comparison protocol designed for the widely used Needleman-Wunsch algorithm, a generalized version of the edit distance algorithm. Particularly, by incorporating the Paillier homomorphic encryption scheme, our protocol ensures that no private information is revealed. Furthermore, we introduce a random processing strategy and a multi-layered masking method to obfuscate the values while preserving the original order of encrypted editing operation costs. This minimizes communication overhead, requiring a single communication round for each iteration of the Needleman-Wunsch process. We demonstrate the efficiency and applicability of our protocol on three publicly available datasets with comprehensive computational performance analyses and make our source code publicly accessible.