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Fingerprint authentication is a popular security mechanism for smartphones and laptops. However, its adoption in web and cloud environments has been limited due to privacy concerns over storing and processing biometric data on servers. This paper introduces Blind-Touch, a novel machine learning-based fingerprint authentication system leveraging homomorphic encryption to address these privacy concerns. Homomorphic encryption allows computations on encrypted data without decrypting. Thus, Blind-Touch can keep fingerprint data encrypted on the server while performing machine learning operations. Blind-Touch combines three strategies to efficiently utilize homomorphic encryption in machine learning: (1) It optimizes the feature vector for a distributed architecture, processing the first fully connected layer (FC-16) in plaintext on the client side and the subsequent layer (FC-1) post-encryption on the server, thereby minimizing encrypted computations; (2) It employs a homomorphic encryption-compatible data compression technique capable of handling 8,192 authentication results concurrently; and (3) It utilizes a clustered server architecture to simultaneously process authentication results, thereby enhancing scalability with increasing user numbers. Blind-Touch achieves high accuracy on two benchmark fingerprint datasets, with a 93.6% F1- score for the PolyU dataset and a 98.2% F1-score for the SOKOTO dataset. Moreover, Blind-Touch can match a fingerprint among 5,000 in about 0.65 seconds. With its privacy-focused design, high accuracy, and efficiency, Blind-Touch is a promising alternative to conventional fingerprint authentication for web and cloud applications.