Human activity recognition (HAR) through gait analysis is a very promising research area for early detection of neurodegenerative diseases because gait abnormalities are typical symptoms of some neurodegenerative diseases, such as early dementia. While working with such biometric data, the performance parameters must be considered along with privacy and security issues. In other words, such biometric data should be processed under specific security and privacy requirements. This work proposes an innovative hybrid protection scheme combining a partially homomorphic encryption scheme and a cancelable biometric technique based on random projection to protect gait features, ensuring patient privacy according to ISO/IEC 24745. The proposed hybrid protection scheme has been implemented along a long short-term memory (LSTM) neural network to realize a secure early dementia diagnosis system. The proposed protection scheme is scalable and implementable with any type of neural network because it is independent of the network's architecture. The conducted experiments demonstrate that the proposed protection scheme enables a high trade-off between safety and performance. The accuracy degradation is at most 1.20% compared with the early dementia recognition system without the protection scheme. Moreover, security and computational analyses of the proposed scheme have been conducted and reported.