The scientific study of privacy-preserving biometrics, represented by the palmprint, face, and iris, has grown tremendously. That being said, there has not been much attention paid to the proper preservation, transmission, and authentication of biometric images used in everyday applications. In this paper, we propose a new complete model for encrypting and decrypting biometric images, including their signing and authentication, using a nested algorithm of 3D Arnold Transform. In addition, the ElGamal Encryption Algorithm for the encryption part and the ElGamal Digital Signature for the signature part are applied. The model is mainly based on the Arnold Transform and Public-Key Cryptosystem, which are convenient for key transfer and fully functional. Here, the model succeeds in encrypting and securing the authentication process for privacy-preserving biometric images. Various tests have been carried out to demonstrate the feasibility and security of the proposed model and have been compared with existing encryption methods to achieve better results. Moreover, the proposed model can also be extended to the storage, transmission, and authentication of biometric data for daily use.