###### Abstract  
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. privacy-preserving biometrics; cryptography; Arnold transform; ElGamal algorithm; digital signature +
Footnote †: journal: Accepted: 11 July 2023  
Currently, there are a number of well-established encryption schemes for image encryption; for example, [PERSON] et al. [2] proposed a Chaos-based encryption algorithm; [PERSON] et al. [3] proposed a fast image encryption algorithm based on parallel computing systems; [PERSON] et al. [4] used optical technology for image encryption; and [PERSON] et al. [5] proposed an image encryption algorithm based on Autoblocking and Electrocardiography. However, biometric data, due to its privacy and uniqueness, is compatible with the high security and integrity assurance features of cryptographic algorithms. The need for separate encryption of biometric data during transmission is more in line with the irreversible and random nature of cryptographic algorithms. In recent years, the rapid development of quantum cryptography has given us a new direction, and many advances have been made in the study of quantum cryptography, such as those made by [PERSON] et al. [6] and [PERSON] et al. [7], which greatly enhanced the security of encryption algorithms and better protected the original data. For example, [PERSON] et al. [8] proposed a model for a complete set of digital signatures and encryption algorithms based on quantum cryptography, opening a new path for many researchers. However, quantum cryptography has more complex hardware requirements, making it difficult to implement. That being said, traditional cryptography is still the mainstream encryption method. Therefore, not all of these schemes are suitable for the storage and transmission of privacy-preserving biometric images, with specific encryption schemes required to securely store this private data. With this in mind, the paper attempts to propose an encryption, decryption, and verification scheme that can solve these problems. Based on the current research in the security of biometric images, especially face, palmprint, and iris, the paper proposes a novel method for encryption, decryption, and authentication in order to achieve better privacy-preserving biometric images. Firstly, the storage and transmission of the face, palmprint, and iris images require the use of an efficient and convenient cryptographic algorithm that needs to be complex and unbreakable, as well as solve specific issues such as the size of the key space and the suitability of the key for storage and transmission. The generation, distribution, and storage of keys are collectively known as key management. Using a single encryption algorithm for encryption is not secure enough and requires nesting of the algorithms to increase the complexity of the model, thus increasing safety. Here, the more common image encryption algorithms are nested with Public-Key Cryptosystems.