Cryptographic Permutation-Based Primitives: Constructing Large, Secure and Efficient Building Blocks for Next-Generation Symmetric Cryptography

Abstract

In the era of big data, cloud computing, and the Internet of Things (IoT), the demand for secure and efficient cryptographic solutions has significantly increased. Permutation-based cryptography has emerged as a versatile and robust paradigm for next-generation symmetric cryptography, offering provable security, high performance, and resistance against both classical and quantum attacks. This paper provides a comprehensive survey of permutation-based cryptography, focusing on public large-state permutations (greater than 256 bits) to meet the demands of post-quantum security and high-throughput applications. We explore key structures such as the Sponge construction (used in SHA-3), the Duplex construction (for authenticated encryption schemes such as Ascon), and the Farfalle construction (enabling parallel processing in pseudorandom functions, e.g., Kravatte), highlighting their security guarantees and efficiency trade-offs. In addition, we analyze the core components of cryptographic permutations, including nonlinear layers (S-boxes and modular addition) and linear diffusion layers (MDS matrices and Column Parity Mixers (CPMs)), together with common design strategies such as Feistel networks, Substitution–Permutation Networks (SPNs), and Addition–Rotation–XOR (ARX) structures. A comparative study of notable permutations, including Keccak-f, Ascon-p, Xoodoo, Gaston, and Gimli, demonstrates their applications in hashing, authenticated encryption, message authentication codes, and pseudorandom number generation. Finally, we discuss emerging challenges related to standardization, quantum resistance, and performance optimization for future cryptographic designs. This work serves as a reference for researchers and practitioners seeking to understand and advance permutation-based cryptography in the post-quantum era.