Design and Implementation of an Encryption Algorithm Based on GF(5m) in Information Security

Abstract

The rapid development of information technology presents serious challenges in maintaining data confidentiality, authenticity, and integrity. Modern cryptography heavily relies on abstract algebraic structures, particularly finite fields or Galois Fields (GF), as the foundation for designing encryption algorithms. Most popular algorithms, such as AES, use GF(2^m) due to its compatibility with binary representation. However, the utilization of non-binary bases, specifically GF(5^m), is relatively under-researched despite its potential to increase algebraic structure diversity and expand the key space. This article discusses the design and implementation of an encryption algorithm based on GF(5^m). The encryption process involves key generation from an irreducible polynomial, substitution via a non-linear S-Box, and diffusion through matrix multiplication in the GF(5^m) domain. Implementation was carried out using Python, with messages represented as polynomials. Test results show that this algorithm can produce ciphertext with a random symbol distribution, has relatively efficient computation time for short messages, and possesses a large key space, making it resistant to brute-force attacks. Furthermore, the non-linear property of the S-Box in GF(5^m) provides better resistance against linear and differential cryptanalysis compared to GF(2^m)-based approaches. However, further analysis is needed regarding computational optimization and resistance to advanced cryptanalytic attacks. Thus, encryption algorithms based on GF(5^m) offer a promising alternative for developing modern information security systems and open opportunities for further research on the use of non-binary finite fields for future cryptography.