Enhanced Cryptographic Security via a Novel Non-Associative Quaternion Operation and Moufang Loop Structure

Abstract

In this paper, we introduce a novel non-associative operation (◦) on quaternions with the purpose of improving cryptographic security by capitalizing on the structural properties of Moufang loops. The operation, given as P ◦ K = P · K + λ(P · K^∗) · (K · P), where · is the standard quaternion multiplication, K^∗ the conjugate of K, and λ is a small positive parameter controlling the degree of non-associativity, is shown to satisfy the Moufang identity for a small value of  λ. This Moufang loop structure, along with its invertibility, offers significant potential advantages in cryptographic applications where non-associativity can be used to increase resistance to certain cryptanalytic attacks. The controlled introduction of non-associativity via λ increases security and practical implementation. This work is building upon existing research in non-associative algebra and applications in cryptography. The increasing need of secure digital transactions further motivates the need for more research work targeted at encryption methods, including those based on non-associative structures.