Solvability and Nilpotency of Lie Algebras in Cryptography and Steganography

This paper investigates the role of solvable and nilpotent Lie algebras in the domains of cryptography and steganography, emphasizing their potential in enhancing security protocols and covert communication methods. In the context of cryptography, we explore their application in public-key infrastructure, secure data verification, and the resolution of commutator-based problems that underpin data protection strategies. In steganography, we examine how the algebraic properties of solvable Lie algebras can be leveraged to embed confidential messages within multimedia content, such as images and video, thereby reinforcing secure communication in dynamic environments. We introduce a key exchange protocol founded on the structural properties of solvable Lie algebras, offering an alternative to traditional number-theoretic approaches. The proposed Lie Exponential Diffie–Hellman Problem (LEDHP) introduces a novel cryptographic challenge based on Lie group structures, offering enhanced security through the complexity of non-commutative algebraic operations. The protocol utilizes the non-commutative nature of Lie brackets and the computational difficulty of certain algebraic problems to ensure secure key agreement between parties. A detailed security analysis is provided, including resistance to classical attacks and discussion of post-quantum considerations. The algebraic complexity inherent to solvable Lie algebras presents promising potential for developing cryptographic protocols resilient to quantum adversaries, positioning these mathematical structures as candidates for future-proof security systems. Additionally, we propose a method for secure message embedding using the Lie algebra in combination with frame deformation techniques in animated objects, offering a novel approach to steganography in motion-based media.