A Study of Cryptographic Algorithms on Special Linear Group Over Ring of Integers Modulo m

Despite the development of numerous cryptographic algorithms over the last decade to ensure greater security for the users, the cyberattack forms are still advancing, which creates a challenge for the research community to build more efficient infrastructure. It motivates us to design more advanced algorithms based on a set of matrices that form a group under matrix multiplication, meaning they satisfy the group axioms of closure, associativity, identity, and irreversibility. Specifically, this study explores the application of three cryptographic algorithms on a special linear group of matrices over a ring of integers modulo m. The first algorithm relies on the RSA key exchange algorithm, which we modify to a special linear group. The second is an asymmetric RSA variant of a cryptographic algorithm defined on the co-adjoint orbit of element K belonging to a special linear group. All matrices from the orbit of matrix K have the same order and can be easily encrypted and decrypted by a pair of secret asymmetric cryptographic keys. We apply the Kronecker product of non-Abelian groups to construct an energy-efficient cryptographic algorithm with possible application on sensor networks. Due to the multiplicative property of the Kronecker product, both the encryption and decryption can be performed in parallel over the matrices of low degree. The third one is another RSA variant of a symmetric cryptographic algorithm, which can be encoded with a special linear group to significantly increase its complexity at the stage of non-negative matrix factorization. To encrypt a given set of numbers, we write it in a matrix form with a secret sequence of generators. As a result, the decryption relies on solving a simple system of nonlinear equations. Security of the algorithm lies in its complexity, aimed at building a correct sequence of generators necessary for encryption.