Hypercomplex Numbers in Cryptosystems

Abstract

A systematic analysis of the use of the hypercomplex numbers in the construction of the cryptographic algorithms is given. The cases of constructing various types of ciphers, both symmetric and asymmetric, with the use of hypercomplex numbers of 4, 8, and 16 dimensions are considered. These include determining the complex computational problem underlying the security of the cipher. Among block ciphers, modifications of the Hill and Feistel algorithms are considered. Attention is drawn to the use of involutory matrices in the development of the encryption algorithms. Of the public key systems, the review includes hybrid elliptic curve ciphers, scheme based on the computational quaternion conjugacy problem, and NTRU-type systems based on lattices. The research defines the basic principles for the application of hypercomplex numbers in cryptography. The article can help researchers to get a general view of the ways to solve an identified problem. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.