Post-quantum public-key cryptoschemes on finite algebras

One direction in the development of practical post-quantum public-key cryptographic algorithms is the use of finite algebras as their algebraic carrier. Two approaches in this direction are considered: 1) construction of electronic digital signature algorithms with a hidden group on non-commutative associative algebras and 2) construction of multidimensional cryptography algorithms using the exponential operation in a vector finite field (in a commutative algebra, which is a finite field) to specify a nonlinear mapping with a secret trapdoor. The first approach involves the development of two types of cryptoschemes: those based on the computational difficulty of a) the hidden discrete logarithm problem and b) solving a large system of quadratic equations. For the second type, problems arise in ensuring complete randomization of the digital signature and specifying non-commutative associative algebras of large dimension. Ways to solve these problems are discussed. The importance of studying the structure of finite non-commutative algebras from the point of view of decomposition into a set of commutative subalgebras is shown. Another direction is aimed at a significant (10 or more times) reduction in the size of the public key in multivariate-cryptography algorithms and is associated with the problem of developing formalized, parameterizable, unified methods for specifying vector finite fields of large dimensions (from 5 to 130) with a sufficiently large number of potentially implementable types and modifications each type (up to 2$^{500}$ or more). Variants of such methods and topologies of nonlinear mappings on finite vector fields of various dimensions are proposed. It is shown that the use of mappings that specify the exponential operation in vector finite fields potentially eliminates the main drawback of known multivariate-cryptography algorithms, which is associated with the large size of the public key.