Vector finite fields of characteristic two as algebraic support of multivariate cryptography

Abstract

The central issue of the development of the multivariate public key algorithms is the design of reversible non-linear mappings of $n$-dimensional vectors over a finite field, which can be represented in a form of a set of power polynomials. For the first time, finite fields $GF\left((2^d)^m\right)$ of characteristic two, represented in the form of $m$-dimensional finite algebras over the fields $GF(2^d)$ are introduced for implementing the said mappings as exponentiation operation. This technique allows one to eliminate the use of masking linear mappings, usually used in the known approaches to the design of multivariate cryptography algorithms and causing the sufficiently large size of the public key. The issues of using the fields $GF\left((2^d)^m\right)$ as algebraic support of non-linear mappings are considered, including selection of appropriate values of $m$ and $d$. In the proposed approach to development of the multivariate cryptography algorithms, a superposition of two non-linear mappings is used to define resultant hard-to-reverse mapping with a secret trap door. The used two non-linear mappings provide mutual masking of the corresponding reverse maps, due to which the size of the public key significantly reduces as compared with the known algorithms-analogues at a given security level.