On new multivariate cryptosystems based on hidden Eulerian equations over finite fields

We propose new multivariate cryptosystems over n-dimensional vector space over a finite field Fq based on idea of hidden discrete logarithm problem for F ∗ q. These cryptosystems are based on hidden eulerian equations x = a, (α, q − 1) = 1. The method is based on the idea of Eulerian transformations, which allow us to use asymmetric algorithms based on families of nonlinear multiplicatively injective maps of prescribed polynomial density and flexible degree. Mathematics Subject Classification (2010): 12Y05 12Y99 05C81 05C85 05C90 94A60 14G50 key words: Multivariate Cryptography, Finite Filds, Eulerian equations, Eulerian transformations, Linguistic Graphs, Multivariate Maps of Low Density 1 On Post Quantum and Multivariate Cryptography Post Quantum Cryptography serves for the research on asymmetrical cryptographical algorithms which can be potentially resistant against attacks based on the use of a quantum computer. The security of currently popular algorithms are based on the complexity of the following 3 known hard problems: integer factorisation, discrete logarithm problem, discrete logarithm for elliptic curves. Each of these problems can be solved in polynomial time by Peter Shor’s algorithm for theoretical quantum computer. Despite that the known nowadays small experimental examples of quantum computer are not able to attack currently used cryptographical algorithm, cryptographers have already started research on postquantum security. They have also count on the new results of general complexity theory. The history of international conferences on Post Quantum Cryptography (PQC) started in 2006. We have to notice that Post Quantum Cryptography differs from Quantum Cryptography, which is based on the idea of usage of quantum phenomena to reach better security. Modern PQC is divided into several directions such as Multivariate Cryptography, Lattice base Cryptography, Hash based Cryptography, Code based Cryptography, studies of isogenies for superelliptic curves.