# On new multivariate cryptosystems based on hidden Eulerian equations

 1Ustimenko, VA1Institute of Telecommunications and Global Information Space of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2017, 5:17-24 https://doi.org/10.15407/dopovidi2017.05.017 Section: Information Science and Cybernetics Language: English Abstract:  We propose new multivariate cryptosystems over an n-dimensional free module over the arithmetical ring Zm based on the idea of hidden discrete logarithm for ${Z_{m}}^{*}$. These cryptosystems are based on the hidden Eulerian equations. If m is a "sufficiently large" product of at least two large primes, then the solution of the equation is hard without knowledge of the decomposition of m. In the Postquantum Era, one can solve the factorization problem for m and the discrete logarithm problem for ${Z_{m}}^{*}$. However, it does not lead to the straightforward break of such cryptosystem, because of the parameter α is unknown. Some examples of such cryptosystems were already proposed. We define their modifications and generalizations based on the idea of Eulerian transformations, which allow us to use asymmetric algorithms based on families of nonlinear multiplicatively injective maps with prescribed polynomial density and degree bounded by constant. Keywords: algebraic graphs, complexity estimates, hidden discrete logarithm problem, hidden Eulerian equations, multivariate cryptography, postquantum cryptography, public keys
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