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In this paper, we propose a key exchange protocol using multivariate polynomial maps whose security relies on the hardness in finding a solution to a certain system of nonlinear polynomial equations. Under the hardness assumption of solving the system of equations, we prove that our protocol is secure against key recovery attacks by passive attackers if the protocol is established honestly.

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Multivariate public-key cryptography (MPKC) is considered a leading candidate for postquantum cryptography (PQC). It is based on the hardness of the multivariate quadratic polynomial (MQ) problem, which is a problem of finding a solution to a system of quadratic equations over a finite field. In this paper, we survey some recent progress in the security analysis of MPKC. Among various existing multivariate schemes, the most important one is the Rainbow signature scheme proposed by Ding et al. in 2005, which was later selected as a finalist in the third round of the PQC standardization project by the National Institute of Standards and Technology. Under the circumstances, some recent research studies in MPKC have focussed on the security analysis of the Rainbow scheme. In this paper, the authors first explain efficient algorithms for solving the MQ problem and the research methodology for estimating their complexity in MPKC. Then, the authors survey some recent results related to the security analysis of the Rainbow scheme. In particular, the authors provide a detailed description of the complexity analysis for solving the bi-graded polynomial systems studied independently by Nakamura et al. and Smith-Tone et al., and then expound the rectangular MinRank attack against Rainbow proposed by Beullens.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.