A key exchange protocol relying on polynomial maps

Abstract

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.

An Improvement of a Key Exchange Protocol Relying on Polynomial Maps

“…Even if the system of equations has multiple solutions in F n q , the solution in Z n p will be unique with high probability, which enables the party to successfully determine the correct common key. Our proposed protocol with sizes of parameters q and n similar to those of [2] indeed reduces the failure probability significantly. Moreover, we propose a parameter set for our proposed protocol that might achieve both failure probability 2 −120 and 128-bit security level.…”

“…In this paper, we give an improvement of the key exchange protocol by Akiyama et al [2] in reducing the failure probability while keeping the security level. Roughly speaking, in the improved version of Akiyama et al's protocol, one of the two parties solves a certain system of polynomial equations defined over a prime field F q in order to determine the common key.…”

“…Section 2 summarizes some notations and basic properties such as properties of Gröbner basis. In Section 3, we recall the construction of the previous protocol in [2] and summarize advantages and disadvantages of the previous protocol. In Section 4, we describe our proposed protocol and give a theoretical estimate of an upper bound for failure probability of the protocol.…”

“…Section 5 summarizes the results of our computer experiments about our proposed protocol. Section 6 summarizes the results of comparison between our proposed protocol and the previous protocol in [2]. Finally, in Section 7, we evaluate the security of our proposed protocol.…”

“…In 2019, Akiyama et al [2] proposed a key exchange protocol which is based on the hardness of solving a system of multivariate non-linear equations but yet has a design strategy different from ordinary multivariate cryptography. Namely, ordinary multivariate cryptography intends to conceal the internal structure by using compositions of maps, while Akiyama et al's scheme uses both composition and addition of maps to conceal the central map ψ.…”