TrES: Tropical Encryption Scheme Based on Double Key Exchange

Abstract

Shor’s quantum algorithm establishes a polynomial time attack on the discrete logarithm problem in any group. Recent announcements of progress in building quantum computers highlight the need for new concepts to create cryptosystems that are resistant to quantum attacks. In this paper, we present а new message encryption scheme. To enhance the security of the scheme, we suggest double key-exchange protocol (KEP). The first stage of the key exchange uses a matrix power function (MPF) in a tropical semiring. These functions are based on the action of a matrix semiring acting on some matrix set. MPFs can be considered as one-way functions because they are based on some generalized satisfiability problems that are potentially NP-complete. The obtained shared secret key at the first stage of the key exchange serves as an input for the second stage. The security of the second phase relies on the difficulty of the semiring action problem. In our protocol, we suggest using left or right action of the tropical semiring (which can be both min-plus and max-plus) on the group of commutative matrices (circulant matrices, in our case). The fact that the key-exchange protocol works in two phases contributes to its security, since an attacker needs to solve two difficult problems in order to break it. The main advantages of the presented protocol are: the increased efficiency and improved security.