Abstract In this article, we survey and analyze the role of tropical semirings in key-exchange protocols proposed in the past decade. We also present new ideas on cryptanalysis of some tropical key agreement techniques suggested during this period.

A review of the tropical approach in cryptography

Abstract In this article, we make use of an algebraic structure that is derived from a tropical semiring and present a key exchange scheme based on it. This is followed by a security analysis of the protocol. We conclude the article by showing that this scheme is more secure than its other tropical counterparts.

Key exchange protocol based upon a modified tropical structure

Because there is no multiplication of numbers in tropical algebra and the problem of solving the systems of polynomial equations in tropical algebra is NP-hard, in recent years some public key cryptography based on tropical semiring has been proposed. But most of them have some defects. This paper proposes new public key cryptosystems based on tropical matrices. The security of the cryptosystem relies on the difficulty of the problem of finding multiple exponentiations of tropical matrices given the product of the matrices powers when the subsemiring is hidden. This problem is a generalization of the discrete logarithm problem. But the problem generally cannot be reduced to discrete logarithm problem or hidden subgroup problem in polynomial time. Since the generating matrix of the used commutative subsemirings is hidden and the public key matrices are the product of more than two unknown matrices, the cryptosystems can resist KU attack and other known attacks. The cryptosystems based on multiple exponentiation problem can be considered as a potential postquantum cryptosystem.

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Tropical Cryptography Based on Multiple Exponentiation Problem of Matrices

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.

TrES: Tropical Encryption Scheme Based on Double Key Exchange

In this paper, we consider the new version of tropical cryptography protocol, i.e the tropical version of ElGamal encryption. We follow the ideas and modify the classical El Gamal encryption using tropical matrices and matrix power in tropical algebra. Then we also provide a toy example for the reader’s understanding.

The tropical Version of ElGamal Encryption

Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansi...

On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product

A tropical version of Stickel’s key exchange protocol was suggested by Grigoriev and Shpilrain (2014) and successfully attacked by Kotov and Ushakov (2018). We suggest some modifications of this scheme that use commuting matrices in tropical algebra and discuss some possibilities of attacks on these new modifications. We suggest some simple heuristic attacks on one of our new protocols, and then we generalize the Kotov and Ushakov attack on tropical Stickel’s protocol and discuss the application of that generalized attack to all our new protocols.

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Modifying the Tropical Version of Stickel’s Key Exchange Protocol

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Modifying The Tropical Version of Stickel's Key Exchange Protocol.

Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansion of tropical matrix powers, we suggest a simple algorithm for the following tropical discrete logarithm problem: "Given that $A=V\otimes F^{\otimes t}$ for a unique $t$ and matrices $A$, $V$, $F$ of appropriate dimensions, find this $t$." We then use this algorithm to suggest a simple attack on a protocol based on the tropical semidirect product. The algorithm and the attack are guaranteed to work in some important special cases and are shown to be efficient in our numerical experiments.

Sergei Sergeev

Papers

On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product

Modifying the Tropical Version of Stickel’s Key Exchange Protocol

Modifying The Tropical Version of Stickel's Key Exchange Protocol.

On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product