Tropical Cryptography Based on Multiple Exponentiation Problem of Matrices

Abstract

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.

Asymmetric Cryptography Based on the Tropical Jones Matrix

“…According to Reference [19], we can give the definition of the ME problem of the tropical Jones matrix.…”

“…Hence, its modifications are not resistant to the generalized KU attack. In 2022, Huang and Li proposed a new key exchange protocol [19] based on the multiple exponentiation problem of matrices, using tropical algebra as a platform and the adjoint matrix of the first polynomial. The analysis results showed that the protocol can resist all known attacks.…”

Secure Key Exchange in Tropical Cryptography: Leveraging Efficiency with Advanced Block Matrix Protocols

“…Diverse matrix forms were incorporated, ranging from singular [2,3] and non-singular matrices [4] to matrices over bit strings [5]; Tribonacci matrices [6]; Hadamar matrices [7]; non-negative [8] and lattice matrices [9]. Tropical matrices are also of interest ( [10][11][12][13][14][15]). Yet, despite these developments, vulnerabilities have persisted, as evidenced by different documented attacks (see, for example, [16][17][18][19]).…”

Secure Key Exchange in Tropical Cryptography: Leveraging Efficiency with Advanced Block Matrix Protocols

“…Various schemes using tropical matrices have been suggested in [11][12][13][14][15][16]; yet, despite these developments, vulnerabilities have persisted, as evidenced by different documented attacks (see for example [17][18][19][20][21][22]).…”