Tropical cryptography.

*(English)*Zbl 1301.94114This paper analyses replacement of classical algebraic operations with ones based on tropical algebras for two selected cryptographic schemas. It shows that this approach has two primary advantages: it eliminates known vulnerability of selected schemas against various linear algebra attacks and has improved efficiency as one does not have to perform any multiplications since tropical multiplication is the usual addition.

Tropical algebras are built over tropical semiring \(S\) with two operations \(x\oplus y=\min(x,y)\) and \(x\otimes y=x+y\). If we replace linear algebra with tropical one in a key exchange protocol built on an idea of E. Stickel [A new method for exchanging secret keys, Proc. Third Int. Conf. on Information Technology and Applications (ICITA ’05), Vol. 2, 426–430 (2005), doi:10.1109/ICITA.2005.33], we are able to show that it now resists the vulnerability of original approach described in [V. Shpilrain, in: Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008, Lect. Notes Comput. Sci. 5010, 283–288 (2008; Zbl 1142.94360)]. Another cryptographic schema analysed in this paper that would be susceptible to linear algebra attack in the classical case is a public key encryption scheme described in [T. T. Moh, Commun. Algebra 27, No. 5, 2207–2222 (1999; Zbl 0933.94022)]. Two known attacks were analysed and proven infeasible.

Tropical algebras are built over tropical semiring \(S\) with two operations \(x\oplus y=\min(x,y)\) and \(x\otimes y=x+y\). If we replace linear algebra with tropical one in a key exchange protocol built on an idea of E. Stickel [A new method for exchanging secret keys, Proc. Third Int. Conf. on Information Technology and Applications (ICITA ’05), Vol. 2, 426–430 (2005), doi:10.1109/ICITA.2005.33], we are able to show that it now resists the vulnerability of original approach described in [V. Shpilrain, in: Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008, Lect. Notes Comput. Sci. 5010, 283–288 (2008; Zbl 1142.94360)]. Another cryptographic schema analysed in this paper that would be susceptible to linear algebra attack in the classical case is a public key encryption scheme described in [T. T. Moh, Commun. Algebra 27, No. 5, 2207–2222 (1999; Zbl 0933.94022)]. Two known attacks were analysed and proven infeasible.

Reviewer: Vladimír Lacko (Košice)

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