The Kernel Matrix Diffie-Hellman assumption

Morillo, Paz; Ràfols, Carla; Villar, Jorge L.

doi:http://dx.doi.org/10.1007/978-3-662-53887-6_27

The Kernel Matrix Diffie-Hellman assumption

Citation

Morillo P, Ràfols C, Villar JL. The Kernel Matrix Diffie-Hellman assumption. In: Cheon J., Takagi T, editors. Advances in Cryptology – ASIACRYPT 2016. 22nd International Conference on the Theory and Application of Cryptology and Information Security, Proceedings, Part I; 2016 Dec 4-6; Hanoi, Vietnam. Berlin: Springer; 2016. p. 729-58. (LNCS; no. 10.031). DOI: 10.1007/978-3-662-53887-6_27

Abstract

We put forward a new family of computational assumptions, the Kernel Matrix Diffi-Hellman Assumption. Given some matrix A sampled from some distribution D, the kernel assumption says that it is hard to find \in the exponent" a nonzero vector in the kernel of A>. This family is a natural computational analogue of the Matrix Decisional Diffie-Hellman Assumption (MDDH), proposed by Escala et al. As such it allows to extend the advantages of their algebraic framework to com- putational assumptions. The k-Decisional Linear Assumption is an example of a family of de- cisional assumptions of strictly increasing hardness when k grows. We show that for any such family of MDDH assumptions, the corresponding Kernel assumptions are also strictly increasingly weaker. This requires ruling out the existence of some black-box reductions between exible problems (i.e., computational problems with a non unique solution).