Paper 2015/353 Original: eprint.iacr.org/2015/353 ↗

Matrix Computational Assumptions in Multilinear Groups

Abstract

We put forward a new family of computational assumptions, the Kernel Matrix Diffie-Hellman Assumption. Given some matrix $\mathbf{A}$ sampled from some distribution $\mathcal{D}$, the kernel assumption says that it is hard to find "in the exponent" a nonzero vector in the kernel of $\mathbf{A}^\top$. This family is the 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 computational assumptions. The $k$-Decisional Linear Assumption is an example of a family of decisional 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 flexible problems (i.e., computational problems with a non unique solution).