The Kernel Matrix Diffie-Hellman assumption
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We put forward a new family of computational assumptions, the Kernel Matrix Diffie-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 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).
The final publication is available at https://link.springer.com/chapter/10.1007%2F978-3-662-53887-6_27
CitationMorillo, M., Rafols, C., Villar, J. The Kernel Matrix Diffie-Hellman assumption. A: Annual International Conference on the Theory and Application of Cryptology and Information Security. "Advances in Cryptology -- ASIACRYPT 2016: 22nd International Conference on the Theory and Application of Cryptology and Information Security: Hanoi, Vietnam: December 4-8, 2016: proceedings, part I". Hanoi: Springer, 2016, p. 729-758.