Equivalences and black-box separations of Matrix Diffie-Hellman problems
Author
Villar, J.
Type of activity
Presentation of work at congresses
Name of edition
20th International Conference on Practice and Theory of Public-Key Cryptography
Date of publication
2017
Presentation's date
2017-03-30
Book of congress proceedings
Public key cryptography (PKC 2017): 20th IACR International Conference on Practice and Theory in Public-Key Cryptography: Amsterdam, The Netherlands: march 28-31, 2017: proceedings
In this paper we provide new algebraic tools to study the relationship between different Matrix Diffie-Hellman (MDDH) Problems, which are recently introduced as a natural generalization of the so-called Linear Problem. Namely, we provide an algebraic criterion to decide whether there exists a generic black-box reduction, and in many cases, when the answer is positive we also build an explicit reduction with the following properties: it only makes a single oracle call, it is tight and it makes us...
In this paper we provide new algebraic tools to study the relationship between different Matrix Diffie-Hellman (MDDH) Problems, which are recently introduced as a natural generalization of the so-called Linear Problem. Namely, we provide an algebraic criterion to decide whether there exists a generic black-box reduction, and in many cases, when the answer is positive we also build an explicit reduction with the following properties: it only makes a single oracle call, it is tight and it makes use only of operations in the base group. It is well known that two MDDH problems described by matrices with a different number of rows are separated by an oracle computing certain multilinear map. Thus, we put the focus on MDDH problems of the same size. Then, we show that MDDH problems described with a different number of parameters are also separated (meaning that a successful reduction cannot decrease the amount of randomness used in the problem instance description). When comparing MDDH problems of the same size and number of parameters, we show that they are either equivalent or incomparable. This suggests that a complete classification into equivalence classes could be done in the future. In this paper we give some positive and negative partial results about equivalence, in particular solving the open problem of whether the Linear and the Cascade MDDH problems are reducible to each other. The results given in the paper are limited by some technical restrictions in the shape of the matrices and in the degree of the polynomials defining them. However, these restrictions are also present in most of the work dealing with MDDH Problems. Therefore, our results apply to all known instances of practical interest.
The final publication is available at link.springer.com
Citation
Villar, J. Equivalences and Black-Box Separations of Matrix Diffie-Hellman Problems. A: International Conference on Practice and Theory in Public Key Cryptography. "Public-Key Cryptography - PKC 2017". Amsterdam: Springer, 2017, p. 435-464.