International Conference on Practice and Theory in Public Key Cryptography

p. 435-464

DOI: 10.1007/978-3-662-54365-8_18

Presentation's date: 2017-03-30

Abstract:

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]]>

Lecture notes in computer science

Vol. 10174, p. 435-464

DOI: 10.1007/978-3-662-54365-8_18

Date of publication: 2017-02

Abstract:

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 cer- tain 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 suc- cessful reduction cannot decrease the amount of randomness used in the problem instance description). When comparing MDDH problems of the same size and number of pa- rameters, 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 par- tial 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]]>

Journal of cryptology

Vol. 30, num. 1, p. 242-288

DOI: 10.1007/s00145-015-9220-6

Date of publication: 2017-01

Abstract:

We put forward a new algebraic framework to generalize and analyze Die-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`;k-MDDH assumption states that it is hard to decide whether a vector in G` is linearly dependent of the columns of some matrix in G`k sampled according to distribution D`;k. It covers known assumptions such as DDH, 2-Lin (linear assumption), and k-Lin (the k-linear assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of D`;k. We use the hardness results to nd new distributions for which the D`;k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for eciency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin Assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH-Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash-proof systems, pseudo-random functions, and Groth-Sahai NIZK and NIWI proofs. As an independent contribution we give more ecient NIZK and NIWI proofs for membership in a subgroup of G`. The results imply very signicant eciency improvements for a large number of schemes.

The final publication is available at Springer via http://dx.doi.org/10.1007/s00145-015-9220-6

We put forward a new algebraic framework to generalize and analyze Di e-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`;k-MDDH assumption states that it is hard to decide whether a vector in G` is linearly dependent of the columns of some matrix in G` k sampled according to distribution D`;k. It covers known assumptions such as DDH, 2-Lin (linear assumption), and k-Lin (the k-linear assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of D`;k. We use the hardness results to nd new distributions for which the D`;k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for e ciency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin Assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH-Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash-proof systems, pseudo-random functions, and Groth-Sahai NIZK and NIWI proofs. As an independent contribution we give more e cient NIZK and NIWI proofs for membership in a subgroup of G`. The results imply very signi cant e ciency improvements for a large number of schemes.]]>

Journal of cryptology

Vol. 30, num. 1, p. 242-288

DOI: 10.1007/s00145-015-9220-6

Date of publication: 2017-01

Abstract:

We put forward a new algebraic framework to generalize and analyze Diffie-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`,k-MDDH assumption states that it is hard to decide whether a vector in ¿ìs linearly dependent of the columns of some matrix in ¿`×k sampled according to distribution D`,k. It covers known assumptions such as DDH, 2-Lin (linear assumption), and k-Lin (the k-linear assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of D`,k. We use the hardness results to find new distributions for which the D`,k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin Assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH-Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash-proof systems, pseudo-random functions, and Groth-Sahai NIZK and NIWI proofs. As an independent contribution we give more efficient NIZK and NIWI proofs for membership in a subgroup of ¿` . The results imply very significant efficiency improvements for a large number of schemes.]]>

Annual International Conference on the Theory and Application of Cryptology and Information Security

p. 729-758

DOI: 10.1007/978-3-662-53887-6_27

Presentation's date: 2016-12

Abstract:

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]]>

Lecture notes in computer science

Vol. 10031, p. 729-758

DOI: 10.1007/978-3-662-53887-6_27

Date of publication: 2016-12

Abstract:

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 link.springer.com]]>

Date: 2015-04-20

Abstract:

We put forward a new family of computational assumptions, the Kernel Matrix Die- Hellman Assumption. Given some matrix A sampled from some distribution D `;k , the kernel as- sumption says that it is hard to nd \in the exponent" a nonzero vector in the kernel of A > . This family is the natural computational analogue of the Matrix Decisional Die-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 the corresponding Kernel Assumption family is also a strictly increasingly weaker family of computational assumptions. This requires ruling out the existence of some black-box reductions between exible problems ( i.e. , com- putational problems with a non unique solution)

We put forward a new family of computational assumptions, the Kernel Matrix Di e- Hellman Assumption. Given some matrix A sampled from some distribution D `;k , the kernel as- sumption says that it is hard to nd \in the exponent]]>