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 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 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 G . The results imply very significant efficiency improvements for a large number of schemes.

We propose the General Sieve Kernel (G6K, pronounced /Ze.si.ka/), an abstract stateful machine supporting a wide variety of lattice reduction strategies based on sieving algorithms. Using the basic instruction set of this abstract stateful machine, we first give concise formulations of previous sieving strategies from the literature and then propose new ones. We then also give a light variant of BKZ exploiting the features of our abstract stateful machine. This encapsulates several recent suggestions (Ducas at Eurocrypt 2018; Laarhoven and Mariano at PQCrypto 2018) to move beyond treating sieving as a blackbox SVP oracle and to utilise strong lattice reduction as preprocessing for sieving. Furthermore, we propose new tricks to minimise the sieving computation required for a given reduction quality with mechanisms such as recycling vectors between sieves, on-the-fly lifting and flexible insertions akin to Deep LLL and recent variants of Random Sampling Reduction. Moreover, we provide a highly optimised, multi-threaded and tweakable implementation of this machine which we make open-source. We then illustrate the performance of this implementation of our sieving strategies by applying G6K to various lattice challenges. In particular, our approach allows us to solve previously unsolved instances of the Darmstadt SVP (151, 153, 155) and LWE (e.g. (75, 0.005)) challenges. Our solution for the SVP-151 challenge was found 400 times faster than the time reported for the SVP-150 challenge, the previous record. For exact SVP, we observe a performance crossover between G6K and FPLLL's state of the art implementation of enumeration at dimension 70.

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 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 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 G . The results imply very significant efficiency improvements for a large number of schemes.

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We provide for the first time an asymptotic comparison of all known algorithms for the search version of the Learning with Errors (LWE) problem. This includes an analysis of several lattice-based approaches as well as the combinatorial BKW algorithm. Our analysis of the lattice-based approaches defines a general framework, in which the algorithms of Babai, Lindner-Peikert and several pruning strategies appear as special cases. We show that within this framework, all lattice algorithms achieve the same asymptotic complexity. For the BKW algorithm, we present a refined analysis for the case of only a polynomial number of samples via amplification, which allows for a fair comparison with lattice-based approaches. Somewhat surprisingly, such a small number of samples does not make the asymptotic complexity significantly inferior, but only affects the constant in the exponent. As the main result we obtain that both, lattice-based techniques and BKW with a polynomial number of samples, achieve running time 2 O(n) for n-dimensional LWE, where we make the constant hidden in the big-O notion explicit as a simple and easy to handle function of all LWE-parameters. In the lattice case this function also depends on the time to compute a BKZ lattice basis with block size Θ (n). Thus, from a theoretical perspective our analysis reveals how LWE's complexity changes as a function of the LWE-parameters, and from a practical perspective our analysis is a useful tool to choose LWE-parameters resistant to all known attacks.