67 research outputs found
Arithmetic properties of blocks of consecutive integers
This paper provides a survey of results on the greatest prime factor, the
number of distinct prime factors, the greatest squarefree factor and the
greatest m-th powerfree part of a block of consecutive integers, both without
any assumption and under assumption of the abc-conjecture. Finally we prove
that the explicit abc-conjecture implies the Erd\H{o}s-Woods conjecture for
each k>2.Comment: A slightly corrected and extended version of a paper which will
appear in January 2017 in the book From Arithmetic to Zeta-functions
published by Springe
Faster Enumeration-based Lattice Reduction:Root Hermite Factor k1/(2k) Time kk/8+o(k)
International audienc
Shortest vector from lattice sieving: A few dimensions for free
Asymptotically, the best known algorithms for solving the Shortest Vector Problem (SVP) in a lattice of dimension n are sieve algorithms, which have heuristic complexity estimates ranging from (4/3)n+o(n) down to (3/2)n/2+o(n) when Locality Sensitive Hashing techniques are used. Sieve algorithms are however outperformed by pruned enumeration algorithms in practice by several orders of magnitude, despite the larger super-exponential asymptotical complexity 2Î(n log n) of the latter. In this work, we show a concrete improvement of sieve-type algorithms. Precisely, we show that a few calls to the sieve algorithm in lattices of dimension less than n - d solves SVP in dimension n, where d = Î(n/ log n). Although our improvement is only sub-exponential, its practical effect in relevant dimensions is quite significant. We implemented it over a simple sieve algorithm with (4/3)n+o(n) complexity, and it outperforms the best sieve algorithms from the literature by a factor of 10 in dimensions 7080. It performs less than an order of magnitude slower than pruned enumeration in the same range. By design, this improvement can also be applied to most other variants of sieve algorithms, including LSH sieve algorithms and tuple-sieve algorithms. In this light, we may expect sieve-techniques to outperform pruned enumeration in practice in the near future
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
Better Algorithms for LWE and LWR
The Learning With Error problem (LWE) is becoming more and more used in cryptography, for instance, in the design of some fully homomorphic encryption schemes. It is thus of primordial importance to find the best algorithms that might solve this problem so that concrete parameters can be proposed. The BKW algorithm was proposed by Blum et al. as an algorithm to solve the Learning Parity with Noise problem (LPN), a subproblem of LWE. This algorithm was then adapted to LWE by Albrecht et al. In this paper, we improve the algorithm proposed by Albrecht et al. by using multidimensional Fourier transforms. Our algorithm is, to the best of our knowledge, the fastest LWE solving algorithm. Compared to the work of Albrecht et al. we greatly simplify the analysis, getting rid of integrals which were hard to evaluate in the final complexity. We also remove some heuristics on rounded Gaussians. Some of our results on rounded Gaussians might be of independent interest. Moreover, we also analyze algorithms solving LWE with discrete Gaussian noise. Finally, we apply the same algorithm to the Learning With Rounding problem (LWR) for prime q, a deterministic counterpart to LWE. This problem is getting more and more attention and is used, for instance, to design pseudorandom functions. To the best of our knowledge, our algorithm is the first algorithm applied directly to LWR. Furthermore, the analysis of LWR contains some technical results of independent interest
The General Sieve Kernel and New Records in Lattice Reduction
textabstractWe 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 FPLLLâs state of the art
implementation of enumeration at dimension 70
Computing a Lattice Basis Revisited
International audienc
Slide reduction, revisitedâfilling the gaps in svp approximation
We show how to generalize Gama and Nguyen's slide reduction algorithm [STOC
'08] for solving the approximate Shortest Vector Problem over lattices (SVP).
As a result, we show the fastest provably correct algorithm for
-approximate SVP for all approximation factors . This is the range of approximation factors most
relevant for cryptography
Lattice Trapdoors and IBE from Middle-Product LWE
Middle-product learning with errors (MP-LWE) was recently introduced by Rosca, Sakzad, Steinfeld and Stehlé (CRYPTO 2017) as a way to combine the efficiency of Ring-LWE with the more robust security guarantees of plain LWE. While Ring-LWE is at the heart of efficient lattice-based cryptosystems, it involves the choice of an underlying ring which is essentially arbitrary. In other words, the effect of this choice on the security of Ring-LWE is poorly understood. On the other hand, Rosca et al. showed that a new LWE variant, called MP-LWE, is as secure as Polynomial-LWE (another variant of Ring-LWE) over any of a broad class of number fields. They also demonstrated the usefulness of MP-LWE by constructing an MP-LWE based public-key encryption scheme whose efficiency is comparable to Ring-LWE based public-key encryption.
In this work, we take this line of research further by showing how to construct Identity-Based Encryption (IBE) schemes that are secure under a variant of the MP-LWE assumption. Our IBE schemes match the efficiency of Ring-LWE based IBE, including a scheme in the random oracle model with keys and ciphertexts of size (for -bit identities).
We construct our IBE scheme following the lattice trapdoors paradigm of [Gentry, Peikert, and Vaikuntanathan, STOC\u2708]; our main technical contributions are introducing a new leftover hash lemma and instantiating a new variant of lattice trapdoors compatible with MP-LWE.
This work demonstrates that the efficiency/security tradeoff gains of MP-LWE can be extended beyond public-key encryption to more complex lattice-based primitives
Accelerating lattice reduction with FPGAs
International audienceWe describe an FPGA accelerator for the KannanÂâFinckeÂâPohst enumeration algorithm (KFP) solving the Shortest Lattice Vector Problem (SVP). This is the first FPGA implementation of KFP specifically targeting cryptographically relevant dimensions. In order to optimize this implementation, we theoretically and experimentally study several facets of KFP, including its efficient parallelization and its underlying arithmetic. Our FPGA accelerator can be used for both solving stand-alone instances of SVP (within a hybrid CPUÂâFPGA compound) or myriads of smaller dimensional SVP instances arising in a BKZ-type algorithm. For devices of comparable costs, our FPGA implementation is faster than a multi-core CPU implementation by a factor around 2.12
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