21,912 research outputs found
Faster tuple lattice sieving using spherical locality-sensitive filters
To overcome the large memory requirement of classical lattice sieving
algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS
2016] studied tuple lattice sieving, where tuples instead of pairs of lattice
vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017]
recently improved upon their results for arbitrary tuple sizes, for example
showing that a triple sieve can solve the shortest vector problem (SVP) in
dimension in time , using a technique similar to
locality-sensitive hashing for finding nearest neighbors.
In this work, we generalize the spherical locality-sensitive filters of
Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near
neighbor searching on dense data sets, and we apply these techniques to tuple
lattice sieving to obtain even better time complexities. For instance, our
triple sieve heuristically solves SVP in time . For
practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this
shows that a triple sieve uses less space and less time than the current best
near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology
ePrint Archive 2017/228, available at https://ia.cr/2017/122
Fast optimization algorithms and the cosmological constant
Denef and Douglas have observed that in certain landscape models the problem
of finding small values of the cosmological constant is a large instance of an
NP-hard problem. The number of elementary operations (quantum gates) needed to
solve this problem by brute force search exceeds the estimated computational
capacity of the observable universe. Here we describe a way out of this
puzzling circumstance: despite being NP-hard, the problem of finding a small
cosmological constant can be attacked by more sophisticated algorithms whose
performance vastly exceeds brute force search. In fact, in some parameter
regimes the average-case complexity is polynomial. We demonstrate this by
explicitly finding a cosmological constant of order in a randomly
generated -dimensional ADK landscape.Comment: 19 pages, 5 figure
Two Compact Incremental Prime Sieves
A prime sieve is an algorithm that finds the primes up to a bound . We say
that a prime sieve is incremental, if it can quickly determine if is
prime after having found all primes up to . We say a sieve is compact if it
uses roughly space or less. In this paper we present two new
results:
(1) We describe the rolling sieve, a practical, incremental prime sieve that
takes time and bits of space, and
(2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a
sieve that is simultaneously sublinear, compact, and incremental.
The second result solves an open problem given by Paul Pritchard in 1994
Sifting data in the real world
In the real world, experimental data are rarely, if ever, distributed as a
normal (Gaussian) distribution. As an example, a large set of data--such as the
cross sections for particle scattering as a function of energy contained in the
archives of the Particle Data Group--is a compendium of all published data, and
hence, unscreened. Inspection of similar data sets quickly shows that, for many
reasons, these data sets have many outliers--points well beyond what is
expected from a normal distribution--thus ruling out the use of conventional
techniques. This note suggests an adaptive algorithm that allows a
phenomenologist to apply to the data sample a sieve whose mesh is coarse enough
to let the background fall through, but fine enough to retain the preponderance
of the signal, thus sifting the data. A prescription is given for finding a
robust estimate of the best-fit model parameters in the presence of a noisy
background, together with a robust estimate of the model parameter errors, as
well as a determination of the goodness-of-fit of the data to the theoretical
hypothesis. Extensive computer simulations are carried out to test the
algorithm for both its accuracy and stability under varying background
conditions.Comment: 29 pages, 13 figures. Version to appear in Nucl. Instr. & Meth.
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
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