92 research outputs found
On the computation of the values of zeta functions of totally real cubic fields
AbstractBased on earlier papers of the first author we give a concise formula for the values of class zeta functions of totally real cubic fields at even positive integers which is the exact analogue of the Barn-Siegel formula for real quadratic fields. For this purpose we use a rather complicated series representation for the aforementioned values depending on a parameter x which is analyzed for x → 0. The final formula is well suited for actual computations; two tables of values of class zeta functions are given at the end of the paper
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
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
A combinatorial model for reversible rational maps over finite fields
We study time-reversal symmetry in dynamical systems with finite phase space,
with applications to birational maps reduced over finite fields. For a
polynomial automorphism with a single family of reversing symmetries, a
universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has
been conjectured to exist, for the normalized cycle lengths of the reduced map
in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18
(2005) 2171-2192). We show that these statistics correspond to those of a
composition of two random involutions, having an appropriate number of fixed
points. This model also explains the experimental observation that,
asymptotically, almost all cycles are symmetrical, and that the probability of
occurrence of repeated periods is governed by a Poisson law.Comment: LaTeX, 19 pages with 1 figure; to be published in Nonlinearit
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 2Θ(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 = Θ(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
Faster Sieving for Shortest Lattice Vectors Using Spherical Locality-Sensitive Hashing
Recently, it was shown that angular locality-sensitive hashing (LSH) can be used to significantly speed up lattice sieving, leading to heuristic time and space complexities for solving the shortest vector problem (SVP) of . We study the possibility of applying other LSH methods to sieving, and show that with the recent spherical LSH method of Andoni et al.\ we can heuristically solve SVP in time and space . We further show that a practical variant of the resulting SphereSieve is very similar to Wang et al.'s two-level sieve, with the key difference that we impose an order on the outer list of centers. Keywords: lattices, shortest vector problem, sieving algorithms, (approximate) nearest neighbor problem, locality-sensitive hashin
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