21,148 research outputs found
Faster Initial Splitting for Small Characteristic Composite Extension Degree Fields
Let be a small prime and be a composite integer.
For the function field sieve algorithm applied to , Guillevic (2019) had proposed an algorithm for initial
splitting of the target in the individual logarithm phase. This algorithm generates polynomials and tests them for -smoothness
for some appropriate value of . The amortised cost of generating each polynomial is multiplications over .
In this work, we propose a new algorithm for performing the initial splitting which also generates and tests polynomials
for -smoothness. The advantage over Guillevic splitting is that in the new algorithm, the cost of generating a polynomial
is multiplications in , where is the relevant smoothness probability
Trapped Rydberg Ions: From Spin Chains to Fast Quantum Gates
We study the dynamics of Rydberg ions trapped in a linear Paul trap, and
discuss the properties of ionic Rydberg states in the presence of the static
and time-dependent electric fields constituting the trap. The interactions in a
system of many ions are investigated and coupled equations of the internal
electronic states and the external oscillator modes of a linear ion chain are
derived. We show that strong dipole-dipole interactions among the ions can be
achieved by microwave dressing fields. Using low-angular momentum states with
large quantum defect the internal dynamics can be mapped onto an effective spin
model of a pair of dressed Rydberg states that describes the dynamics of
Rydberg excitations in the ion crystal. We demonstrate that excitation transfer
through the ion chain can be achieved on a nanosecond timescale and discuss the
implementation of a fast two-qubit gate in the ion chain.Comment: 26 pages, 9 figure
Faster individual discrete logarithms in finite fields of composite extension degree
International audienceComputing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms in large and medium characteristic fields (e.g., {GF}, {GF}) are the Number Field Sieve and its variants (special, high-degree, tower). The best algorithms in small characteristic finite fields (e.g., {GF}) are the Function Field Sieve, Joux's algorithm, and the quasipolynomial-time algorithm. The last step of this family of algorithms is the individual logarithm computation. It computes a smooth decomposition of a given target in two phases: an initial splitting, then a descent tree. While new improvements have been made to reduce the complexity of the dominating relation collection and linear algebra steps, resulting in a smaller factor basis (database of known logarithms of small elements), the last step remains at the same level of difficulty. Indeed, we have to find a smooth decomposition of a typically large element in the finite field. This work improves the initial splitting phase and applies to any nonprime finite field. It is very efficient when the extension degree is composite. It exploits the proper subfields, resulting in a much more smooth decomposition of the target. This leads to a new trade-off between the initial splitting step and the descent step in small characteristic. Moreover it reduces the width and the height of the subsequent descent tree
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Eutectic colony formation: A phase field study
Eutectic two-phase cells, also known as eutectic colonies, are commonly
observed during the solidification of ternary alloys when the composition is
close to a binary eutectic valley. In analogy with the solidification cells
formed in dilute binary alloys, colony formation is triggered by a
morphological instability of a macroscopically planar eutectic solidification
front due to the rejection by both solid phases of a ternary impurity that
diffuses in the liquid. Here we develop a phase-field model of a binary
eutectic with a dilute ternary impurity and we investigate by dynamical
simulations both the initial linear regime of this instability, and the
subsequent highly nonlinear evolution of the interface that leads to fully
developed two-phase cells with a spacing much larger than the lamellar spacing.
We find a good overall agreement with our recent linear stability analysis [M.
Plapp and A. Karma, Phys. Rev. E 60, 6865 (1999)], which predicts a
destabilization of the front by long-wavelength modes that may be stationary or
oscillatory. A fine comparison, however, reveals that the assumption commonly
attributed to Cahn that lamella grow perpendicular to the envelope of the
solidification front is weakly violated in the phase-field simulations. We show
that, even though weak, this violation has an important quantitative effect on
the stability properties of the eutectic front. We also investigate the
dynamics of fully developed colonies and find that the large-scale envelope of
the composite eutectic front does not converge to a steady state, but exhibits
cell elimination and tip-splitting events up to the largest times simulated.Comment: 18 pages, 18 EPS figures, RevTeX twocolumn, submitted to Phys. Rev.
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