9,659 research outputs found
Rotation symmetric Boolean functions---count and cryptographic properties
The article of record as published may be located at http://dx.doi.org/10.1.1.137.6388Rotation symmetric (RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnsideï¾’s lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2gn, where gn = 1 nPt|n (t) 2n t , and (.) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in Fn2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree > 2. Further, we studied the RotS functions on 5, 6, 7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier
On the Systematic Constructions of Rotation Symmetric Bent Functions with Any Possible Algebraic Degrees
In the literature, few constructions of -variable rotation symmetric bent
functions have been presented, which either have restriction on or have
algebraic degree no more than . In this paper, for any even integer
, a first systemic construction of -variable rotation symmetric
bent functions, with any possible algebraic degrees ranging from to , is
proposed
Bose-Einstein condensates with a bent vortex in rotating traps
We consider a 3D dilute Bose-Einstein condensate at thermal equilibrium in a
rotating harmonic trap. The condensate wavefunction is a local minimum of the
Gross-Pitaevskii energy functional and we determine it numerically with the
very efficient conjugate gradient method. For single vortex configurations in a
cigar-shaped harmonic trap we find that the vortex line is bent, in agreement
with the numerical prediction of Garcia-Ripoll and Perez-Garcia, Phys.Rev.A 63,
041603 (2001). We derive a simple energy functional for the vortex line in a
cigar-shaped condensate which allows to understand physically why the vortex
line bends and to predict analytically the minimal rotation frequency required
to stabilize the bent vortex line. This analytical prediction is in excellent
agreement with the numerical results. It also allows to find in a simple way a
saddle point of the energy, where the vortex line is in a stationary
configuration in the rotating frame but not a local minimum of energy. Finally
we investigate numerically the effect of thermal fluctuations on the vortex
line for a condensate with a straight vortex: we can predict what happens in a
single realization of the experiment by a Monte Carlo sampling of an atomic
field quasi-distribution function of the density operator of the gas at thermal
equilibrium in the Bogoliubov approximation.Comment: 25 pages, 20 figures, version including discussion of energy saddle
poin
Tetrahedratic mesophases, chiral order, and helical domains induced by quadrupolar and octupolar interactions
We present an exhaustive account of phases and phase transitions that can be stabilized in the recently introduced generalized Lebwohl-Lasher model with quadrupolar and octupolar microscopic interactions [ L. Longa, G. PajÄ…k and T. Wydro Phys. Rev. E 79 040701 (2009)]. A complete mean-field analysis of the model, along with Monte Carlo simulations allows us to identify four distinct classes of the phase diagrams with a number of multicritical points where, in addition to the standard uniaxial and biaxial nematic phases, the other nematic like phases are stabilized. These involve, among the others, tetrahedratic (T), nematic tetrahedratic (NT), and chiral nematic tetrahedratic (NT*) phases of global Td, D2d, and D2 symmetry, respectively. Molecular order parameters and correlation functions in these phases are determined. We conclude with generalizations of the model that give a simple molecular interpretation of macroscopic regions with opposite optical activity (ambidextrous chirality), observed, e.g., in bent-core systems. An estimate of the helical pitch in the NT* phase is also given
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