Nicolas Mordant

Benjamin Miquel

Intermittency and emergence of coherent structures in wave turbulence of a vibrating plate

We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long time numerical simulations makes this system extremely valuable for wave turbulence studies. The purely 2D character of dynamics of the elastic plate makes it much simpler to handle compared to much more complex 3D physical systems that are typical of geo-and astrophysical issues (ocean surface or internal waves, magnetized plasmas or strongly rotating and/or stratified flows). When the forcing is small the observed wave turbulence is consistent with the predictions of the Weak Turbulent Theory. Here we focus on the case of stronger forcing for which coherent structures can be observed. These structures look similar to the folds and D-cones that are commonly observed for strongly deformed static thin elastic sheets (crumpled paper) except that they evolve dynamically in our forced system. We describe their evolution and show that their emergence is associated with statistical intermittency (lack of self similarity) of strongly nonlinear wave turbulence. This behavior is reminiscent of intermittency in Navier-Stokes turbulence. Experimental data show hints of the weak to strong turbulence transition. However, due to technical limitations and dissipation, the strong nonlinear regime remains out of reach of experiments and therefore has been explored numerically.

I. INTRODUCTION

Historical investigations of wave turbulence were motivated by the quest for a statistical modeling of the sea surface or turbulence in magnetized plasmas [1,[START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF]. It led to the development of the Weak Turbulence Theory (WTT): a statistical theory of wave turbulence in the limits of asymptotically large systems and vanishingly small nonlinearity [START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF][START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | [END_REF]. In this theoretical framework, it is possible to demonstrate the occurrence of an energy cascade in forced, out of equilibrium, dispersive wave systems. This result is remarkable in comparison with the theoretical developments of a statistical theory of turbulence where such an energy cascade is accepted for a century but its justification relies mostly on phenomenological arguments [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF].

Although the theory exists for about half a century, clear experimental and numerical evidence of weak turbulence in laboratory experiments is much more recent [START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | [END_REF][START_REF] Falcon | Discrete and continuous dynamical system series B 13[END_REF][START_REF] Nazarenko | [END_REF][8][START_REF] Cadot | Wave Turbulence in Vibrating Plates[END_REF][START_REF] Yarom | [END_REF][11][12]. Part of the difficulties come from the two above mentioned hypotheses underlying the theory which are difficult to fulfill in the laboratory where the systems are of finite size and dissipative. Dissipation must be overcome by strong enough a forcing in order to develop a non linear stage and this may not be compatible with the hypothesis of weak non linearity. Another difficulty lies in the measurements. Indeed waves are characterized by their propagation. Probing the presence of waves in the non linear system and discriminating them from other structures demands a space and time resolved mea- * nicolas.mordant@univ-grenoble-alpes.fr surement of the turbulent field [START_REF] Nazarenko | [END_REF][START_REF] Yarom | [END_REF]13,14]. This is usually challenging due to the very nature of turbulence: a wide range of time and space scales. This is particularly difficult in 3D systems. Note that truly weakly non linear numerical simulations are also challenging due to the scale separation between the linear wave period and the much longer nonlinear timescales. This separation makes numerical simulations extremely costly.

In that respect, the vibrating elastic plate has been shown to be a fruitful physical model for studies of wave turbulence. Thin elastic plates can support the propagation of flexion waves. These waves become non linear when the in-plane stretching due to large amplitude deformation becomes significant. As the non linear interactions grow, a turbulent state can develop [13,[15][16][17][18]. This state is actually responsible for the noise radiated from a shaken plate and has been utilized for centuries to mimic thunder in theaters [START_REF] Mott | Radio sound effects[END_REF]. The intrinsic 2D character of thin plates made possible experimental studies by means of a space and time resolved profilometry technique as well as numerical simulations either of idealized spatially periodic configurations [8,[START_REF] Yokoyama | [END_REF][21][22] or more realistic ones [23]. These studies investigated stationary out of equilibrium cases and pointed out the consistency with the description offered by the WTT as well as limitations in the applicability of this theory, the most serious of which being due to dissipation in experimental systems [8,24,25]. Non stationary situations either of turbulence decay or turbulence buildup have also been studied that are also consistent with the WTT [START_REF] Humbert | Turbulence d'onde dans les plaques minces en vibration[END_REF][START_REF] Miquel | [END_REF][28]. Furthermore, when the forcing is increased, the vibrating plate displays a transition to a regime of stronger non linearity characterized by the generation of strongly non linear coherent structures that induce deviations from the WTT in terms of power spectra as well as the emergence of intermittency [20-22, 29, 30]. In the present article we investigate both experimentally and numerically this strong forcing regime in much more details. We show that the occurrence of singular structures changes the power spectrum with an additional component on top of the propagating waves. We quantify the observed intermittency and show a very good agreement between experiments and numerical simulations.

The article is organized as follow: in section 2, we introduce the governing equations of thin plates and we give a description of the experimental setup and of the numerical methods. The different regimes potentially encountered as the forcing intensifies are presented and qualitatively described in section 3. We identify in section 4 the space-time spectral signature of the coherent structures. From this signature, we infer a first estimate of their mean velocity. A dual analysis in physical space is carried out in section 5, which confirms the physical picture of coherent singularities propagating following a velocity distribution that we relate to the space-time spectrum of the motion. Finally, we demonstrate in section 6 that the emergence of singular structures is associated with the apparition of intermittency in the system. The last section is dedicated to concluding remarks.

II. METHODS

A. The wave equation

The dynamics of the deformation ζ of a shaken thin elastic plate (flat at rest) follows the Föppl-von Karman equations:

∂ tt ζ = - Eh 2 12ρ(1 -σ 2 ) ∆ 2 ζ + 1 ρ {ζ, χ} + F + D , (1a) ∆ 2 χ = -E {ζ, ζ} 2 , (1b)
where the physical properties of the material are described by the following coefficients: Young's modulus E, Poisson's ratio σ, the density ρ. (

) 2
In the limit of vanishingly small amplitude deformations, pure flexion yields the linear part of equation (1a) which translates into the flexion waves dispersion relation:

ω = ±ck 2 with c = Eh 2 12ρ(1-σ 2 )
. As the amplitude of the deformation grows, in-plane stretching comes into play and generates non linear terms in the FvK equations. Despite being in the limit of linear elasticity for the bulk material, non linear terms arise for geometrical reasons: the local stretching induced in an initially flat surface is proportional to the Gaussian curvature G of the deformation ζ whose full expression and leading order in the limit of a weak slope are respectively:

G(ζ) = ∂ xx ζ∂ yy ζ -(∂ xy ζ) 2 (1 + (∂ x ζ) 2 + (∂ y ζ) 2 ) 2 (3a) = G 0 (ζ) 1 + O |∇ζ| 2 where G 0 (ζ) = {ζ, ζ} 2 (3b)
Formally, in-plane stretching is conveniently taken into account at the lowest order by introducing the auxiliary Airy function χ. The Airy function obeys equation (1b) where the leading order of the Gaussian curvature G 0 acts as a source term. For this reason, the statistical properties of the Gaussian curvature are detailed in sections 5 and 6 as the wave turbulence intensity increases in a vibrating plate.

B. Experimental setup

The description of the experiment is briefly recalled here and the reader is referred to earlier publications for more details (see [16,18]). A 1×2 m 2 stainless steel plate of thickness 0.4 mm is hanging vertically under its own weight. The plate is set in motion by an electromagnetic shaker oscillating at 30 Hz anchored in the lower part of the plate. The vibration is recorded by a Fourier transform profilometry technique [31,32] that provides the deformation of the plate resolved both in time (at frequencies up to a 5 kHz) and space (over about a square meter). Seven experiments (run A to G) have been realized and the relative range of injected power P is shown in table I. The maximum power is 50 times the smallest. For stainless steel, E ≈ 2 10 11 Pa, σ ≈ 0.3, ρ = 7800 kg/m 3 thus c ≈ 0.61 m 2 /s.

C. Numerical simulations

The numerical simulation has been described in [8]. The forced Föppl-Von Karman equations of a square plate with periodic boundary conditions are time-stepped using a pseudo-spectral scheme. The linear dissipation D k = -γ k ∂ t ζ k can be chosen either similar to the experimentally measured one or as an "ideal" dissipation that is zero below a cutoff wavenumber (chosen here as k c /2π = 100 m -1 ). The former case has been shown to reproduce qualitatively the measurements [8] whereas the latter case reproduces the scaling properties of the theoretical predictions from the WTT. In this article we consider only the second sort of ideal dissipation.

The system is driven out-of-equilibrium by a spectral forcing of the form:

F k (t) = F 0 e iψ k exp - (|k| -k f ) 2σ 2 k cos (ω k t + φ k ) (4)
which corresponds to forcing with an amplitude F 0 an isotropic crown of wavenumbers centered around the value k f = 5π m -1 with a width σ k = 1 m -1 . Each mode is forced to its linear frequency ω k and the random phases ψ k and φ k ensure a disordered forcing. In the eight runs (named run 1 to 8) reported in this paper, we kept the parameters of the equation constant such that to correspond to a 2 × 2 m 2 plate of steel of thickness 0.4 mm. The magnitude of the forcing F 0 has been increased by factors of 2 (i.e. the relative amplitude spans 1 to 128). The corresponding injected powers (proportional to F 2 0 ) span four orders of magnitude (table I) which is much larger than what is possible in the experiments.

III. MORPHOLOGICAL EVOLUTION OF THE DEFORMATION OF THE VIBRATING PLATE WITH THE FORCING INTENSITY

We characterize in this section the different dynamical regimes observed as the forcing intensity increases. The shape of the Fourier power spectrum E η (k) of the deformation η (integrated over the direction of the wavevector k, assuming isotropy) has been previously observed to evolve with the forcing intensity both experimentally and numerically [START_REF] Yokoyama | [END_REF][21][22]29]. We illustrate these previously published observations with a new set of numerical simulations with a higher forcing magnitude. As is seen in fig. 1, the low wavenumber part of the spectrum steepens progressively as the forcing increases. The same trend is visible on both experiment and numerical simulation. In comparison to the experiment, the numerical simulation is forced much more strongly. Qualitatively the strongest experiment (run G) appears similar to the numerical run 4 as far as the low wavenumber part is concerned. A direct quantitative comparison is not possible as the dissipation processes are very different as well as the boundary conditions. These observations have been related to the creation of strongly nonlinear structures that can be observed in physical space [29]. Indeed fig. 2 shows four snapshots of the deformation of a vibrating plate at four increasing values of the forcing amplitude. The spatial structure of the rugosity of the deformation is obviously changing when the forcing is increased. Large scale structures incorporating ridges and developable cones [33] are getting more and more visible when the amplitude of the forcing grows. At the highest forcing intensity the deformation resembles crumpled paper (be aware though that the magnitude of the vertical deformation has been strongly amplified in fig. 2). These structures can also be unit) from bottom to top and thus evolves on a much larger range than the experiment. The dissipation is also quite different (see text). The vertical dashed line corresponds to the wavenumber over which the dissipation operates. The forcing operates at k/2π = 2.5 m -1 (that corresponds to 30 Hz for linear waves as in the experiment). Black squares show the change of behavior of the spectrum between structure dominated regime (at low k) and weak turbulence at high k.

identified in spectral space: the emergence of the steeper part of the spectrum at low wavenumber has been shown to correspond to the emergence of ridges and cones [29].

We illustrate this phenomenon in figure 3(a) where we display the deformation spectra obtained numerically in runs 1-8 and normalized by k 3 /P 1/3 . This scaling corresponds to the WTT prediction which derives a stationary spectrum of the form: Observe that in absence of coherent structures (weakest runs 1 and 2) the wave turbulence scaling is observed as the spectra collapse on top of each other. They also follow quite convincingly the logarithmic trend predicted by WTT. At intermediate forcing and for low wavenumbers, the spectra departs from the weak case with a large amplitude steeper part. It means that the low k part of the spectrum follows a distinct scaling both in k and P . At higher wavenumber these spectra recover the weak amplitude logarithmic scaling from below and the P 1/3 scaling. For stronger forcing where coherent structures are observed (runs 5-7), three different regions are identified successively in the spectra as frequency increases: the low frequency part is heavily affected by the apparition of ridges and cones and hence develops further a scaling dubbed "Coherent Structures" (CS) scaling which is distinct from the P 1/3 WTT scaling; the small scales remain potentially in a weak wave turbulence regime, as the high frequencies still follow the weak scaling; the highest frequencies at the end of the cascades exhibit a dissipation dominated dynamics associated with a viscous cutoff of the spectrum. This transition in the regimes observed in the spectrum are summarized schematically in figure 4.

E η (k) ∝ P 1/3 log 1/3 (k * /k) k 3 . (5)
Note that the existence of the intermediate regime of weak wave turbulence depends on the relative intensities of the forcing and the dissipation: we observed with our most vigorous forcing (run8) a direct transition from the CS scaling to the viscous cutoff and a complete absence of the WTT scaling (see upper curve of figure 3). Finally, we identify the low frequency CS scaling with P to be proportional to P as demonstrated by displaying the deformation spectra normalized by k 3 /P on fig. 3(b). The spectrum of the singularities has been predicted in [17] to be proportional to 1/k -5 and observed to be proportional to P (thus a global scaling P/k 5 ). This scaling has been first observed in [29] and is confirmed in fig. 3(b). The KZ spectrum is scaling as P 1/3 /k 3 . The former dominates at low k whereas the latter dominates at large k. The two spectra are equal at a critical wavenumber k c such that P 1/3 /k 3 c ≈ P/k 5 c which corresponds to k c ∼ P 1/3 . This behavior is confirmed in fig. 4 where the boundary of the WT and CS regimes scales as P 1/3 .

Injected

(k) ∝ P k -5 ) to the KZ spectrum (E η (k) ∝ P 1/3 log 1/3 (k * /k) k 3
). Thus the transition wavenumber kc is expected to follow roughly kc ∝ P 1/3 , discarding logarithmic corrections.

IV. SPATIO-TEMPORAL SPECTRAL SIGNATURE OF COHERENT STRUCTURES

The natural way to investigate the dynamics of propagating waves is the space-time energy spectral density of the wave elevation E η (k, ω), which we use in this section to obtain a more subtle characterization of the structures presented above. E η (k, ω) is obtained by computing the Fourier transform of the plate deformation η(x, y, t) both in time and space so that to obtain η(k, ω) and thus

E η (k, ω) = |η(k, ω)| 2 .
In order to get a 2D picture that is easier to represent and comprehend than a full 3D spectrum, one takes advantage of the isotropy of the system by computing E η (k, ω) the spectrum summed over the directions of the wave vectors. This spectrum is shown in fig. 5 for a collection of increasing forcing magnitudes in the numerical simulations. The case of weakest forcing is typical of weak turbulence, for which energy is localized in the very vicinity of the linear dispersion relation. A spectacular effect of the increase of the forcing magnitude is the emergence of a "tongue" of energy centered around the zero frequency, in addition to the dispersion relation. This tongue is more and more present when increasing the forcing at its highest magnitudes. This extra component of the spectrum is the most obvious trace of the presence of the structures in the dynamics.

The growth of the tongue of energy is also visible in fig. 6 that shows cuts of E η (k, ω) at two given values of the wavenumber k. At low forcing the spectrum is strongly peaked at frequencies close to the linear dispersion relation with the previously mentioned slight non linear shift. As the non linearity of the waves in the turbulent state is decreasing with the wavenumber, so does the non linear shift: the frequency of the peak is almost equal to the linear frequency at the highest wavenumbers. The widening of the peak with increasing forcing is also visible. At low wavenumber, the width of the peak gets of similar order of magnitude than its central frequency so that at the highest forcing magnitude the peaks are merging as seen for k/2π = 10 m -1 (top curves of top panel of fig. 6). In the range of wavenumbers for which the peaks have merged, a continuous band of wavenumbers and frequencies can be excited, which actually opens the possibility of observing more complex structures than propagating sine waves. At the highest forcing magnitude, a third peak is observed centered on zero frequency. This peak corresponds to non propagative structures.

V. IDENTIFICATION OF THE COHERENT STRUCTURES, STATISTICAL PROPERTIES AND INTERMITTENCY

A. Structures and curvature

As we mentioned when we introduced the Föppl-von Kàrmàn equations, the source of the non linearity is the Gaussian curvature G that induces stretching of the plate which eventually generates coherent structures. Thus we will focus on the study of the statistical properties of this curvature (or rather its simplified form {ζ, ζ}). In addition to the Gaussian curvature we also study the average curvature

L(ζ) = (1 + ∂ x ζ 2 )∂ xx ζ -2∂ x ζ∂ y ζ∂ xy ζ + (1 + ∂ y ζ 2 )∂ yy ζ (1 + ∂ x ζ 2 + ∂ y ζ 2 ) 3/2 , ( 6a
) = L 0 (ζ) 1 + O |∇ζ| 2 where L 0 (ζ) = ∂ xx ζ + ∂ yy ζ . (6b)
Since L 0 is a linear function of ζ, we use it to estimate L. The rms value of the gradient of the deformation is at most 0.36 for run 8 and significantly lower for the other runs. Hence the square gradient is of the order of 10% which justifies the use of the approximate expressions of L ≈ L 0 and G ≈ G 0 in the following. Figure 7 displays some snapshots of both sorts of curvature for weak and strong forcing intensities. The cur- vatures are obviously very small scale quantities as expected from their expression that involve second order derivatives. At the weakest forcing, the deformation field is close to Gaussian statistics and both curvatures look totally disordered. When increasing the forcing a spatial organisation of the curvature becomes obvious. At intermediate forcing an organisation under the form of stripes appear. At the strongest forcing, very intense localized structures dominate over the background. These localized events show a spatial structure with crescent shapes. These crescent shapes have been also observed for G in the case of static developable cones, near the tip of the singularity and they are the footprint of stress focussing in the plate's deformation [33][34][START_REF] Amar | Proc. R. Soc. Lond. A[END_REF]. This observation is confirmed by the shape of the probability density functions (PDF) shown in fig. 8. At weak forcing, the distribution of L is close to Gaussian as expected for weak turbulence. In very weak turbulence, the statistics are expected to show very little departure from Gaussian statistics. The distribution of G is not Gaussian in the weak case which is expected since G is quadratic in the second order derivatives of ζ. The distribution of G is seen to be asymmetric with power law tails, the positive events being more probable than the negative ones. A sharp cusp is also observed for G close to zero. This shape is reminiscent of the distribution of the product of two correlated Gaussian random variables [START_REF] Falcon | [END_REF] that displays such an asymmetric shape with exponential tails. Here the leading order of the Gaussian curvature 2 is more complex than just the product of two random variables as it is the difference between two quadratic terms, nevertheless the shape of its distribution is clearly related to this simple case. When the intensity of the forcing is increased, the shape deviates from the Gaussian case. The distribution of both curvature displays very wide tails. These tails are symmetric for L as expected from the ζ ↔ -ζ symmetry of the FvK equations and from L being an odd function of ζ. By contrast the distribution of G is not symmetric: this can be understood by noticing that G is an even function of ζ so that negative events stem from different relative contributions of the two terms ∂ xx ζ∂ yy ζ and ∂ xy ζ 2 instead of being merely obtained by symmetry ζ → -ζ. Extreme negative curvatures (saddle-like structures) are more probable than the positive ones.

G 0 = ∂ xx ζ∂ yy ζ -(∂ xy ζ)
Thus our observations show the appearance of localized intense structures at strong forcing that are consistent with our previous interpretation of these structures being developable cones. At intermediate values of the forcing, a spatial organization of the curvature is also particularly visible in the case of the average curvature L in fig. 7. Strong events are organized spatially in stripes of either strong positive or negative values. This spatial organisation is the trace of the large scale folds that can be seen on the snapshots of the deformation in fig. 2. The traces of the folds are less visible on G as for a simple fold the Gaussian curvature is expected to be zero. The folds are visible nonetheless because of the fluctuations at the smallest scales that are sensitive to the flexion present in the vicinity of the crest of the folds.

B. Motion of the structures

We perform a detection and tracking in time of the strong events of curvature. D-cones are characterized by strong values of the Gaussian curvature. One technical issue for detection of D-cones is that the Gaussian curvature is extremely sensitive to the presence of small scale waves that propagate on top of the structures. They induce very strong small scale fluctuations of the Gaussian curvature that oscillate between very large negative and positive values. The detection of the position of the structures is thus very noisy making the tracking of the structures quite difficult. As the D-cones are always surrounded by a crescent of strong average curvature (related to the folds that connect at the cones) that appears to be much smoother in space, detection using large values of |L| is more efficient. We then apply a tracking procedure using the algorithm developed by N. Ouellette [37] [38]. In this way we obtain trajectories of the strong events. These trajectories have a finite duration as these structures have a finite lifetime and ultimately fall below the detection threshold. By differentiating the trajectories in time, we obtain their velocity. This step requires a low-pass filtering operation to remove the noise around the actual trajectories due to the errors on the estimate of the position of the structure. Determining accurately the cutoff frequency of the filter is an issue and remains somewhat arbitrary. The PDF of the velocity of the strong events is shown in fig. 9. Such tracking is possible only at the highest forcing intensities for which the strong events are clearly overcoming the fluctuating background due to the small scale waves. We see that the distribution of the velocity is close to Gaussian. The velocity variance seems to increase slightly with the forcing intensity and is of the order of 20 m/s. We have no clue so far to explain this particular value for the velocity. II. Values of the root mean square velocities of the singularities. The major source of uncertainty on the evaluation of the velocity is the presence of noise in the trajectories. This noise has to be filtered [40] before differentiating the trajectories to obtain the velocity. Getting the adequate cutoff frequency of the filter is not obvious in this case so that the values of the velocity given above should be taken as an order of magnitude rather than an accurate estimate. consider the strongest forcing as observed in the numerical simulations in run 8 (fig. 10), for which the feature is the most visible. The frequency bandwidth of this peak is seen to increase with the wavenumber so that the peaks observed at various given wavenumbers do not collapse in fig. 10(a). By contrast, when the frequency is normalized by the wavelength λ = 2π k , the central peaks for various values of k collapse fairly on each other. In this representation, the width of the central peak defines a quantity homogeneous to a velocity. The dashed line in fig. 10(b) is a Gaussian shape of width 20 m/s which reproduces well the shape of the peak. This figure can also be drawn for the second highest forcing and gives a similar result. At lower forcing intensities the central peak is not strong enough so that the estimate of its width is not possible with enough accuracy. The consistency between our spectral estimate and our direct measurement in physical space supports strongly the fact the peak observed in fig. 10 is indeed associated to the motion of coherent structures. Furthermore, the velocity of sweeping of these structures can be estimated precisely using the spectral analysis described above. This central peak is actually similar to the space-time spectrum of isotropic homogeneous Navier-Stokes turbulence in which no waves are present and only sweeping of vortices is observed (see fig. 2 of reference [39]).

We propose below a simple 1D model of the random motion of such structures in the spirit of previous work by Kuznetsov [41]. Let us assume that some spatial structures f (r) (assumed identical and isotropic for simplicity) are placed randomly in space (following a Poisson distribution) with an average density n. The structures translate with velocities that are distributed following a isotropic distribution. P (u) is the probability distribution of the velocity u. Although we are not able to demonstrate it rigorously, Monte-Carlo-like simulations suggest that the space-time spectrum of such signal may be:

E η (k, ω) ∝ n| f (k)| 2 k P (ω/k) ( 7
)
where f (k) is the Fourier transform of a single structure f (r).

If we trust this expression, then the shape of the collapsed cuts of the spectrum (at given values of k) shown in fig. 10(b) is reminiscent of the distribution of the ve-The shape of the observed peak is seen to follow a Gaussian variation. As has been shown above the distribution of the velocity of the structures is indeed very close to a Gaussian distribution. If one takes the case ω = 0 in the formula [START_REF] Nazarenko | [END_REF] we obtain E(k, 0) ∝ n| f (k)| 2 k P (0) which is mainly the spectrum of the structure f (r). Similar spectra at ω = 0 for the numerical simulation are shown in fig. 11. The spectrum E(k, 0)/ E η (k, ω)dkdω that is shown has been summed over directions which induces an extra k factor so that the curves that are shown should correspond to | f (k)| 2 following (7). The 3 curves corresponding to the lowest forcing are almost superimposed and correspond to cases without structures. They are not zero because of the slight spread of the energy concentrated around the dispersion relation. When the forcing is further increased the spectrum at zero frequency shows the appearance of the structures and their spectrum is then extending more and to small scales. At the strongest forcing they tend to go to a limit shape that appears to follow a power law which exponent is close to k -20/3 . This observation is slightly steeper yet comparable to the spectrum of the D-cones which has been predicted by Düring et al. to be proportional to k -6 [15]. The discrepancy may be explained by the fact that the D-cones are actually 2D structures that are regularized by dissipation at small scales. Furthermore they are not isotropic at small scale and/or that they are most likely distributed in size.

VI. STATISTICAL PROPERTIES AND INTERMITTENCY A. Numerical simulations

The strong events of curvature seem to have an internal structure and a spatial organisation. These are the ingredients for statistical intermittency as was observed in Navier-Stokes turbulence [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]. The statistical tools to study this intermittency are for instance the structure functions that probe the statistics at various scales. For Navier-Stokes turbulence, one often studies the spatial increments of velocity δ r v = v(x + r) -v(x) or the gradients of the velocity gradient averaged over a sphere of radius r. Following a similar approach we study the evolution of the statistics of the curvature when first smoothing the picture of the deformation by low-pass filtering. In practice, the deformation field is low-pass filtered by a Gaussian kernel. The curvature of the filtered images is then computed. Note that the order of the operations of filtering and calculation of the curvature does not matter for the average curvature which is approximated by the linear operator L 0 . By contrast, the Gaussian curvature, even evaluated using the simplified form G 0 , is not a linear operator and the order of the calculation matters. After computing the curvatures G and L of the smoothed deformation with different low-pass filter cutoff scales, we compute the distributions and the moments of these curvatures.

Examples of the curvature of such low pass filtered deformation are shown in fig. 12. At small forcing the picture of the curvature remains disorganized and close to Gaussian statistics regardless of the scale of smoothing. At strong forcing the intense localized structures visible at small scales are progressively filtered and filamentary organisation in the picture appears at intermediate smoothing scale. These observations are the trace of a spatial organisation of the curvature that is not self similar in scale.

The lack of self similarity is visible when computing the distribution of the curvature of the smoothed deformation. In fig. 13, the curvatures are computed for run5 (top line). The distribution of L is seen to be close to Gaussian at small scales, develops wider tails at intemediate scales and then evolves back to Gaussian at the largest scales. The distribution of G also develops wider tails at intermediate scales. For run1 (weak turbulence, bottom line of fig. 13), the PDF are almost unchanged whatever the smoothing scale). This shows that the intermittency (understood as the lack of self similarity of the deformation field) is due to the existence of the coherent structures.

A way to quantify the change of the shape of the distribution is to compute its moments made dimensionless by normalizing by the variance. For instance the flatness F = L 4 L 2 2 is a measure of the evolution of the width of the tails of the distribution. The evolution of the flat- ness with the smoothing scale is shown in fig. 14. At low forcing the flatness only shows a slight evolution when increasing the scale: The average curvature evolves from 3, the value for a Gaussian random field, to about 2.5 and back to 3 at large scales. The Gaussian curvature displays a similar trend. At larger forcing intensity, the evolution of the flatness displays a maximum at intermediate scales (or order 1 cm). This shows indeed that the distribution of the PDF displays wider tails at these intermediate scales. At the stronger forcing the flatness takes very large values at small scales and decays continuously to the Gaussian value at large scale.

B. Experiments

The PDF of the Gaussian curvature of the smoothed experimental deformations are shown in fig. 15 for two forcing intensities. Computing the curvature of a real image is quite challenging as it involves differentiating twice an image which contains measurement noise. Thus this cannot be achieved at the smallest forcing intensities for which the noise has a big impact. For this reason, we restrict ourselves to the analyse of a medium intensity forcing and of the largest experimental forcing. The figure 15 should be compared to the equivalent figure 13. A similar trend emerges: almost no evolution of the shape is observed at the smallest forcing whereas the tails of the PDF are wider at intermediate scales at the highest forc-ing. This widening of the tails is much less pronounced on the experiments than on the numerical simulation due to not as strongly non linear a motion, resulting itself from experimental limitations on the forcing [8,[START_REF] Miquel | [END_REF]. Indeed as seen in fig. 1, the experiment with the strongest forcing is qualitatively similar to the numerical run 4. Run 4 corresponds to a mild forcing for which most of the features discussed above are not clearly visible. Nevertheless the emergence of the structures in the experiment can be observed clearly from the change of shape of the spectrum (fig. 1). Concerning the distribution of the curvature in fig. 15 the emergence of structure is visible only in the very far tails that correspond to extremely rare events. unfortunately we could not achieve a stronger forcing in the experiment because the instantaneous input power is strongly fluctuating. Thus the shaker must intermittently provide an instantaneous (electrical) power several orders of magnitude larger than the average, which the power supply unit can not provide despite being strongly oversized compared to the average power required (see discussion on the statistics of input power by Cadot et al. [42]).

This evolution of the shape of the PDF can be quantified by the flatness and hyperflatness (order 6 moment). As the change in shape of the PDF occurs in the very far tails the evolution is best seen on the hyperflatness which is a higher order moment and thus more sensitive to rare events. The change of shape of the PDF can be seen only on the statistics of the Gaussian curva- ture which is the only one shown. At low forcing (blue curve) the flatness and hyperflatness are quite flat except at small scales at which the noise dominates over the signal. When the forcing is increased a local maximum of the flatness (barely visible) and the hyperflatness (more visible) emerges which is in qualitative agreement with the observations for numerical simulations although not as strong (see fig. 14).

VII. CONCLUDING REMARKS

We have described in this paper the gradual emergence of coherent structures in a wave turbulence system: in the case of flexural waves, coherent structures first appear at large wavelength and march toward small scales as the forcing is increased. These structures are composed of developable cones, which imprint at small scale is the crescent shape structure observed in the numerical simulation, which are connected by folds [33]. At the smallest scales though, these structures are regularized and dominated by a bath of weakly non linear waves with small wavelength (except possibly for the highest forcing case) or by dissipation. Observe that all these structures appear, evolve dynamically and die. This network of interconnections between cones generates some intermediate scale structures, namely ridges and facets, resulting in a crumpled paper appearance. The weak turbulence scaling is confined to a range of small scales that diminishes and eventually disappears at the highest forcing. A change in the statistical properties of the system accompanies this transition: intermittency appears.

In a thin elastic plate, we associate the occurence of intermittency with the observation of coherent structures. Indeed the apparition of intermittency at large forcing intensity is correlated with the development of wide tails of the curvature distribution at small scale, a signature of developable cones. Our observations can be compared to the case of fluid turbulence. It was suggested in the last decades that the intermittency observed in Navier-Stokes turbulence could be associated with the observed intense coherent vorticity filaments but this issue is still a matter of debate.

The vibrating plate appears to be a relatively simple physical setup in which to observe the transition from weakly non linear wave turbulence compatible with the theory of Weak Turbulence to a regime of stronger nonlinearity that generate coherent structures. Although such behavior maybe expected generically in other wave systems the variety of possible coherent structures probably makes each system specific. In other words, this transition results from the "breakdown of weak turbulence" as worded by Newell et al. [43]. The widening of the dispersion relation is related to the nonlinear time scale. The nonlinear timescale T N L (k) can be defined as the correlation time of the modulation of the Fourier modes as in ref. [25]. One can write that the Fourier mode of the plate deformation as ζ k (t) = A k (t)e iω k t where A k (t) is the modulation. The width (in frequency) of the deformation spectrum (around the dispersion relation) shown in fig. 5 is thus directly related to the characteristic timescale of A k (t) that can be extracted from the temporal correlation of A k (t) [25]. As seen in fig. 5, at strong forcing the widening is so high that the two branches of the dispersion relation overlap at low frequency. This is the scenario predicted by Newell et al.: the hypothesis of scale separation between the nonlinear time scale T N L (k) and the linear period of the wave T (k) must breakdown either at large scale or small scale depending on wether T N L /T is increasing of decreasing with k on the KZ spectrum. For surface gravity waves the ratio T N L /T is increasing with k so that wave turbulence becomes more and more nonlinear as the cascade proceeds to small scales and may lead ultimately to whitecapping. By contrast, for elastic plates, the ratio T N L /T is decreasing when the frequency is increasing (as for capillary waves for instance). So the breakdown of weak turbulence occurs at low frequency (near the forcing scales) but the time scale separation remains valid at large enough frequency. We confirmed experimentally this breakdown in fig. 11 of [25] where we observed indeed that the nonlinear time scale is indeed comparable to the wave period at the strongest forcing (run G). Note that the coherent structures appear at large scale in fig. 2 but are originating from stress focussing at the smallest scales. Thus the vibrating plate corresponds to another class of systems than the case of surface gravity waves. Newell et al. suggest that for surface gravity waves the breakdown should lead to the observation of the Philipps spectrum. This spectrum does not depend on the input power P anymore as it corresponds to a saturated spectrum. Whitecap events dissipate extra energy so that to remain on the Philipps spectrum. This is not what is observed in the ideal elastic plate. The spectrum changes from being proportional to P 1/3 in the weak case to being proportional to P in the strong case. This corresponds to the behavior predicted by Kuznetsov [41] for the spectrum of point singularities that is expected to be proportional to P . Increasing the forcing generates more and more singularities. It should be noted though that dissipation may alter this scenario in real plates. In our sim-ulations, the dissipation is simply linear in the velocity and only present at small scales so that strongly nonlinear dissipation analogous to that occurring in whitecap events of sea waves cannot exist. We also assume the Föppl-von Karman equation to remain valid, as a toy model of weak turbulence, rather than to simulate real plates. In actual plates, it is likely that, at extremely strong forcing, strongly non linear deformation can lead to nonlinear dissipation and even irreversible events due to plasticity. It may be possible that for plates that would be large enough another transition could occurs from the Kuznetsov-like regime to a saturated Philipps-like spectrum at large scales and thus a spectrum that becomes independent of P but this remains very speculative and clearly beyond the possibilities of our simulations where the linear dissipation is enough to regularize the singularities at small scale.

F

  and D are forcing and dissipation respectively. The brackets {•, •} denote the bilinear differential operator {ζ, χ} = ∂ xx ζ∂ yy χ + ∂ yy ζ∂ xx χ -2∂ xy ζ∂ xy χ .

FIG. 1 .

 1 FIG.1. Evolution of the Fourier power spectrum E η (k) of the deformation η of the plate (integrated over the directions of the wavevector k) as a function of the forcing intensity. (a) experiment. The average forcing power is P =[1,[START_REF] Newell | [END_REF][START_REF] Cadot | Wave Turbulence in Vibrating Plates[END_REF]16,25,[START_REF] Falcon | [END_REF], 50] (arb. unit) from bottom to top. The power is proportional to the square of the force magnitude (at a given excitation frequency). (b) numerical simulations. The input power is P = [1, 3.5, 14, 60, 245, 910, 3760, 11100] (arb. unit) from bottom to top and thus evolves on a much larger range than the experiment. The dissipation is also quite different (see text). The vertical dashed line corresponds to the wavenumber over which the dissipation operates. The forcing operates at k/2π = 2.5 m -1 (that corresponds to 30 Hz for linear waves as in the experiment). Black squares show the change of behavior of the spectrum between structure dominated regime (at low k) and weak turbulence at high k.

FIG. 2 .

 2 FIG. 2. Snapshots of the deformation of the plate for various forcing intensities (numerical simulation).The corresponding run number is displayed in the title. The input power is increasing with the run number and the relative values(1,14, 245, 3760) from (a) to (d) respectively.

FIG. 3 . 3 .

 33 FIG. 3. Scaling properties of the spectrum of the deformation. The color code is the same as in fig. 1. (a) spectra normalized by k 3 /P 1/3 in order to compare to the prediction of the WTT that predicts a stationary spectrum E η (k) ∝ P 1/3 log 1/3 (k * /k) k 3 . The dashed line correspond to a fitted log 1/3 (k * /k) with k * /2π = 90 m -1 . The insert is a zoom of the logarithmic region in semilogarithmic scale. (b) Spectrum normalized by k 3 /P . This normalisation was chosen to illustrate the linear scaling with P of the low frequency part of the spectrum. Black squares show the transition as in fig. 1.

FIG. 5 .

 5 FIG.5. Examples of the spectrum E η (k, ω) obtained from the numerical simulations, for run1 (weakest forcing), run6, run7 and run8 (strongest) (P = 1, 910, 3760 & 11100, from (a) to (d) respectively). For a better contrast the displayed quantity is actually the normalized spectrum k 4 E η (k, ω).

FIG. 6 .

 6 FIG. 6. Cuts of the k, ω spectrum for two given values of k ((a) k = 20π and (b) k = 80π m -1 ) and for all the values of the forcing magnitude. The linear values of the linear wave frequency at the given values of k are shown with vertical dashed lines.

FIG. 7 .FIG. 8 .

 78 FIG. 7. Snapshots of both sorts of curvature L and G at various forcing intensities in the DNS. Top line: L, bottom line G. (a)&(b): run 1 (weak forcing), (c)&(d): run 6 (intermediate forcing), (e)&(f): run 8 (strongest forcing). The input power are respectively P = 1, 910 and 11100.

FIG. 9 .

 9 FIG. 9. Distribution of the velocity of the strong events of |L| at the 3 highest forcing intensities (run 6 to run 8, blue, red and green resp.). The dashed line is a Gaussian distribution. The corresponding forcing powers are respectively P = 910, 3760 & 11100.

FIG. 10 .

 10 FIG. 10. Cuts of the (k, ω) spectrum at various values of k/2π = 15, 20, 25, 30, 35, 40, 45 m -1 for run 8 (strongest forcing). Main figure: E η (k, ω)/E(k, 0) as a function of ω/k. The dashed line is a Gaussian shape of width 20 m/s. Insert: E η (k, ω)/E(k, 0) as a function of ω

FIG. 11 .

 11 FIG. 11. Decay of the space-time spectrum at zero frequency k 6 E(k, ω = 0)/ E η (k, ω)dkdω for all simulations. The normalization of the spectrum means that the deformation has been normalized to be of variance 1. The dashed line shows a decay k. -2/3 . The forcing increases from bottom to top.

FIG. 12 .

 12 FIG. 12. Examples principal curvature L of the low pass filtered deformation. Top line, (a), (b), (c): weakest forcing (run 1, P = 1). Bottom line, (d), (e), (f): strongest forcing (run 8, P = 11100). (a) & (d): curvature of the deformation. (b) & (e): curvature of the deformation after smoothing over 5 pixels (1.3 cm). (c) & (f): curvature of the deformation after smoothing over 32 pixels (8.3 cm). The images display only a portion of the full plate.

  FIG. 13. PDF of both curvature L and G as a function of the smoothing for run5 ((a) & (b), P = 245) and run1 ((c) & (d), P = 1). (a) & (c): L, the dashed line shows a Gaussian distribution, (b) & (d): G.In each subfigure, the lengthscale is increasing from top to bottom and the curves have been shifted vertically for clarity. The smoothing scales are from bottom to top: 0.05, 0.14, 0.29, 0.43, 0.57, 0.86, 1.1, 1.7, 2.3, 3.4, 4.6, 6.9, 9.2, 13 and 18 cm.

FIG. 14 .FIG. 15 .

 1415 FIG. 14. Evolution of the flatness of (a) L and (b) G with scale.

FIG. 16 .

 16 FIG.16. Evolution with scale of the flatness (a) and hyperflatness (b) of the curvature for the experiment for runs E, F, G (colors blue, red and black resp., input power P = 25, 36 & 50 resp.).

TABLE I .

 I Parameters of the experiments (top) and numerical simulations (bottom)

  Power P

	(run:) 1 (run:) (run:) (run:) (run:) (run:) (run:) (run:)	2	3	4	5	6	7	8
					Dissipative cutoff
		10 2
	Wave number k	10 1	Weak Turbulence (WT)		Coherent Structures (CS)
					Forcing scales
		10 0			10 2				10 4

FIG.

4

. Sketch of the different regimes encountered in the spectrum of the motion as the injected power increases. The different numerical runs labeled from 1 to 8 (see table

I

) are represented by vertical dotted lines. The largest scales and the smallest scales are dominated by the forcing and the dissipation, respectively. The intermediate scales exhibit a Weak Turbulence (WT) scaling for weak forcing, or are dominated by coherent structures (CS) for very large forcing. At intermediate forcing, the large scales are populated by the coherent structure while the small scales follow a WT dynamics. The transition observed in the spectra represented in figure

1 and

3 is denoted by plain squares. They correspond to the change from the the strongly nonlinear regime (E η

VIII. AKNOWLEDGEMENTS

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 647018-WATU). NM was supported by Institut Universitaire de France. We thank O. De Marchi, C. Bonamy and G. Moreau for their help to run the numerical simulations.


Kubo Akira

Sato Hiroyuki

Design of Graded Trusts by Using Dynamic Path Validation

In modern information service architectures, security is one of the most critical criteria. Almost every standard on information security is concerned with internal control of an organization, and particularly with authentication. If an RP (relying party) has valuable information assets, and requires a high level to authentication for accepting access to the valuable assets, then a strong mechanism is required. Here, we focus on a trust model of certificate authentication. Conventionally, a trust model of certificates is defined as a validation of chains of certificates. However, today, this trust model does not function well because of complexity of paths and of requirement of security levels. In this paper, we propose "dynamic path validation," together with another trust model of PKI for controlling this situation. First, we propose Policy Authority. Policy Authority assigns a level of compliance (LoC) to CAs in its domain. LoC is evaluated in terms of a common criteria of Policy Authority. Moreover, it controls the path building with considerations of LoC. Therefore, we can flexibly evaluate levels of CP/CPS's in one server. In a typical bridge model, we need as many bridge CAs as the number of required levels of CP/CPS's. In our framework, instead, we can do the same task in a single server, by which we can save the cost of maintaining lists of trust anchors of multiple levels.

Introduction

In modern information service architectures, security is one of the most critical criteria. Today, security is discussed in terms of computer security, network security, and information security. It is not long before information security is considered to be important. Information security is concerned with controls of behaviors of systems and humans for protecting information assets. As one of major differences of information security to others, we must consider organizations as major players of security. "Internal control" is discussed organization-wise. Security policies are also organization-wise defined and published.

Information security is closely related to the concept of information assets. An organization recognizes that information itself has and produces value, whenever it experiences information theft, leakage, and insider trading. Moreover, information security is related to legal issues such as privacy. Privacy related information is often accumulated in an organization. In such a situation, the organization is required by law to protect such information that is considered to be "owned" by individuals, not by the organization. Thus, protection of information assets is demanded. By controlling systems and humans that handle information assets, information security gives some guarantee to such protection.

There are defined several standards on information security. For example, ISMS, or ISO 27001, is commonly used as a criteria of system and information security. Actually, almost every standard such as ISMS is concerned with internal control of an organization. Among several issues of internal control, authentication is the most critical. Allowing access of critical information assets is guaranteed by how assured the used authentication is. A Level of assurance associated with an authentication differs in its mechanism. For example, certificate authentication certainly provides higher level mechanism than password authentication. Even in certificate authentication, its strength differs in CPs of certificates.

If an RP (relying party) has valuable information assets, and requires a high level to authentication for accepting access of the valuable assets, then a strong mechanism is required. Although it is of course that the strength of authentication is brought by its mechanism, initial setups and lifecycle management of credentials and IDs must also be considered. In certificate authentication, they are defined and published in CP/CPS's of certificates. Matching of the strength of IdP (ID providers) and requirements by RPs are the source of trust in a federation.

In this paper, we focus on a trust model of certificate authentication. Conventionally, a trust model of certificates is defined as a chain of certificates. A certificate chain is constructed so that an issuing CA is endorsed, or given its digital signature by another CA. If the anchor of the chain is contained in a list of trusted CAs, then the target CA of the chain can be trusted. These chains are central in constructing the trust of PKI. However, today, this trust model does not function very well. Its reasons are classified in twofold: one is that there can be constructed an arbitrary complex chain, in which chains are hard to control. Although there are defined three trust models, hierarchical, mutual, and bridge for taming this complexity, they are only partially implemented to validate complex chains. The other reason is more critical: because there are provided several levels of certificate policies, CAs of the same levels are fragmented into small groups. This means that we need as many as CAs as levels, which proliferates the number of CAs. Actually, major commercial PKI vendors operate as many CAs of different assurance as required even for the same usage such as client authentication. Although the difference of levels can be inferred by checking CP/CPS's, it must manually be done. This will cause a long negotiation in building a bridge CA. To control such fragmentation is strongly required. This paper proposes dynamic path validation, together with another trust model of PKI for controlling this situation. First, we propose a policy management server. This server assigns a level to CP/CPS of a given CA. The assignment may mutually be done in an agreement of the policy management server with the CA. Or, some criteria approved by a group of CAs may be used. Second, we propose an extended path validation based on the levels provided by the policy evaluation server. In the path construction, levels are used together with certificate chains. The consistency of levels is also discussed.

Our framework assumes one policy management server, which plays as a pivot among policies of CAs. Instead of mutually agreeing or fighting on CP/CPS of a bridge CA, this policy management server accepts multiple levels of securities of CAs. Therefore, we can flexibly evaluate levels of CP/CPS's in one server. In a typical bridge model, we need as many bridge CAs as the number of required levels of CP/CPS's. In our framework, instead, we can do the same task in a single server.

The rest of this paper is organized as: Section 2 studies scenarios in which efficiently handling multiple levels is important. Section 3 proposes dynamic path validation as our solution. Policy Authority is introduced. Furthermore, path validation is extended in the way that levels of CP/CPS's are reflected. Section 4 surveys related work. Section 5 summarizes this paper.

Stratified Paths Depending on LoA

Recently, many of critical services have been implemented as Web applications. Accordingly, there are many services of various levels of significance. Today, even in a single organization, there are provided many services that have various levels of significance. The significance is evaluated in the information assets handled by the service. For example, if a service handles privacy, it must be treated with care. If a service handles medical information, it must be treated with the highest security.

Authentication is a key mechanism that implements the levels of significance. Its idea is to control the access by identifying end users with how assured the authentication can be. Generally, they are called "level of assurance (LoA)." There are defined some standards of LoAs such as NIST 800-63 [START_REF] Burr | Electronic Authentication Guidelines[END_REF] for evaluating the levels. In such situations, an SP(service provider) requires an appropriate LoA to an IdP(ID provider) for accessing its information assets. In a fixed trust circle, it is common that its member IdPs and SPs are under some agreement as for keeping LoA of IdPs, which actually gives trust in the circle.

Among various authentication mechanisms, certificate authentication is usually given the highest LoA. Certificate authentication includes a process of path validation: a path between two CAs is constructed if a CA trusts the other CA. If the root of the path constructed in the validation process is in the domain of trusted CAs, then the validation, and therefore authentication succeeds. Thus, the trust of certificates is reduced to path construction whose root is trusted.

Trust brought by path validation causes a problem: the "LoA" of the trusted domains. In general, CAs are operated in various levels of CP/CPS. Some CP/CPS can be stricter than others. More strictly operated CAs can provide a higher LoA to SPs. SPs that require a higher LoA can trust only strictly operated CAs. A problem is that high level requirements result in inconvenience to users and high cost in operations. Today, a solution to such trade-offs is given in such a way that an organization operates CAs of multiple roots that correspond to multiple levels of operations. Looser certificates are used to access less important information assets, but with less cost than strictly issued certificates.

Typical examples of multiple roots can be seen in server certificates. Today, most major browsers classify server certificates as three: EV certificates [START_REF]CA/Browser Forum: Guidelines for the Issuance and Management of Extended Validation Certificates[END_REF], web trust [START_REF]Trust Services Principles, Criteria and Illustrations for Security, Availability, Processing Integrity[END_REF], and others. In the path validation of server certificates, the three never intersect. Even in client certificates, major vendors such as Verisign provide multiple roots of different levels of trust. In a complex organization, the situation is very similar to the real world. There are many organizational units that have various levels of independence. There are many services that require various LoAs. The result would be many CAs of various LoAs to cope with the various requirements of services.

This kind of scenarios causes fragmentation and maintenance problems. In the real world, many CAs are established to provide required LoAs. In such a situation, the cost of maintenance of trust domain is high. The domain is fragmented according to LoAs. Moreover, if a requirement level of an SP changes, the list of trusted CAs must be accordingly modified, causing a problem in maintenance. Even if no change occurs to an SP, because the world of CAs continuously changes, the maintenance is still a problem.

Dynamic Path Validation

In this paper, we propose "dynamic path validation." to tame the fragmentation and maintenance problems stated above. Dynamic path validation is a kind of delegated path validation in which Policy Authority plays as a key component.

Architecture of Dynamic Path Validation

In our scenario, there are three players: end entities, RP(or SP), and Policy Authority. An end entity requests authentication for a service with his/her certificates. An RP (Replying Party) or an SP (Service Provider) is a server that authenticates a user. In this scenario, an RP requires that a certificate of an end entity has a certain level of assurance. The RP delegates the path validation to Policy Authority. The Policy Authority checks whether a path provided by an end entity is valid. Policy Authority dynamically builds the path by using levels of CP/CPS's. Specifically, it checks whether requirements of an RP given as a level of certificates is satisfied in the certificate chain. In other words, this framework checks conditions of path validation in a way not related to information statically embedded in certificates. This is the name of "dynamic" path validation We illustrate our architecture in Fig. 1. 1. to decide and publish the common criteria of CP/CPS, 2. to register CAs that comply with the published criteria of CP/CPS, and 3. to validate paths on behalf of RPs.

In Fig. 1,1. corresponds to "Common Criteria," 2. to DB of Levels, and 3. to Responder, respectively.

Policy Authority must be operated under an agreement with participating CAs. It assumes that participating CAs agree on some predefined criteria. This at least includes those on audit and delegation of assignment of levels to subordinate CAs. Today, audit is considered to be a standard way to assure the quality of operations. Therefore, we demand audit to assure the compliance with the criteria. Moreover, delegation must be operated in an appropriate way.

In general, delegation is one of major solutions of distributed system management in the case that specific tasks are hard to control or to maintain. In this case, path validation is a heavy task, and hard to maintain in a single client.

In this way, with Policy Authority as the core, CAs and RPs participate in the circle, which simulates the circle of trust in Liberty-like federations. The difference is that in the latter (Liberty), IdPs and RPs mutually evaluate the quality of their services, while in the former (ours), they refer to the criteria via Policy Authority. In this meaning, Policy Authority plays as a pivot in the circle. We illustrate our concept of circle in Fig. 2. Actually, building circles of trust is one of key issues in federations. by Policy Authority acting as the pivot of the circle, we can save cost of building multiple circles. This scenario resembles putting bridge CAs as a pivot in path building.

Fig. 2. Circle of Trust consisting of servers of various LoCs as Policy Authority as the Pivot

RP and CA An RP delegates path validation to Policy Authority. In the delegation, maintaining the list of trust anchors is a task of the delegated server. In our framework, an RP maintains its requirement to levels of certificates. Policy Authority, or the delegated server receives the requirement together with a path, then validates it. This means that it can house multiple path validation methods in one server. Policy Authority controls path validation by using required levels together with trust anchors (i.e. registered CAs) as illustrated in Fig. 3 We see that in the figure, instead of having as many bridge CAs as the number of levels, we can house multiple levels of path building in one Policy Authority.

CP Certification

The relation that a CA trusts another CA is determined by some kind of evaluation of CP of the target CA. The evaluation must be based on a common criteria such as EV, Web Trust and RFC 3647.

Levels of Compliance of Certificates

Conventionally, a level of assurance is given to a CA according to a specific criteria and certification based on the criteria and related audit. Typical criteria include WTCA [START_REF]Trust Services Principles, Criteria and Illustrations for Security, Availability, Processing Integrity[END_REF], EV [START_REF]CA/Browser Forum: Guidelines for the Issuance and Management of Extended Validation Certificates[END_REF], and "Specified Certification Business" [22] in Japan. All of these criteria require audit to assure the quality of CA operations. However, the audit is done for checking compliance with CP/CPS's of given CAs. Compliance of a CP/CPS with a given criteria must be proved as another process.

In this paper, we define "level of compliance (LoC)." Definition: We define a level of compliance (LoC) for a criteria as a numeric value that represents how strictly a server is operated in compliance with the given criteria.

As a criteria of LoC, we have some standard templates for CP/CPS's such as EV, WTCA, and RFC3647. Conventionally, because they define the minimum set of requirements, compliance levels are just 0/1 (yes or no). In our extended compliance, we may define optional criteria that enhances the level of security in the same template. Therefore, we have more than two levels as for compliance: enhanced compliance/minimum compliance/no. In this meaning, LoC can be considered as an extension of conventional certification.

Moreover, LoC can be defined as an extension of LoA. Usually, certificate authentication is given the highest level. In LoC, the level of operations of CA is in concern, which is the same framework as the assignment of LoA in which the level of ID providers is in concern.

In evaluating LoC, operations of a given CA are audited for a given criteria. In LoC, audit is done not only for CP/CPS, but also for a predefined criteria. The evaluation must be done by an authority for the criteria. Here, Policy Authority plays as an auditor.

Assigned Levels and Derived Levels In our framework, in addition to Policy Authority, which assigns a level to a CA, a CA can assign derived levels to its subordinate CAs. This delegation is essential in saving cost of Policy Authority operations. It is a fundamental assumption that a CA must control its subordinate CAs in path validation. This means that a level of a given CA is inherited to its subordinate CAs. In this paper, this control is extended to derived level assignment. This assignment is done under the restriction that the derived level must not be greater than the level of the parent CA.

Evaluation Axis of Criteria In NIST standard [START_REF] Burr | Electronic Authentication Guidelines[END_REF], we see four axis for evaluating authentications. In this paper, we borrow related two axes of evaluation: levels of initial identification (more generally, ID lifecycle management in [START_REF] Sato | A Service Framework based on Grades of IdPs and SPs[END_REF][START_REF] Sato | N ± ε: Reflecting Local Risk Assessment in LoA[END_REF]), and levels of tokens. For example, Verisign defines three levels as for the levels of assurance depending on the methods of initial identification and the coverage of assurance. In this meaning, our proposal is already implemented in the real world.

Moreover, a CA must be operated under a certain security constraint. RFC 3647 also defines security constraint in operations. [START_REF] Sato | A Service Framework based on Grades of IdPs and SPs[END_REF] proposes criteria for both IdPs and SPs. Here, we propose our evaluation axes of criteria in terms of [START_REF] Sato | A Service Framework based on Grades of IdPs and SPs[END_REF]:

1. ID lifecycle management, 2. levels of tokens 3. Quality of management of the server:

(a) Management of access control (b) Control of physical security.

(c) Management of privileges in operation

In addition to these axes, we require audit as the mechanism that guarantees the quality in terms of published criteria. To control the quality of operations, audit is considered to be very effective. It is mandatory that Policy Authority audits participating CAs.

Extended Path Validation by using Dynamic Path Validation

Now we have two components: LoC and Policy Authority. We extend path validation so that a CA of a lower LoC can trust a CA of a higher LoC, even if there is no path between the two in conventional meaning. We call this extension as "dynamic path validation (DyPV)."

Our DyPV is processed as follows: first, all CAs registered at the given Policy Authority are considered to be in its domain. In other words, a CA in the domain is given an LoC under the common criteria. Second, the path validation is extended by using levels: if a certificate issued by CA 1 is presented at an RP that requires n 2 as LoC, and n 1 is given to CA 1 , then the certificate is validated if n 1 ≥ n 2 . We extend this validation to a general certificate chain. If a path is built whose root is CA 1 , and CA 1 • • • CA n , are registered, then we compare their LoCs with the required LoC.

The algorithm of DyPV is given in Fig. 4. A validating RP delegates the validation to Policy Authority. Policy Authority responds with true/false depending on whether DyPV succeeds or not. The given inputs are CC[], a certificate chain given by a validatee, and LoC, a required LoC given as a policy of the validating RP. This algorithm partially extends CC[] so that the root of the path is in the domain. Here, Policy Authority builds a path in the conventional way so that its root is in the domain. Then, Policy Authority compares the required LoC with LoCs in the domain. In other words, registered CAs play as trust anchors in a conventional sense. Instead of maintaining the list of trust anchors, an RP just makes an inquiry of LoC as its requirement, and Policy Authority returns yes/no to the inquiry.

A problem arises in the algorithm: the path extension. There can be a case that there are two or more possibilities of extension, and in one extension, the extended validation succeeds, and in another extension, it fails. Our algorithm requires that the extension must be done so that the extended certificate chain DD[] satisfies the condition DD [START_REF] Alterman | Interfederation Initiatives for Identity Authentication[END_REF] ≥ LoC. By this restriction, we can eliminate false cases. If validation fails in a path extended in this way, we restart the path building.

If we can validate a path, then we must guarantee that a validation of any extension of the path also succeeds. Therefore, we require that if there is a path from CA 1 -→ CA 2 , meaning that CA 1 issues a certificate to CA 2 , then their LoCs must satisfy LoC(CA 1 ) ≥ LoC(CA 2 ). This consistency must be maintained by Policy Authority.

Comparison with RFC 5280

Conventionally, path validation is defined as RFC 5280 [START_REF] Cooper | Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile[END_REF]. In RFC5280, there is defined control of path building via policy extension fields in certificates. In our framework, for representing policies of CAs, we use LoC under a common criteria of Policy Authority. Our idea is that lifecycles of CAs and of their policies are not the same. Policies and operations can continuously be enhanced even in the same CA. We separate the two lifecycles, and manage policies by using Policy Authority on-line.

Control of Subordinate CAs

In our framework, Policy Authority allows registered CAs to assign levels to subordinate CAs. This delegation is a key to save the cost of operations of Policy Authority. Assignment of LoC assumes that subordinate CAs have lower or equal LoCs than those of their parents. Policy Authority must enforce this restriction on every participating CAs Audit must also be effective for this enforcement.

Related Work

Path building [START_REF] Cooper | Internet X.509 Public Key Infrastructure: Certification Path Building[END_REF], and validation [START_REF] Cooper | Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile[END_REF] have been central issues in PKI domain extensions. There have been proposed three major methods of path construction models: hierarchical, mutual, and bridge models. Although the bridge model has been considered to scale, and has been implemented on some major domains, there are found some problems other than technical ones to hinder its growth. Furthermore, delegation of tasks related to them is studied because they are too heavy for general RPs. The discussions are summarized as RFC 3379 [START_REF] Pinkas | Delegated Path Validation and Delegated Path Discovery Protocol Requirements[END_REF]. OCSP [START_REF] Myers | X.509 Internet Public Key Infrastructure Online Certificate Status Protocol -OCSP[END_REF] and SCVP [START_REF] Freeman | Server-Based Certificate Validation Protocol[END_REF] are also classified as protocols partly delegating validation. Our framework is also classified as delegation. Ours considers LoCs in path validation.

It is commonly understood that operations of CAs can differ in their CPs. They include usage, profile, and security. There are some standard templates of CPs such as RFC 3647 [START_REF] Chokbani | Internet X.509 Public Key Infrastructure Certificate Policy and Certification Practices Framework[END_REF], and PKI lite [21].

Evaluating IdPs and assigning specific LoA is required by some securitysensitive SPs [START_REF] Alterman | InCommon Federation: Progress, Partnerships, Opportunities. Internet2 2007 Fall Meeting[END_REF]. There have been proposed several systems that use LoA. As major federated identity systems, both Liberty and OpenID provide a mechanism of sending LoA of IdPs to SPs [START_REF]OASIS: Level of Assurance Authentication Context Profiles for SAML 2[END_REF][START_REF]OpenID Provider Authentication Policy Extension 1[END_REF].

Moreover, in Grid, there are established policy management authorities[23] to enforce the policies of Grid on participants.

Although discussions of LoA [START_REF] Nedanic | Levels of Authentication Assurance: an Investigation[END_REF] have been limited to ID and authentication, they are very fruitful in assuring security level in building federations. In particular, they are essential in the framework that ID information is provided to an SP by IdPs in multiple organizations via SSO. OMB guidance [START_REF]E-Authentication Guidance for Federal Agencies[END_REF] and NIST standard [START_REF] Burr | Electronic Authentication Guidelines[END_REF] are milestones in the discussion. They are also the driving force to define LoA to large federations. Today, LoA is widely discussed in many organizations, grids, federations [START_REF]InCommon Federation: Identity Assurance Profiles Bronze and Silver[END_REF], and inter-federations [START_REF] Alterman | Interfederation Initiatives for Identity Authentication[END_REF].

LoA can be generalized to SPs. [START_REF] Sato | A Service Framework based on Grades of IdPs and SPs[END_REF][START_REF] Sato | N ± ε: Reflecting Local Risk Assessment in LoA[END_REF] propose a consistent assignment of LoA to SPs in terms of security policies of organizations.

Note that all of these must be done as a part of risk management. [START_REF] Stoneburner | Risk Management Guide for Information Technology Systems[END_REF][START_REF] Yan | Password Memorability and Security: Empirical Results[END_REF] discuss authentication in terms of risk management.

Concluding Remarks

In this paper, we have proposed dynamic path validation (DyPV). In DyPV, CAs in a domain are registered in Policy Authority, which plays as a pivot. Moreover, according to a common criteria, LoC is assigned to each CA. In this way, Policy Authority houses multiple levels of compliance in one server. Furthermore, path validation has been extended so that an LoC, or a level of CP/CPS's is reflected.

Our framework uses LoC instead of a list of trust anchors. CAs are not required to issue unnecessary certificates for path building, but Policy Authority checks whether the validation in terms of LoC requirement succeeds. Operations under a common criteria of Policy Authority is easier than maintaining lists of trust anchors of multiple levels in multiple bridges.

Fig. 1 .

 1 Fig. 1. Architecture of Dynamic Path Validation

Fig. 3 .

 3 Fig. 3. Housing Multiple Path Validations in One Policy Authority