Jose Ordonez-Miranda

Younès Ezzahri
email: younes.ezzahri@univ-poitiers.fr

Jose A Tiburcio-Moreno

Karl Joulain

Jérémie Drevillon

Radiative Thermal Memristor

Keywords: numbers: 65.60.+a, 65.80.-g, 65.90.+i

Based on the thermal hysteresis of a phase change material exchanging radiative heat with a phase invariable one, we propose a radiative thermal memristor characterized by a Lissajous curve between their exchanged heat flux and temperature difference periodically modulated in time. For a memristor with terminals of VO2 and a blackbody, it is shown that: (i) the temperature variations of its memristance follow a closed loop determined by the thermal hysteresis width of VO2. (ii) The thermal memristance on/off ratio is determined by the contrast of VO2 emissivities for its insulating and metallic phases and is equal to 3.59. The analogy of the proposed memristor to its electrical counterpart makes it promising to lay the foundations of the thermal computing with photons.

The last fifteen years have witnessed the emergence of the new research area thermotronics devoted to the development of thermal analogues of the fundamental electronic components, by means of the heat transport via phonons, electrons, and photons [1][2][3][4][5][6][7][8][9][10][11][12]. Several conceptual devices, such as thermal rectifiers/diodes [13][14][15][16][17][18][19], thermal transistors [20][21][22][23][24][25][26], thermal logical gates [27] and thermal memories [28] have been proposed and the working principles of some of them have experimentally been proven [29][30][31], which makes it possible to process information through heat currents.

The electrical memristor, a two-terminal device relating electrical charge with magnetic flux in electronics circuits, has recently attracted an enormous interest, due to its tremendous and revolutionary potential in microelectronics, informatics, spintronics, and neuromorphic networks [9,[START_REF] Tetzla | Memristors and Memristive Systems[END_REF]. Based on symmetry arguments, Chua [START_REF] Chua | [END_REF] first conceived this device in 1971 as the fourth fundamental electronic component besides the resistor, capacitor and inductor. Almost forty years later, Strukov et al. [34] developed the physical model for describing the hysteretic current-voltage behavior that characterizes an electrical memristor and showed that its realization can be achieved with doped metal-oxide semiconductors.

Despite of the vast research on and development of the electrical memristor [START_REF] Tetzla | Memristors and Memristive Systems[END_REF], its thermal counterpart has received little attention. Based on the asymmetric temperature response of the thermal conductivity of VO 2 during its heating and cooling, Ben-Abdallah [35] proposed a conductive thermal memristor, whose Lissajous curve between the heat flux and temperature difference of its terminals was recently reported by Yang et al. [36]. The operation of this promising memristor is driven by the hysteretic metal-insulator transition of VO 2 , which is one of the most investigated Phase Change Materials (PCM), due to its ability to change its thermal, optical, and electrical properties in a narrow interval of temperatures, near room temperature [37][38][39][40][41]. The performance of the conductive memristor is, however, limited by the inevitable presence of interface thermal resistances and the speeds of the involved phonons and electrons, which are typically 3 orders of magnitude smaller than that of photons. These limitations could thus be overcome by a radiative memristor with contactless terminals.

The purpose of this letter is to demonstrate the proof of principle of a radiative thermal memristor, an anologue of the electrical memristor defined by the functional relation T = M (x, q)q between the temperature difference (T ) and heat flux q of a system with a generalized thermal resistance M (x, q) determined by q and a state variable x driving the internal state of the system [START_REF] Chua | [END_REF]34]. The q-T characteristic curve of this memristive system is generally a Lissajous figure crossing the origin (q = 0 for T = 0), which distinguishes it from an arbitrary dynamical system, as is the case of an electrical memristor [34].

Let us consider a PCM exchanging heat by far-field thermal radiation with a non-PCM, due to their temperature difference ∆T (t) periodically modulated in time t, as shown in Fig. 1. The emissivity ε 1 of the PCM depends on its temperature T 0 + ∆T (t), while the one ε 2 of the non-PCM is considered to be independent of its corresponding steady-state temperature T 0 , as is the case of SiO 2 and SiC, for T 0 < 600 K [42]. In order to ensure the thermalization of the PCM, we consider that the modulation period τ of ∆T (t) is much longer than its thermalization time (time required to reach the thermal equilibrium between its electrons and phonons), which tipically ranges from a few to a hundred microseconds, for the case of VO 2 [37,43] used in this work. This condition is good enough to guarantee that the heating frequency (τ -1 < ∼ 1 MHz) is much smaller than the spectral frequencies of the photons emitted by the two materials, which are in the order of GHz, for temperatures near room temperature [START_REF] Modest | Radiative Heat Transfer[END_REF]. Under this quasi-steady state, the far-field heat flux q exchanged by the two plates is e 1 (T 0 +DT(t)) q(T 0 , e 1 , e 2 , DT(t)) e 2, T 0 FIG. 1. Scheme of a radiative thermal memristor: a PCM exchanging radiative heat with a non-PCM through a vacuum gap, due to their temperature difference ∆T (t) periodically modulated in time t.

given by the Stefan-Boltzmann law, as follows [START_REF] Modest | Radiative Heat Transfer[END_REF]

q = σε (T 0 + ∆T (t)) (T 0 + ∆T (t)) 4 -T 4 0 , (1)
where σ is the Stefan-Boltzmann constant and the effec-

tive emissivity ε = ε -1 1 + ε -1 2 -1 -1 is determined by
the emissivities of the PCM and non-PCM [START_REF] Modest | Radiative Heat Transfer[END_REF]. For the sake of simplicity, we will consider that this non-PCM is a blackbody (ε 2 = 1), which reduces the effective emissivity to that of the PCM (ε = ε 1 ). We consider that the emissivity ε of the PCM changes for temperatures within its phase transition, such that its value (ε h ) during the heating process is different than that (ε c ) for the cooling one, at a given temperature T , as is the case of VO 2 [37,[START_REF] Lee | [END_REF][46][47]. This thermal hysteresis of the PCM emissivity is driven by correlated atomic interactions and establishes that ε is given by

ε(T ) = ε h (T )δ(T ↑) + ε c (T )δ(T ↓), (2)
where δ(T ↑) and δ(T ↓) take the value of unity as the PCM temperature raises and lowers, respectively, otherwise they reduce to zero. For VO 2 , the emissivitities ε h and ε c can be well described by the following model [48] ε

n (T ) = ε i + ε m -ε i 1 + e -β(T -T0n) , (3)
where ε i = 0.79 and ε m = 0.22 are the emissitivities of VO 2 in its insulating and metallic phases, respectively, T 0h = 342.5 K and T 0c = 334.5 K are its respective transition temperatures for the heating (n = h) and cooling (n = c) processes, and β = 1.57 K -1 . Equations ( 2) and ( 3) thus take into account the VO 2 thermal hysteresis of T 0h -T 0c = 8 K experimentally observed by Qazilbash et al. [40] and they indicate that the VO 2 emissivity is bounded between its values for the metallic and insulating phases (ε m ≤ ε 1 ≤ ε i ), as shown in Fig. 2. Note that at low temperature (T < 332 K), VO 2 behaves as a insulator with high emissivity, while for high temperatures (T > 345 K), it becomes a metal with low emissivity, such that its emissivity changes a factor of ε i /ε m = 3.59 across its phase transition. This significant emissivity contrast in a relatively small temperature interval, allows to enhance (reduce) the radiative heat flux with VO 2 in its insulating (metallic) phase, as established by Eq. ( 1).

T 0c =334.5 K 8 K T 0h =342.5 K 0.22 VO 2 mean emissivity, Temperature, T (K) FIG.
2. Mean emissivity of VO2 for its heating and cooling processes. The dots stand for experimental data [40,41,48], while the lines represent the predictions of Eq. ( 3).

According to Eq. ( 1), the emissitivy variations of VO 2 between its insulating and metallic phases along with its thermal hysteresis generate a closed loop for the evolution of heat flux q, as the temperature difference ∆T periodically changes through time. This loop unfolds in two coupled loops (Lissajous curve) by properly modulating ∆T through positive and negative values, as shown below. The effective thermal resistance ∆T /q of the system (Fig. 1) will thus depends on ∆T and its sign (heating or cooling), which represents the history of the input temperature difference. This thermal memory effect is analogous to that of an electrical memristor characterized by a Lissajous I -V curve that induces the dependence of its electrical resistance on the input voltage difference and its polarity [34]. The radiative system shown in Fig. 1 can thus play the role of a thermal memory resistor (thermal memristor) with a memory resistance (memristance) M = ∆T /q given by Eq. ( 1), as follows

M -1 = σε (T 0 + ∆T ) (2T 0 + ∆T ) (T 0 + ∆T ) 2 + T 2 0 . (4
) Taking into account that the PCM temperature T 0 + ∆T determining the values of M also drives the appearance of metallic (insulating) domains in an insulating (a metallic) matrix during the heating (cooling) processes [37], the memristance can also be described in terms of the volume fraction of these microscopic domains [40,41].

For instance, the volume fraction f h of the metallic domains appearing during the heating of VO 2 at temperature T is given by [37]

f h (T ) = (-T /U )W 0 {-(U/2T ) exp(-U/T )ercf [2β(T 0h -T )/5]},
where W 0 (.) is the main branch of the Lambert W function, erfc(.) is the complementary error function and U is the activation energy (in units of Kelvin) of the metallic domains. Taking into account that U T [37], for all temperatures within the VO 2 phase transition, and that W 0 (x 1) ≈ x, this later relation reduces to

f h (T ) ≈ 0.5ercf [2β(T 0h -T )/5] ≈ [1 + exp (-β(T -T 0h ))]
-1 , which in combination with Eq. ( 3), yields

f h (T ) = ε i -ε h (T ) ε i -ε m . (5)
Equation ( 5) thus relates the microstructural parameter f h of VO 2 with its emissivity ε h for the heating process. For the cooling process, on the other hand, one can show that the known volume fraction f c [37] of the metallic matrix of VO 2 containing its insulating domains, also reduces to Eq. ( 3), provided that U T and the subindex h is replaced by c. The common dependence of f n , ε n and M on the PCM temperature T 0 + ∆T allows therefore to consider the ensemble of Eqs. ( 2)-(5) as a system of parametric equations for the evolution of the thermal memristance M (f ) with the volume fraction f = f h δ(T ↑) + f c δ(T ↓) describing the degree of metallization of VO 2 , across its phase transition. This control of the thermal memristance by the state variable f reinforces its analogy with the electrical memristance, whose values are driven by the electrical charge [34], which is a state variable in electricity.

The heat flux and memristance are now numerically analyzed for a thermal memristor made up of a VO 2 plate exchanging radiative heat with a blackbody. To capitalize on the thermal hysteresis of the VO 2 emissivity, the reference temperature T 0 = (T 0h + T 0c )/2 = 338.5 K is set as the average of the insulating-to-metal and metalto-insulating transition temperatures. Furthermore, we consider a temperature difference ∆T (t) = θ sin(2πt/τ ) varying sinusoidally with time, in a similar way to the electrical memristor, which is usually excited with an alternating voltage difference [START_REF] Chua | [END_REF]34]. The amplitude θ is chosen large enough (θ ≥ T 0h -T 0c = 8 K) to ensure that VO 2 completes its hysteresis loop after one period τ . In practice, this periodic oscillation of the PCM temperature can be set with a Peltier cell [49], while the non-PCM is kept at constant temperature with a second Peltier cell. These cells typically operate with a resolution better than 0.1 K [50] and modulation frequencies from 0.1 Hz up to 100 Hz, which are much smaller than the infrared ones driving the thermal radiation at temperatures around T 0 . This is the reason why the oscilating temperature of the PCM is not expected to significantly alter the equilibrium state of the emitted photons in the radiative thermal memristor shown in Fig. 1.

Figure 3 shows the heat flux q, temperature difference ∆T and VO 2 emissivity ε driving the thermal radiation in the thermal memristor, as functions of time. Note that ε exhibits four different behaviors, within one period τ . In the initial case 1 (0 ≤ t/τ < 0.19), ε drastically reduces due to the insulating-to-metal transition of VO 2 , as ∆T increases. An additional increase and then decrease of ∆T keeps the VO 2 in its metallic phase with constant and low emissivity, as revealed by case 2 (0.19 ≤ t/τ < 0.50). In case 3 (0.50 ≤ t/τ < 0.69), ∆T continues reducing and triggers the metal-to-insulating transition of VO 2 characterized by the significant increases of ε, as shown in Fig. 2. The subsequent variations of ∆T maintain the VO 2 in its insulating phase with constant and high emissivity, as displayed by case 4 (0.69 ≤ t/τ < 1). The combined time variations of ∆T and ε for cases 1 and 3, induce a heat flux with a strong nonlinear dependence on the temperature difference, despite of the fact that ∆T T 0 = 338.5 K. For cases 2 and 4, on the other hand, ε takes constant values and q ∝ ∆T , as shown in Fig. 3 by the occurrence of the maximum and minimun values q and ∆T at the same times, and is established by Eq. ( 1), for ∆T T 0 . It is thus clear that the sharp variations of the heat flux with respect to those of the temperature difference, are generated by the phase transitions of VO 2 .

The Lissajous q -∆T curve of the proposed thermal memristor is shown in Fig. 4. The cycle direction is indicated by the arrows associated to the four behaviors illustrated in Fig. 3. The straight lines in cases 2 and 4 confirm the proportionality (q = G∆T ) between the heat flux and temperature difference, due to the absence of VO 2 phase transitions. The slope G = M -1 is the thermal conductance of the system and its value (G(2) ≈ 4σε m T 3 0 ) for case 2 is smaller than the one (G(4) ≈ 4σε i T 3 0 ) for case 4, because of ε m < ε i . The nonlinear behaviors in cases 1 and 3 are due to the VO 2 phase transitions driving the emissivity variations displayed in Fig. 3. The absence of these changes near

)

Temperature difference, T (K)

1 T 0h -T 0c = 8 K 2 -6 -4 -2 0 2 4 6 -40 -30 -20 -10 0 10 T 0h -T 0c = 6 K T 0h -T 0c = 4 K T 0h -T 0c = 0 K q (Wm -2 ) T (K)
FIG. 4. Lisssajous q -∆T curve characterizing the thermal memristor. The regimes 1, 2, 3, and 4 correspond to the four characteristic zones highlighed in Fig. 3. Calculations were done for θ = 8 K and different hysteresis widths T 0h -T0c.

the extreme values (±8 K) of ∆T induces the pinched shape of the Lissajous curve. Note that the amplitude (differences between the curves 1 -2 and 3 -4) of this defining curve of the thermal memristor, is driven by the hysteresis (T 0h -T 0c = 8 K) of the VO 2 emissivity, such that it disappears for T 0h -T 0c = 0 K, as shown in the inset of Fig. 4. This amplitude also increases with the emissivity contrast ε i /ε m of VO 2 , as shown in Fig. S2 of the supplementary material. These facts make evident that the existence of the Lissajous q -∆T curve is determined by the combined impact of the thermal hysteresis (T 0h = T 0c ) and emissivity contrast (ε i /ε m = 1) of the involved PCM. This is confirmed by the area of the loops 1 -2 and 3 -4, which are equal and proportional to (ε i -ε m ) sinh [β(T 0h -T 0c )/2], as shown in the supplementary material. This latter relation establishes the material requirements to optimize the performance and applications of the proposed thermal memristor. Figure 5 shows the thermal memristances M related to the Lissajous q -∆T curves displayed in Fig. 4, as functions of the temperature difference ∆T . The linear portions of the top (2) and bottom (4) lines correspond to the regimes without a phase transition of VO 2 , while the complementary right (1) and left (3) lines results from its phase changes, in agreement with Figs. 3 and4. The height of the M -∆T loops is determined by the VO 2 emissivity contrast ε i /ε m = 3.6, which defines the ON/OFF states of the memristance (Fig. 5) and is higher than the one (1.6) of a conductive memristor [36]. The width of the memristance loop increases with the hysteresis T 0h -T 0c > 0, such that it dissapears for T 0h -T 0c = 0, in agreement with Fig. 4. In addition, the closed M -∆T loops are similar to those of the electrical memristance [START_REF] Tetzla | Memristors and Memristive Systems[END_REF], which indicates that what is done with electrical currents through the electrical memristor, could also be done with thermal currents, by means of the thermal memristor. For instance, the thermal memristor could be used as a binary memory, in which the ON and OFF values of the memristance (Fig. 5) are chosen to code the binary states 0 and 1, as is done with its electrical counterpart [START_REF] Tetzla | Memristors and Memristive Systems[END_REF]. In addition, given that the proposed radiative memristor can be scaled down to nanometric dimensions through near-field radiation, this device has great potential for developing thermal memories that could complement the electronic flash ones.

Finally, we point out that although the Lissajous q -∆T curve reported in this letter has been obtained with one PCM, we anticipate that this memristor effet can also be achieved with more complex systems involving a non-PCM with oscilating temperature (supplementary material) or two PCMs with temperatures properly modulated in time with equal or different frequencies. Furthermore, the development of the radiative memristor in the near field, would allow not only its miniaturization, but also the modulation of its memristance with the separation distance and steady-state temperature component T 0 of its terminals, due to the strong dependence of the heat flux on these two parameters [51]. Radiative thermal memristors could thus be useful to tailor the heat transfer at different spatial scales.

In summary, the proof of principle of a radiative thermal memristor has been demonstrated, by exploiting the thermal hysteresis of the emissivity of a phase change material exchanging radiative heat with a phase invariable one. For the case of VO 2 , it has been shown that the Lissajous q -∆T curve of the memristor yields a memristance, whose temperature variations in a closed loop are determined by the hysteresis width of VO 2 and the ratio of emissivities for its insulating and metallic phases. Considering the unique thermal properties of the proposed thermal memristor, the obtained results can open new research avenues in the development of thermal neurons, energy storage, and thermal management.

2 FIG. 3 .

 23 FIG.3. Time evolution of the heat flux, temperature difference and VO2 emissivity driving the thermal radiation in the memristor shown in Fig.1. The colored zones 1 and 3 stand for phase transitions, while the zones 2 and 4 represent no phase changes. Calculations were done for θ = 8 K.

2 KW - 1 ) 1 T

 211 FIG. 5. Closed loops of the thermal memristance associated to the Lissajous curves shown in Fig. 4. The regimes 1, 2, 3, and 4 correspond to the four characteristic time zones highlighed in Fig. 3.


Laurent Tranchant
email: tranchant.laurent191@mail.kyutech.jp

Satoki Hamamura

Jose Ordonez-Miranda

Tomohide Yabuki

Alejandro Vega-Flick

Fernando Cervantes-Alvarez

Juan Jose Alvarado-Gil

Sebastian Volz

Koji Miyazaki
email: miyazaki.koji055@mail.kyutech.jp

la diffusion de documents scientifiques de niveau

Keywords: In-plane thermal conductivity, surface electromagnetic waves, silica thin film, 3ω method, transient grating technique

recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Surface phonon polaritons (SPhPs) are able to propagate along the surface of polar dielectric nanomaterials, over distances that are more than 4 orders of magnitude longer than the typical mean free paths of phonons at the same frequency, and therefore, they can be considered as powerful heat carriers at the nanoscale, as was theoretically predicted in different structures. [START_REF] Ordonez-Miranda | Anomalous Thermal Conductivity by Surface Phonon-Polaritons of Polar Nano Thin Films due to their Asymmetric Surrounding Media[END_REF][START_REF] Gluchko | Focusing of Surface Phonon-Polaritons Along Conical and Wedge Polar Nanostructures[END_REF] Experimentally, SPhPs can be excited optically or thermally. For the optical excitation, the material surface is illuminated with a monochromatic light beam of a given frequency, which generates SPhPs propagating with the same frequency along the material, as established by the energy and momentum conservations. [START_REF] Ocelic | Subwavelength-Scale Tailoring of Surface Phonon Polaritons by Focused Ion-Beam Implantation[END_REF] By contrast, for the thermal excitation, the material is heated and SPhPs are simultaneously excited in a broad frequency range. Because of the possibility of exciting a single mode at a time, the optical method is the easiest one to highlight the characteristics (dispersion curve, propagation length, lifetime) of these surface electromagnetic waves, [START_REF] Agranovich | Surface Polaritons. Electromagnetic Waves at Surface and Interfaces[END_REF][START_REF] Huber | Near-Field Imaging of Mid-Infrared Surface Phonon Polariton Propagation[END_REF][START_REF] Sumikura | Highly Confined and Switchable Mid-Infrared Surface Phonon Polariton Resonances of Planar Circular Cavities with a Phase Change Material[END_REF] whereas the observation of thermally excited SPhPs requires different techniques, such as surface grating, 7,8 attenuated total reflection, [START_REF] Agranovich | Surface Polaritons. Electromagnetic Waves at Surface and Interfaces[END_REF] or thermal scanning tunneling microscopy. [START_REF] De Wilde | Thermal Radiation Scanning Tunnelling Microscopy[END_REF] The thermal energy carried by SPhPs propagating along packed beds of silica nanoparticles [START_REF] Tervo | High Thermal Conductivity in Polaritonic SiO 2 Nanoparticle Beds[END_REF] and hexagonal boron nitride encapsulated graphene [START_REF] Tielrooij | Out-of-Plane Heat Transfer in Van Der Waals Stacks Through Electron-Hyperbolic Phonon Coupling[END_REF] was measured. In the first work, the obtained thermal conductivity increase (κ = 0.01-0.03 W/(m•K)) does not agree with the predictions of two theoretical models: the wave model, where wave solutions to the electric field interactions between nanoparticles are developed and the radiation model, where the nanopaticle network is viewed as a system of discrete dipoles interacting through many-body radiations. However, it partially agrees with a ballistic propagation model of SPhPs. Indeed, some of the high-temperature thermal conductivity results are still underestimated by this model. In the second work, the cooling time determined by ultrafast time-resolved photocurrent measurements for hexagonal boron nitride encapsulated graphene shows lower values than expected and cannot be explained with in-plane cooling by scattering with graphene acoustic phonons. These latter results are explained with the out-of-plane heat transfer: coupling between hot carriers in graphene and hyperbolic phonon polaritons supported by boron nitride; 12 however, quantitatively, the energy contribution of SPhPs is not isolated. On the other hand, the enhancement of the in-plane thermal conductivity of silica thin films, due to the propagation of SPhPs, has already been predicted theoretically [START_REF] Ordonez-Miranda | Anomalous Thermal Conductivity by Surface Phonon-Polaritons of Polar Nano Thin Films due to their Asymmetric Surrounding Media[END_REF][START_REF] Chen | Surface Phonon-Polariton Mediated Ther-mal Conductivity Enhancement of Amorphous Thin Films[END_REF][START_REF] Yang | Long-Range Surface Modes Supported by Thin Films[END_REF] but not yet confirmed experimentally. Our present work demonstrates the existence of this new heat channel.

The in-plane thermal conductivities of suspended membranes of conductors, [START_REF] Zhang | Thermal and Electrical Conductivity of a Suspended Platinum Nanofilm[END_REF][START_REF] Balandin | Superior Thermal Conductivity of Single-Layer Graphene[END_REF] semiconductors, [START_REF] Chávez-Ángel | Reduction of the Thermal Conductivity in Free-Standing Silicon Nano-Membranes Investigated by Non-Invasive Raman Thermometry[END_REF][START_REF] Vega-Flick | Thermal Transport in Suspended Silicon Membranes Measured by Laser-Induced Transient Gratings[END_REF][START_REF] Nomura | Thermal Phonon Transport in Silicon Nanowires and Two-Dimensional Phononic Crystal Nanostructures[END_REF] and insulators [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF][START_REF] Sikora | Highly Sensitive Thermal Conductivity Measurements of Suspended Membranes (SiN and Diamond) Using a 3ω-Völklein Method[END_REF] were measured by different techniques, such as the central line heater method, [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF][START_REF] Sikora | Highly Sensitive Thermal Conductivity Measurements of Suspended Membranes (SiN and Diamond) Using a 3ω-Völklein Method[END_REF] the distributed self-heating method, [START_REF] Zhang | Thermal and Electrical Conductivity of a Suspended Platinum Nanofilm[END_REF] time domain thermoreflectance, [START_REF] Nomura | Thermal Phonon Transport in Silicon Nanowires and Two-Dimensional Phononic Crystal Nanostructures[END_REF] Raman thermometry, 17 and transient grating. [START_REF] Vega-Flick | Thermal Transport in Suspended Silicon Membranes Measured by Laser-Induced Transient Gratings[END_REF] However, these methods have not been applied to specifically suspended amorphous silica films, even though the crossplane thermal conductivity of silica films deposited on a substrate was already reported in the literature. [START_REF] Kato | Thermal Conductivity and Interfacial Thermal Resistance: Measurements of Thermally Oxidized SiO 2 Films on a Silicon Wafer Using a Thermo-Reflectance Technique[END_REF][START_REF] Yamane | Measurement of Thermal Conductivity of Silicon Dioxide Thin Films Using a 3ω Method[END_REF] This cross-plane thermal conductivity depends strongly on the sample purity, and its values tend to be independent of the sample thickness, [START_REF] Yamane | Measurement of Thermal Conductivity of Silicon Dioxide Thin Films Using a 3ω Method[END_REF] even for thicknesses as small as a few tens of nanometers. Based on these previous works, we decided to fabricate our amorphous silica films by thermal oxidation to obtain ultrathin suspended silica films with high purity [START_REF] Yamane | Measurement of Thermal Conductivity of Silicon Dioxide Thin Films Using a 3ω Method[END_REF] (see Figure S4 in the Supporting Information), as well as to measure with a high sensitivity their in-plane thermal conductivity and thermal diffusivity with and without a central line heater deposited on the samples, through the 3ω [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF][START_REF] Sikora | Highly Sensitive Thermal Conductivity Measurements of Suspended Membranes (SiN and Diamond) Using a 3ω-Völklein Method[END_REF] (see Figure 1) and transient grating methods, [START_REF] Vega-Flick | Thermal Transport in Suspended Silicon Membranes Measured by Laser-Induced Transient Gratings[END_REF] respectively. We explain the theoretical estimation of the SPhP thermal conductivity first proposed by Chen et al. [START_REF] Chen | Surface Phonon-Polariton Mediated Ther-mal Conductivity Enhancement of Amorphous Thin Films[END_REF] and analyze its application to our suspended silica samples of thickness t, halfwidth a, and length b, as shown in Figure 2 and Figure S2 in the Supporting Information.

The film is surrounded by air, has a frequency-dependent dielectric constant ε SiO 2 (ω), and its temperature gradient in the z-direction can be neglected, as its thickness is very small (smaller than 300 nm). In this condition, the Boltzmann transport equation under the relaxation time approximation predicts that the SPhP contribution to the in-plane thermal conductivity of the film is given by 24

κ = 1 4πt ω H ω L ωΛβ R ∂f 0 ∂T dω, (1)
where f 0 is the Bose-Einstein distribution function, T is the film temperature,

β R = Re(β)
is the real part of the in-plane wave vector β, Λ = (2Im(β)) -1 is the SPhP propagation length (mean free path), is Planck's constant divided by 2π, and ω L and ω H are the lowest and highest frequencies, respectively, supporting the propagation of SPhPs. Equation ( 1) is derived in detail in the section "Thermal conductivity due to surface waves" of the Supporting

Information and shows that the values of κ are determined by the SPhP dispersion relation β = β(ω) as well as by the frequency range of integration, which, for ideal lossless materials, is defined by Re(ε SiO 2 (ω)) < -1. [START_REF] Yang | Long-Range Surface Modes Supported by Thin Films[END_REF] For a lossy material as an amorphous silica film, on the other hand, the SPhP dispersion relation establishes that the SPhPs propagate in a frequency range much broader than the one for a lossless material 1 (see Figure S1 of the Supporting Information), as was recently demonstrated experimentally. [START_REF] Gluchko | Thermal Excitation of Broadband and Long-Range Surface Waves on SiO 2 Submicron Films[END_REF] The surface modes propagating with frequencies higher and lower than the ones in a lossless medium have been referred as Zenneck and transverse-magnetic (TM)-guided modes, respectively. [START_REF] Gluchko | Thermal Excitation of Broadband and Long-Range Surface Waves on SiO 2 Submicron Films[END_REF] An additional theoretical analysis of this frequency broadening can be found in the section "Broadening and spatial confinement of surface waves" of the Supporting Information. It is commonly accepted to name phonon polariton the hybridation between the electromagnetic field and the atomic motions in amorphous silica. This general conception was allowed because the resonances in the silica dielectric permittivity appear at the same frequencies as the ones of the quartz dielectric permittivity.

Taking into account that the thermal excitation of surface electromagnetic waves requires the excitation of optical phonons, which appears in amorphous silica for frequencies smaller than ω H = 258 Trad/s only, as was demonstrated through the experimental and theoretical determination of their density of states, 25 all of the TM-guided modes with frequencies higher than ω H cannot be thermally excited. This means that the propagation in amorphous silica of these TM-guided modes does not rely on a phonon-photon coupling; however, their optical excitation is still possible, as was already reported. [START_REF] Gluchko | Thermal Excitation of Broadband and Long-Range Surface Waves on SiO 2 Submicron Films[END_REF] In the low-frequency regime, on the other hand, we limit the integration in eq (1) to the smallest frequency, ω L = 7.6 Trad/s, for which a nonzero density of state has been experimentally observed. [START_REF] Sarnthein | Origin of the High-Frequency Doublet in the Vibrational Spectrum of Vitreous SiO 2[END_REF] From now on, we are going to refer to surface electromagnetic waves as both SPhP and Zenneck modes.

Taking into account that the intrinsic propagation length, Λ, of surface electromagnetic waves supported by a 20 nm thick amorphous silica film can exceed 1 m, 1 the effective propagation length, Λ e , used in eq (1) has been calculated through the Mathiessen's rule: 26

Λ -1 e = Λ -1 + a -1
. This relation holds when the surface waves only propagate in the ydirection, which is expected to be the case when the heater temperature gradient is not too significant (∆T ≤ 85 K). However, if the temperature gradient along the heater is significant, the excitation of these waves is not uniform and they can propagate in diagonal directions in the x-y plane. In this latter case, the effective propagation length Λ e increases (Λ e > Λ e )

and is given by

1 Λ e = 1 Λ + 1 L max -L min ln L max L min (2)
where L min = a and L max = a 2 + (b/2) 2 is the half-diagonal of our samples (see Figure 2). These two effective propagation lengths Λ e and Λ e will enable us to know whether only the half-width or also the length of our samples play a role on their in-plane thermal conductivity enhancement, and they are referred as models I and II, respectively. Due to the strong asymmetry between the substrate (silicon) and the superstrate (air) 1,27 (Figure 2), we consider that the surface waves cannot propagate beyond the borders of the free-standing films. In general, the propagation of these surface modes will be partially diffusive and partially ballistic; however, because of the limited propagation of the ballistic surface waves to the borders of the films, their mean free path is decreased to the film half-width a (model I) or the diagonal distance in the x-y plane (model II).

We have fabricated free-standing silica films with thicknesses between 300 and 20 nm through the thermal oxidation method. This range of thicknesses is wide enough to fully highlight and quantify the surface electromagnetic waves' thermal energy without ambiguities, as previous theoretical studies predicted. [START_REF] Ordonez-Miranda | Anomalous Thermal Conductivity by Surface Phonon-Polaritons of Polar Nano Thin Films due to their Asymmetric Surrounding Media[END_REF][START_REF] Chen | Surface Phonon-Polariton Mediated Ther-mal Conductivity Enhancement of Amorphous Thin Films[END_REF] For the 3ω measurements, the dimensions of the aluminum wires used as heaters (100 nm thick and 6 µm wide) and of the rectangular silica films (250-700 µm wide and 4-5 mm long) have been carefully chosen to avoid excess heat losses, as shown in Figure 1. On the other hand, for the transient grating measurements, a thin layer of AuPd (2.8 nm thick) was deposited on each square silica sample (600-680 µm long) in order to ensure the conversion of the laser light into heat at the illuminated surface.

See the section "Transient grating technique" of the Supporting Information for details.

The first method used to determine the in-plane thermal diffusivity and thermal conductivity of the samples is based on the 3ω method previously reported by Jain and Goodson. [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF] An aluminum wire deposited at the center of the suspended silica film acts as a heater and a temperature sensor due to its temperature-dependent electrical resistance (Figure 1c and Figures S6 and S8a of the Supporting Information). By applying a DC and then an AC voltage across the wire, the thermal conductivity and thermal diffusivity of the samples can be separately determined. When a DC voltage is applied to the wire, the average steady-state temperature increase of the metallic heater is given by θDC

= 4RI 2 0 κt +∞ n=0 tanh [(2n + 1)πa/b] [(2n + 1)π] 3 , ( 3
)
where R is the electrical resistance of the sample at the measured temperature and I 0 is the applied electric current. Equation (3) shows that the in-plane thermal conductivity κ can be easily extracted from the plot between the input electrical power RI 2 0 and the temperature increase θDC (see Figure S8b in the Supporting Information), which is obtained from the voltage measured across the metal wire. [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF] By applying an oscillatory voltage of frequency, ω, the temperature increase of the heater has a steady-state component associated with eq (3) and an oscillatory one, θAC , whose average amplitude over one period is given by

| θAC | = 2RI 2 0 κtb +∞ n=0 tanh(∆ 2n+1 a) ∆ 2n+1 [(2n + 1)π] 2 , (4)
where

∆ 2n+1 = (2n + 1)π b 2 + 2iω α 2 , (5)
| θAC | is the amplitude of the 2ω component of the temperature and is directly proportionnal to the 3ω one of the voltage across the wire. [START_REF] Jain | Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3ω Method[END_REF] Equation 4 is used to determine the thermal diffusivity α of samples (see Figure S9 in the Supporting Information). Details of the DC and AC calculations can be found in the section of the Supporting Information "Thermal modeling of the 3ω experiment". Implicitly, we neglected the radiation losses in this thermal model because the emissivity of the films is very low (see Figure S7 in the Supporting Information), and even its values are not negligible, this model still applies for a not-so-high heater temperature increase, as explained in the section "Study of the radiation losses" in the Supporting Information.

In order to further corroborate the in-plane heat transport results from the 3ω method, we performed measurements using a time-resolved optical spectroscopy technique known as laser induced transient grating. [START_REF] Eichler | Laser-Induced Dynamic Gratings[END_REF] In this technique, two short laser pulses (≈ 8 ns) are crossed at an angle 2θ in order to produce an intensity pattern with a periodicity Λ = λ/(2sin(θ )), were λ is the optical wavelength (see Figure S12 in the Supporting Information). Absorption of the laser light by the sample creates a spatially periodic temperature profile, which will remain until the thermal energy is redistributed from peak to null. This temperature decay will induce temperature-dependent changes in the refractive index that can be monitored via the diffraction of a continuous wave laser. For the complete technical details regarding the implementation of the transient grating method, we refer the reader to the section "Transient grating technique" in the Supporting Information. In the case of the semitransparent silica films, the temperature profile induced by the transient grating technique decays as an exponential function [START_REF] Eichler | Laser-Induced Dynamic Gratings[END_REF] (see Figure S13):

T (t) = T 0 e -t/τ , τ = Γ 2 /4π 2 α ( 6
)
where Γ is the transient grating periodicity and α is the thermal diffusivity of the film.

The in-plane thermal conductivity (κ) and thermal diffusivity (α) of our silica films measured by means of the 3ω method are respectively shown in Figures 33a and44a, as functions of their thicknesses. For thicknesses around 300 nm, both κ and α have values similar to those of their corresponding bulk counterparts, but they exhibit an increase of more than 50% for thicknesses smaller than 50 nm. This sizable increase is attributed to the energy contribution of surface electromagnetic waves due to the following reasons: (i) it is higher for thinner films, as established by the theoretical description of surface effects, and (ii) the thermal conductivity of bulk amorphous silica only increases about 0.17 W/(m•K) (×12%) [START_REF] Zeller | Thermal Conductivity and Specific Heat of Noncrystalline Solids[END_REF] when its temperature increases from 289 to 381 K, which is comparable to or higher than the temperature increase undergone by the metal heater used in our 3ω experiments.

The experimental data do not completely follow either model I or II in Figure 3a, but they are located in between. The experimental results obtained for the samples with a halfwidth smaller than 230 µm are better described by the predictions of model I, whereas those measured for wider samples follow a trend better predicted by model II. These observations are difficult to explain, but we can note that wider films have a higher density of wrinkles, which deviates the heat transport from the y-direction assumed by model I. However, narrow films can also exhibit a thermal conductivity higher than the one predicted by model I as is the case of the 45 nm thick film (green dot). This trend is not experimentally observed for thicknesses greater than 50 nm, due to the limited presence and propagation of surface waves, as established theoretically (see Figure S3 in the Supporting Information presenting silica at room temperature. [START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF] This correlated enhancement of κ and α is confirmed by the transient grating method (details in the Supporting Information), which was applied to measure these thermal properties for three additional silica films with thicknesses of 141.2, 87.3, and 27.5 nm. The measured thermal conductivities and thermal diffusivities are much higher than their corresponding values obtained by the 3ω method for films of similar thicknesses (see Figures S14-16 in the Supporting Information), due to the AuPd nanolayer (2.8 nm thick) deposited on these three samples and the differences on the size of the films (square shape whose half-size lies between 300 and 340 µm, enabling a relatively longer propagation length of surface waves than most of the 3ω samples). However, by normalizing the κ and α values obtained by each technique with their corresponding ones (κ 0 and α 0 ) measured for the thickest film (see Figures 3c and4b), we observe that both techniques yield a consistent enhancement of κ/κ 0 and α/α 0 as the film thickness decreases, which represents the fingerprint of the surface electromagnetic waves. Interestingly, the relatively high values of κ and α measured with the transient grating method suggest that these surface electromagnetic waves are able to couple with the free electrons of the highly conductive AuPd nanolayer to enhance even more their energy transport. This enhancement is corroborated by some studies proving that the presence of a metallic layer will not destroy the long-range propagation of the surface electromagnetic waves 8 and that the total thermal conductivity of such a two-layer system increases with the thickness of the metal layer without taking into account the presence of surface waves. [START_REF] Ordonez-Miranda | The Effect of the Electron-Phonon Coupling on the Effective Thermal Conductivity of Metal-Nonmetal Multilayers[END_REF] This latter result may thus open a new avenue for theoretical and experimental studies of the propagation of surface electromagnetic waves along polar/metal interfaces.

Finally, it is worthwhile to mention that the thermal energy of polaritons could also be observed in polar nanowires, whose polariton thermal conductance is quantized and given by G = (π 2 k 2 B T )/3h, with k B and h being the Boltzmann and Planck constants, respectively. [START_REF] Ordonez-Miranda | Quantized Thermal Conductance of Nanowires at Room Temperature due to Zenneck Surface-Phonon Polaritons[END_REF] At room temperature (T = 300 K), G = 0.3 nW/K, which indicates that the polariton thermal conductivity of a SiO 2 nanowire of length L and cross section A = πa 2 becomes comparable to that of its phonon counterpart (k pol 1.4 W/(m•K)), for L/A = k pol /G ≥ (14/3) nm -1 . Taking into account that this condition can be fulfilled with long enough and/or thin enough nanowires (i.e., L ≥ 147 µ m (L ≥ 1.47 µ m) for a = 100 nm (a = 10 nm)), with aspect ratios (L/a > 10 2 ) that can currently be fabricated, these polar nanowires could also be used to probe the thermal energy of polaritons.

In summary, we have experimentally quantified the thermal energy carried by surface electromagnetic waves propagating along silica nanofilms based on the 3ω and transient grating measurements of their in-plane thermal conductivity and thermal diffusivity. We have shown that both thermal properties increase more than 50% of the corresponding bulk values for film thinner than 50 nm. This enhancement represents the signature of the surface waves energy, which increases with the film temperature and confirms the generation of these thermally excited waves. The obtained results are consistent with the theoretical predictions for the thermal conductivity of long-range and broad frequency band surface electromagnetic waves, and therefore, they demonstrate the existence of a new (surface) channel for the heat transport along polar nanomaterials.

Figure 1 :

 1 Figure 1: Fabricated suspended silica films with Al wires. Pictures (a) and (b) show a top view of a 30 nm thick suspended film, for two different magnifications. (c) Top view and side view schemes of a film sample. During the 3ω measurements, we use four different Al pads to separately measure the current intensity (two lower ones) and the voltage (two upper ones) of the electric current flowing along the Al wire.

Figure 2 :

 2 Figure 2: Thermal excitation and propagation of surface waves. Scheme of the half of the suspended film. Surface waves are thermally excited by the hot metal wire placed at the center of the film and propagate along both sides of the membrane, before being scattered at its edges. The top view shows the minimum and the maximum propagation length used in the theoretical calculations (models I and II).

Figure 3 :

 3 Figure 3: Enhancement of the in-plane thermal conductivity of suspended silica membranes. (a) Plot of the in-plane thermal conductivity as a function of the film thickness. The experimental data are compared with the theoretical predictions considering that the propagation of the surface waves is limited by the film half-width (model I in solid lines) or occurs in all directions of the film plane (model II in dashed lines). The dotted line represents the bulk thermal conductivity of amorphous silica at room temperature. 23 (b) Plot of the normalized in-plane thermal conductivity obtained by the 3ω and transient grating methods, as a function of the film thickness. The values κ 0 = 1.55 W/(m•K) (t = 138.1 nm) and κ 0 = 3.37 W/(m•K) (t = 141.2 nm) have been used for the normalization of the 3ω and transient grating data, respectively. (c) Plot of the in-plane thermal conductivity as a function of the film temperature. The experimental data are compared with previous measurements of bulk silicon dioxide as a function of the temperature 29 and with the same data to which are added the predictions of models I and II.

a 1 Figure 4 :

 14 Figure 4: Enhancement of the thermal diffusivity of suspended silica membranes.(a) Plot of the thermal diffusivity as a function of the silica film thickness, obtained by the 3ω method. The dashed line corresponds to the bulk thermal diffusivity of amorphous silica at room temperature.[START_REF] Yamane | Measurement of Thermal Conductivity of Silicon Dioxide Thin Films Using a 3ω Method[END_REF][START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF] (b) Plot of the normalized thermal diffusivity obtained by the 3ω and transient grating methods, as a function of the film thickness. The values α 0 = 0.869 mm 2 /s (t = 135.7 nm) and α 0 = 1.69 mm 2 /s (t = 141.2 nm) have been used for the normalization of the 3ω and transient grating data, respectively.

Acknowledgement

This work was partially supported by JSPS KAKENHI Grant No. JP16K18031, Japanese Research foundation for Opto-science and Technology, the Conacyt Projects 192 Fronteras de la ciencia and 251882 Investigación Científica Básica 2015.

the same data as a function of the membrane half-width). Therefore, the deviations between the experimental data and theoretical curves (less than 10% for the large majority of the experimental points) can be associated with the wrinkled surfaces, which may add some propagation distortions that are difficult to predict and also to the purity of the suspended silica films. The theoretical computations (detailed in the Supporting Information) were indeed performed with the dielectric constant ε SiO 2 reported by Palik [START_REF] Palik | CRC Handbook of Optical Constants of Solids[END_REF] for pure amorphous silica, which is not exactly the case of our samples, as shown by the energy-dispersive Xray spectroscopy analysis reported in the Supporting Information (Figure S5b and Table S1). The purity of the samples explains also the experimental data below the bulk value for samples having a short half-width for which no large enhancement of the thermal conductivity is expected. Consequently, the final result will strongly depend on their intrinsic thermal properties.

The dependence of the thermal conductivity on the film temperature is displayed on Figure 3c for a 45 nm thick film. The experimental values are compared with some referenced data obtained in accordance to model I or II. We observe an increase of the thermal conductivity, which is consistent with the theoretical predictions for the thermal energy of surface electromagnetic waves. We did not heat the sample to temperatures greater than 600 K because the glass dielectric constant changes dramatically [START_REF] Joulain | Radiative Thermal Rectification Between SiC and SiO 2[END_REF] and the suspended film could undergo some deformations (see Figure S10 showing the experimental setup). Additional data of the dependence of the thermal conductivity on the film temperature are presented in Figure S11 of the Supporting Information.

The enhancement of the thermal conductivity is also present for the thermal diffusivity, as shown in Figures 4a. The consistency between the thermal conductivity and thermal diffusivity measurements is further strengthened by their ratio (see Supporting Information Figure S5a), which is nearly independent of the film thickness and takes values of about (1.5-2)×10 6 J/(m 3 •K) corresponding to the volumetric heat capacity reported for amorphous

Competing interests

The authors declare no competing financial interest.

Supporting Information Available

The Supporting Information is available free of charge on ACS Publications website at DOI:


Léo Robert
email: leo.robert@uca.fr

Daiki Miyahara
email: daiki.miyahara.q4@dc.tohoku.ac.jp

Pascal Lafourcade
email: pascal.lafourcade@uca.fr

Takaaki Mizuki
email: mizukilncs@tohoku.ac.jp

Physical Zero-Knowledge Proof for Suguru Puzzle

Keywords: Physical zero-knowledge proof, Suguru, Security, Completeness, Extractability, and Zero-knowledge

Suguru is a paper and pencil puzzle invented by Naoki Inaba. The goal of the game is to fulfil a grid with numbers between 1 and 5 and to respect three simple constraints. In this paper we design a physical Zero-Knowledge Proof (ZKP) protocol for Suguru. A ZKP protocol allows a prover (P ) to prove that he knows a solution of a Suguru grid to a verifier (V ) without leaking any information on the solution. For constructing such a physical ZKP protocol, we only rely on a small number of physical cards and an adapted encoding. For a grid of Suguru with n cells, we only use 5n + 5 cards. Moreover, we prove the three classical security properties of a ZKP: completeness, extractability, and zero-knowledge.

Introduction

Zero-Knowledge Proofs (ZKP) were introduced in 1985 by Goldwasser et al. [START_REF] Goldwasser | Knowledge complexity of interactive proofsystems[END_REF]. Two parties are involved in such a ZKP protocol: a prover P and a verifier V . At the end of the protocol, the verifier V is convinced that P knows the solution s to the instance I of a problem P, without revealing any information about s. A zero-knowledge proof prevents the verifier from gaining any knowledge about the solution other than its correctness. In fact, when both randomization and interaction are allowed, the proofs that can be verified in polynomial time are exactly those proofs that can be generated within polynomial space [START_REF] Shamir | IP = PSPACE[END_REF].

Formally, for a solution s to any instance I of a problem P , a convincing interactive zero-knowledge protocol between P and V must then satisfy the three following properties : Completeness: If P knows s, then he is able to convince V . Extractability ‡ : If P does not know s, then he is not able to convince V except with some small probability. More precisely, we want a negligible probability, i.e., the probability should be a function f of a security parameter λ (for example the number of repetitions of the protocol) such that f is negligible, that is for every polynomial Q, there exists n 0 > 0 such that:

∀ x > n 0 , f (x) < 1 Q(x)
.

Zero-knowledge: V learns nothing about s except I, i.e. there exists a probabilistic polynomial time algorithm Sim(I) (called the simulator) such that outputs of the real protocol and outputs of Sim(I) follow the same probability distribution.

There exist two kinds of ZKP: interactive and non-interactive. In an interactive ZKP the prover can exchange messages with the verifier in order to convince him, while in the non-interactive case the prover can just create the proof in order to convince the verifier.

ZKPs are usually executed by computers. They are often used in electronic voting to prove that some parties correctly mix some ballots without cheating, or in multi-party computation [START_REF] Cramer | Multiparty computation from threshold homomorphic encryption[END_REF][START_REF] Damgård | Secure two-party computation with low communication[END_REF][START_REF] Romero-Tris | Multi-party private web search with untrusted partners[END_REF]. Moreover, there exist generic cryptographic zero-knowledge proofs for all problems in NP [START_REF] Goldreich | Proofs that yield nothing but their validity and a methodology of cryptographic protocol design[END_REF], via a reduction to an NP-complete problem with a known zero-knowledge proof.

In [START_REF] Quisquater | How to explain zero-knowledge protocols to your children[END_REF], the authors explained simply this concept to some children using a circular cave. This was the first proposition of a physical ZKP. Later, Gradwohl et al. [START_REF] Gradwohl | Cryptographic and physical zero-knowledge proof systems for solutions of Sudoku puzzles[END_REF] proposed a ZKP for the famous Nikoli's puzzle called Sudoku § . They just used some physical cards to construct a ZKP protocol. It was one of the first interactive physical ZKP protocols for such puzzles. Our aim is to design a ZKP protocol for Suguru puzzles in the same spirit as the one done for Sudoku.

Suguru: It was designed by Naoki Inaba, the original name of the game was "Nanba Burokku' but it is also known as Tectonics or Number Blocks. Suguru is a paper and pencil puzzle in which a grid is divided into outlined blocks called region. Each region containing up to five cells. Every cell of the grid must contain a number from 1 to 5 (according to the number of cells in the region). Each cell should be filled such that no two identical numbers touch -not even diagonally.

Suguru's rule: This puzzle is formed by a rectangular grid where blocks divide the overall area. Those blocks called region contain up to five cells. The goal is to fill all the cells with integers under the following constraints:

-Number region rule: A region composed of k cells must be filled with integers 1, . . . , k. -Neighbour rule: For every cell, all of its eight neighbours must have different values from the cell's value.

In Figure 1, we give an example of an initial Suguru grid and in Figure 2 we give its unique solution.

Contributions: We propose a simple ZKP protocol for Suguru using a small number of cards. Our construction is simple and can be used as a pedagogical example to explain the role of ZKP protocols. We propose an encoding of the number using simple cards. Using this encoding, the prover places some cards on the grid according to its solution. We use these cards to prove that the two rules of Suguru are satisfied. We start with the first number of cell in a region, after having verified the validity of all regions and replacing the cards placed by the prover, we reuse them to prove the second rule of Suguru about the eight neighbours of each cell of the grid. Here is the difficulty of Suguru, since we need to prove that all values of the eight neighbours of each cell are different without revealing any information to the verifier. Here, we use a trick of our encoding of the values of the cards in order not to leak any information. Our encoding requires five cards per cell; therefore if a Suguru grid has n cells to guess, our protocol only requires 5n+5 cards. Finally, we prove the three security properties of our construction i.e., completeness, extractability, and zeroknowledge.

Related Work: In [START_REF] Sasaki | Efficient card-based zero-knowledge proof for Sudoku[END_REF] the authors proposed an improved ZKP protocol for Sudoku that follows the pioneer work of [START_REF] Gradwohl | Cryptographic and physical zero-knowledge proof systems for solutions of Sudoku puzzles[END_REF]. In [START_REF] Dumas | Interactive physical zero-knowledge proof for Norinori[END_REF], the authors proposed a ZKP protocol for the Nikoli's puzzle Norinori.

In [START_REF] Lafourcade | A physical ZKP for Slitherlink: How to perform physical topology-preserving computation[END_REF], a method to take into account one feature of several puzzles that consists to construct a single loop, has been invented. This technique used a topological approach with successive interactive transformations.

Recently several ZKP proofs have been proposed for different Nikoli's puzzles. In [START_REF] Miyahara | Card-based ZKP protocols for Takuzu and Juosan[END_REF], card-based ZKP Protocols for Takuzu and Juosan have been proposed. In [START_REF] Ruangwises | Physical zero-knowledge proof for Numberlink[END_REF] a physical ZKP proof for Numberlink has been designed. In [START_REF] Miyahara | Card-based physical zeroknowledge proof for Kakuro[END_REF], a card-based physical ZKP for Kakuro have been given that improves the first version proposed by Bultel et al. in [START_REF] Bultel | Physical zero-knowledge proofs for Akari, Takuzu, Kakuro and KenKen[END_REF] with the ZKP protocols for three other Nikoli's games: Akari, Takuzu, and Kenken.

All these works clearly demonstrate that designing physical ZKP is clearly an interesting topic of research. Each game has its own particular rules and requires an adapted construction.

Although the existence of all those previous works, one cannot reuse or adapt directly them for the Suguru game. Indeed, the main reason is the "strong" neighbour rule where no cell can have its eight neighbours with the same value. Other puzzles have a similar rule but with relaxed restrictions. For instance, Makaro has a neighbour rule but only for adjacent cells (and not in diagonal). Thus a naive adaptation would imply a loss in terms of efficiency and of zero-knowledge (no information about the solution can be leaked). Furthermore, it is worth noting that the encoding for the proof of NP-completness of Makaro cannot be applied for Suguru. Thus ZKP for Suguru cannot be directly adapted from the ZKP of Makaro. Further discussion is given in Section 5.

Outline: In Section 2, we present our notations, and all subprotocols needed to construct our ZKP. In Section 3, we design our ZKP protocol for Suguru. In Section 4, we prove the security of our protocol. In Section 5, we discuss a complexity of Suguru and of Makaro. In the last section, we conclude the paper.

Preliminaries

We introduce some notations of cards and shuffles used in our construction.

Notations

Card: A deck of cards used in our protocol consists of blacks ♣ and reds ♥ whose back sides are identical ? . Each integer i ∈ {1, . . . , 5} is encoded as:

1 ♥ 2 ♣ 3 ♣ 4 ♣ 5 ♣ 1 , 1 ♣ 2 ♥ 3 ♣ 4 ♣ 5 ♣ 2 , 1 ♣ 2 ♣ 3 ♥ 4 ♣ 5 ♣ 3 , 1 ♣ 2 ♣ 3 ♣ 4 ♥ 5 ♣ 4 , 1 ♣ 2 ♣ 3 ♣ 4 ♣ 5 ♥ 5 .
We call such face-down five cards ? ? ? ? ? corresponding to an integer according to the above encoding rule a commitment to the respective integer.

We also use numbered cards such as 1 2 3 4 5 whose backs are identical ? .

Neighbour cell: Consider a target cell denoted c t on a grid. A cell is a neighbour of c t if it is next to c t . It can be on the left, the right, the top, or the bottom of c t , and also on its diagonal: c t Thus, a cell can have at most eight neighbours.

Pile-scramble shuffle: A shuffle used in our protocol is a pile-scramble shuffle, which was first used by Ishikawa et al. [START_REF] Ishikawa | Efficient card-based protocols for generating a hidden random permutation without fixed points[END_REF] and was used in other physical ZKP protocols for puzzles (e.g., Sudoku [START_REF] Sasaki | Efficient card-based zero-knowledge proof for Sudoku[END_REF]). Consider that we have a sequence of piles of cards, each of which consists of the same number of face-down cards, denoted by (p 1 , p 2 , . . . , p ) for some positive integer . Applying a pile-scramble shuffle to the sequence results in (p r -1 (1) , p r -1 (2) , . . . , p r -1 ( ) ) where permutation r is uniformly and randomly chosen from the symmetric group of degree . That is, it randomly permutes a sequence of piles and nobody knows the order of the resulting sequence.

One can easily implement a pile-scramble shuffle by using physical tools that can fix each pile of cards such as rubber bands and envelopes; a player (or players) randomly shuffle them until nobody traces the order of the piles.

ZKP Protocol for Suguru

We propose a ZKP protocol for Suguru composed of two phases, the setup phase and the verification phase.

Setup phase

The verifier V and the prover P place commitments corresponding to the integers on the initial grid of a Suguru puzzle. In addition, when a region of k cells is already filled with k -1 cells, then P and V agreed on the last cell to complete and place the commitment accordingly ¶ .

Then, P continues to place commitments on all the remaining cells by himself according to the solution of the puzzle.

Verification phase

There are two verifications to ensure the number region rule and the neighbour rule.

Number region rule: V wants to check that a region of k cells contains all the consecutive integers from 1 to k.

1. For every i, 1 ≤ i ≤ k, V picks all cards of the i-th cell (in any ordering) to form a pile p i . Then, V attaches a numbered card i to p i . Thus, there are p 1 , . . . , p k piles, each of which consists of six cards. 2. Apply the pile-scramble shuffle [START_REF] Ishikawa | Efficient card-based protocols for generating a hidden random permutation without fixed points[END_REF]. 3. V reveals the cards of each pile except for the numbered card. The revealed output is of the form (up to a permutation in the rows), i.e., all the k (opened) commitments corresponding to 1 through k should appear. For example, if k = 4, the revealed output should be of the following form (up to a permutation in the rows):

♥ ♣ ♣ ♣ ♣ ? ♣ ♥ ♣ ♣ ♣ ? ♣ ♣ ♥ ♣ ♣ ? ♣ ♣ ♣ ♥ ♣ ?
, where the face-down cards on the right side are the numbered cards.

If the revealed output is not of this form, V aborts. 4. Turn over the face-up cards and apply the pile-scramble shuffle again to the piles. 5. Reveal only the numbered cards of all piles. Because these revealed cards indicate the initial positions for each pile, V rearranges each pile back to their initial place. The revealed numbered cards can be reused for the remaining verifications. ¶ For example, in Figure 1 the upper left region can be directly completed with a 1.

Neighbour rule: V wants to check if a given cell has no neighbour having the same integer as the cell.

1. V picks the first card of the target commitment and then picks each first card of the commitments on its neighbour cells (in any ordering) to form the pile p 1 . The following is an example when there are eight neighbours:

7 ? ? ? ? ? .

2. V repeats the same operation until the pile p 5 is formed.

3. V attaches a numbered card i to p i . (If the target cell is the last one, V does not perform this step.) 4. If an integer is written on the target cell, then go to the next step.

Otherwise, apply the pile-scramble shuffle to the piles. 5. V reveals the first card of each pile, which corresponds to the target commitment. Let p t denotes the pile where a red card appears. 6. V reveals all the cards in the pile p t except for the numbered cards. If there are two red cards in the pile then V aborts; otherwise, V goes to the next step. 7. As Steps 4 and 5 in the previous verification, V rearranges all the cards in the piles back to their initial places. (If the target cell is the last one, V does not perform this step.)

Evaluation

If the size of the grid is represented by n then the number of cards used in this protocol is equal to 5n + 5. Indeed, each cell must be encoded with 5 cards (4 blacks and 1 red) and 5 numbered cards are used for all target cell.

We could have encoded each cell with a total of cards where is the number of cells in the region (thus, a region with two cells has its cell encoded with only two cards, a red and a black). Yet, this would lead to inconstancy in the encoding rule which is required in the neighbour verification.

Security proofs

We give the theorems along with their proofs to provide that our protocol respects the security properties. A ZKP protocol is secure if the following three properties are satisfied:

Correctness: If the prover P commits its cards according to the actual solution, then all verifications will not abort. Hence, if P knows a solution, then it can always convince the verifier V . Correctness is proven in Theorem 1. Extractability: If P 's input is invalid, the protocol will point errors out to V . Therefore, if P does not know the solution, it cannot convince V . Extractability is proven in Theorem 2. Zero-knowledge: V learns nothing about P 's solution. Zero-knowledge is proven in Theorem 3.

Theorem 1 (Completeness). If P knows a solution of a Suguru grid, then it can convince V .

Proof. Suppose that the prover P knows the solution of the Suguru grid.

It runs with the verifier V the Setup phase (Section 3.1). We show that P can perform both verification phases without aborting.

Number region verification:

In this phase, the goal of P is to show that each region of size k contains consecutive integers from 1 to k (note that the lower bound of k is 1 and its upper bound 5). Since P places the cards accordingly with the solution, each region of size k contains the numbers 1 to k. Without loss of generality, suppose that the pile p i corresponds to the number i with i = 1 . . . k. The pile p 1 is composed of the sequence (in this order):

♥ ♣ ♣ ♣ ♣ .
The pile p 2 is composed of the sequence (in this order):

♣ ♥ ♣ ♣ ♣ .
More generally, the pile p i is a sequence of black cards where the red card is placed on position i.

Since the pile-scramble shuffle applied on Step 2 does not modify the order of the sequence, the red card of pile p i is on position i. As i = 1 . . . k, all numbers from 1 to k are represented. Thus, V is convinced that the number region rule is verified by revealing the piles in Step 3.

Neighbour verification:

The goal of P is to convince V that no cell has the same number of its neighbours (there are eight neighbours as defined in Section 2). Let c t be the target cell placed on the center of the 3 × 3 square. Since P placed the commitments according to the solution, there is no cell with the same value of c t in this square. Let i be the position of the red card of c t (with i = 1 . . . 5). Since no neighbour cell has the same value of c t , there is no other red card with position i. Since each pile is composed of card with the same indices, the pile p i (before the shuffle) contains exactly one red card. Hence, V is convinced that the neighbour rule is verified.

Finally, since all verifications are checked, we proved that if P has the solution then the verifications will always succeed.

Theorem 2 (Extractability). If P does not provide a solution of the Suguru puzzle, it is not able to convince V .

Proof. Suppose that P does not know a solution for the puzzle. We want to show that V will always detect it.

Since P cannot provide the solution, at least one of the two rules is not verified (if both can be verified, this is the solution). We can distinguish two cases corresponding to each verification:

-The number region rule is not respected. That is, suppose w.l.o.g. that a region of size k with k > 1 does not contain the number 1. Hence, the sequence corresponding to this number is missing, meaning that V cannot reveal at Step 3 the sequence:

♥ ♣ ♣ ♣ ♣ .
Thus, V will abort the protocol and detect that P cannot provide the solution.

-The neighbour rule is not respected. Suppose that we have the following configuration: 2 2 with blank cards of unimportant value but different from 2. We encode number 2 as:

♣ ♥ ♣ ♣ ♣ .
Thus, the pile p 2 corresponding to all the cards with indices 2 (before the shuffle) will contain exactly two red cards. Thus, V will abort the protocol.

We proved that if P does not have the solution then the verifications will abort in both cases meaning that P cannot convince V .

Theorem 3 (Zero-knowledge). V learns nothing about P 's solution of the given grid G.

Proof. We use the same proof technique as in [START_REF] Gradwohl | Cryptographic and physical zero-knowledge proof systems for solutions of Sudoku puzzles[END_REF]: zero-knowledge is caused by a description of an efficient simulator which simulates interaction between a cheating verifier and a real prover. However, the simulator does not have a solution but it can swap cards for different ones during shuffles. The simulator acts as follows:

-During the number region verification at Step 2, the simulator swaps the piles to replace them by the sequences (up to a permutation in the rows):

♥ ♣ ♣ ♣ ♣ ♣ ♥ ♣ ♣ ♣ ♣ ♣ ♥ ♣ ♣ ♣ ♣ ♣ ♥ ♣
-During the neighbour verification, when revealing the cards at Step 6, the simulator swaps the pile with a pile containing k -1 black cards and 1 red card.

The simulated proofs and the real proofs are indistinguishable; thus, V learns nothing about P 's solution.

Discussion

Among In [START_REF] Iwamoto | Herugolf and Makaro are NP-complete[END_REF], the authors proved that the two games Herugolf and Makaro are NP-complete. Solving Makaro was shown to be NP-complete via a reduction from 3-SAT.

Makaro is a game close to Suguru, with different constraints as follows:

1. Room condition: Each room contains all the numbers from 1 up to the number of cells in the room. 2. Neighbour condition: A number cannot be next (adjacent) to the same number in another room. 3. Arrow condition: Every black arrow cell must point at the largest number among the numbers in the adjacent cells of the black cell (possibly the fours cells: right, left, above, and bottom).

In Figure 3, we give a simple example of a Makaro game, where all black cells are arrow cells and all white cells are empty cells except for one filled cell with three. It is easy to verify that the three constraints are satisfied in the solution given in Figure 4. We remark that in a solution all white cells are filled with numbers between 1 and k, where k is the maximum size of all the rooms of the grid.

The room condition is the same, however the neighbour condition is different in Suguru. In Suguru for each cell, we consider the eight neigh-bours while in Makaro we consider only the four neighbours. Moreover in Suguru the size of the regions is limited to five while in Makaro there is no limit on the size of the rooms.

These differences avoid us to reuse the 3-SAT encoding used in the proof of NP-completeness of Makaro [START_REF] Iwamoto | Herugolf and Makaro are NP-complete[END_REF]. If we remove the limit of the maximum of five cells in each region, we can use the same gadgets as the ones used for Makaro to prove in a similar way that Suguru is also NP-complete. But with the limit of five cells per regions and the change of four neighbours into eight neighbours, it is not clear if we can prove the NP-completeness of Suguru. These extra constraints seem to remove some difficulty in finding some solutions for Suguru. At the moment we are not able to prove NP-completeness and we conjecture that solving a Suguru grid should be in P.

Even if solving Suguru is in P, our physical ZKP is an interesting approach since it requires only 5n + 5 cards as we show when we present our ZKP protocol in Section 3. (In addition, it could always happen that one cannot solve a Suguru puzzle.) This is clearly a real ZKP protocol that can be used by Suguru players in practice.

Conclusion

In this paper we propose a simple card-based physical ZKP for Suguru. Our solution is simple and efficient since it relies on only 5n + 5 cards. One open question left for the future is to demonstrate the conjecture states in the introduction i.e., proving that solving a Suguru grid is in P. We clearly cannot adapt the proof of NP-completeness of Makaro with the rules of Suguru, it is why we conjecture that Suguru is in P.

Moreover, our long term research direction is to design physical ZKP protocols for all Nikoli's games. However some rules of some games like Shakashaka † † that requires to draw rectangles which is not easy to model without leaking any information. Another example of challenging game is Shikaku ‡ ‡ , where the rules are simple: 1) Divide the grid into rectangles with the numbers in the cells. 2) Each rectangle is to contain only one number showing the number of cells in the rectangle. However it remains a challenging open question to design a physical ZKP for this game without revealing any information on the positions of the rectangles.

Fig. 1 .

 1 Fig. 1. Initial Suguru grid

Fig. 2 .

 2 Fig.2. Solution of the Suguru grid of Figure1

Fig. 3 .Fig. 4 .

 34 Fig. 3. Example of a Makaro grid

† † http://www.nikoli.co.jp/en/puzzles/shakashaka.html ‡ ‡ https://www.nikoli.co.jp/en/puzzles/shikaku.html

Acknowledgements. We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. This work was supported in part by JSPS KAKENHI Grant Number JP19J21153.