Ibrahim B Malham

Lionel Bureau

Density effects on collapse, compression and adhesion of thermoresponsive polymer brushes

Keywords:

We probe, using the Surface Forces Apparatus, the thermal response of poly(Nisopropylacrylamide) (PNIPAM) brushes of various grafting densities, grown from plasma-activated mica by means of surface-initiated polymerization. We thus show that dense thermoresponsive brushes collapse gradually as temperature is increased, and that grafting density greatly affects their ability to swell: the swelling ratio of the brushes, which characterizes the thickness variation between the swollen and the collapsed state, is found to decrease from ∼ 7 to ∼ 3 as the number of grafted chains per unit area increases. Such a result, obtained with an unprecedented resolution in grafting density, provides qualitative support to calculations by Mendez et al. [Macromolecules 2005 38, 174]. We further show that, in contrast to swelling, adhesion between two PNIPAM brushes appears to be rather insensitive to their molecular structure.

I. INTRODUCTION

Temperature sensitive polymers attract a growing interest for their use in designing 'smart' surfaces or interfaces [1,2,3], which properties (wettability [4], adhesion [5], friction [6,7],...) can be deeply altered by an external thermal stimulus. Such smart systems rely on the fact that thermosensitive polymers exhibit a lower critical solution temperature (LCST) when mixed with a solvent [8]. Below the LCST, polymer chains are well solvated and adopt a swollen coil conformation. As the temperature is increased across the LCST, the solvent quality goes from good to poor, phase separation occurs, and macromolecules collapse to display a dense globular conformation. Such a behaviour is commonly encountered in liquid binary mixtures in which specific interactions (e.g. hydrogen bonding) exist between constituents [9,10], as in aqueous solutions of poly(N-isopropylacrylamide) (PNIPAM), one of the most studied among thermosensitive systems [11,12]. PNIPAM in water exhibits a LCST around 32 • C [11,12], above which polymer chains become hydrophobic. The coil-to-globule transition in bulk solutions occurs over a narrow range of temperature and involves large variations in chain dimension [12,13]. These properties, which have been recently exploited in microfluidics for the design of switches [14] and valves [15] for flow control, have found their main applications in bioengineering [2,3,16]. The use of PNIPAM coatings on solid surfaces is indeed a widespread technique for cell adhesion control in tissue engineering [17,18], or for separation of biomolecules in temperature selective chromatography [18].

Grafted PNIPAM brushes are viewed as promising surface modifiers in order to finely tune the interactions * Electronic address: bureau@insp.jussieu.fr between cells or proteins and the surfaces of biomedical devices [18]. However, empirical design of such thermoresponsive sustrates might reveal insufficient. This appears for instance from two recent studies on harvesting of cell cultures grown on thermoresponsive substrates, that reached opposite conclusions about the influence of brush thickness on cell proliferation/detachment [17,19]. This underlines that reliable control of interactions requires a deeper understanding of how the brush properties (surface energy, ability to swell/collapse, thermal response,...) depend on molecular parameters such as grafting density or chain length [3].

The structure of PNIPAM brushes in water has been the subject of recent experimental investigations. Force sensing techniques (Atomic Force Microscopy and Surface Forces Apparatus) have been used to measure (i) the range of repulsive forces resulting from brush compression [20,21,22,23,24], and (ii) adhesion forces between PNIPAM brushes and various countersurfaces [5,21,22,23,24]. Most of these studies have been performed at only two temperatures, below and above the bulk LCST. On the other hand, neutron reflectivity [25], quartz-cristal microbalance [20,26,27] or surface-plasmon resonance [28] have been used to follow in detail the evolution with temperature of the thickness of brushes of different densities and molecular weights. Bringing together the conclusions of these studies, two important results may be put forward:

(i) the collapse of densely grafted chains occurs over a broader temperature range [25,26,27,28] than that of dilute chains in solution [13] or brushes with low grafting density [20],

(ii) the swelling ratio, i.e. the ratio of the swollen to the collapsed thickness, is affected by both chain length and grafting density [23,24,25].

The former result has been obtained in several independent studies, and is in qualitative agreement with numerical and theoretical predictions about the collapse of brushes in poor solvent [29,30,31]. However, open questions remain regarding point (ii), due to the limited number of studies which systematically investigated the effect of molecular parameters on the magnitude of brush collapse [23,24,25]. In particular, whether a high grafting density favors or prevents large swelling ratio is still to be clarified. Furthermore, the role of brush density on the magnitude of adhesion forces with PNIPAM surfaces has remained unexplored.

These points, which are crucial for the design of smart surfaces based on thermoresponsive thin films, have led us to investigate in further details the effect of grafting density on the properties of PNIPAM brushes.

We have used the surface forces apparatus to study the thermal response of PNIPAM brushes grown on mica substrates by the grafting from method.

We focus on brushes of given molecular weight (M w ≈ 475 kg.mol -1 ) exhibiting densities ranging from 2 × 10 -4 to 4 × 10 -3 chain. Å-2 . We show that, in the range of temperatures explored (20-40 • C):

(i) for all grafting densities, the brush thickness decreases slowly as the temperature is increased up to 30 • C, then exhibits a marker decrease between 30 and 35 • C, temperature above which the collapsed thickness is reached. This confirms the broadening of the collapse transition previously observed for dense brushes [25,26,27,28].

(ii) the swelling ratio decreases noticeably with increasing surface coverage. This trend, which we observe with an unprecedented resolution over a large range of grafting densities, is in good qualitative agreement with recent predictions of Mendez et al. [32].

(iii) in contrast to the density dependence of collapse, adhesive forces between two identical brushes, which build up for temperature above ∼ 30 • C, appear to be rather insensitive to grafting density.

II. MATERIALS AND METHODS

A. Materials

Substrates were cleaved from muscovite mica plates purchased from JBG-Metafix (France). The monomer Nisopropylacrylamide (NIPAM 99%, Acros Organics) was recrystallized three times in n-hexane and stored in a desiccator under vacuum until use. n-hexane, toluene and absolute ethanol (Normapur, VWR France) were used as received. 3-aminopropyl-triethoxysilane (APTES, ≥ 98%) and propyl-trimethoxysilane (PTMS, 98%) were obtained from Sigma-Aldrich. Triethylamine (TEA, 99% pure), copper (I) bromide (CuBr 98%, extra pure), copper (II) bromide (CuBr 2 , 99% extra pure), 1,1,7,7-Pentamethyldiethylenetriamine (PMDETA, 99%) and 2bromo-2-methylpropionyl bromide (98%, pure) were purchased from Acros Organics. All organic solvents were filtered over 0.20 µm Teflon membranes prior to use. All aqueous solutions were prepared in deionized water (18 MΩ) filtered over a 0.20 µm cellulose acetate membrane before use.

B. Surface Forces Apparatus

Experiments reported in the present paper were performed using a home-built surface forces apparatus (SFA) [33]. The apparatus has been modified to include a stainless steel cell of volume 1.5 mL that can be filled with a solvent into which the surfaces under study are fully immersed. We have used this instrument for time-resolved measurements of force-thickness curves during compression/decompression of PNIPAM brushes in water, confined between two atomically smooth mica surfaces. Such measurements were performed at different temperatures, below and above the bulk collapse temperature. The experimental configuration, involving two facing brushes grafted on curved mica sheets, is sketched on Fig. 1.

Normal forces are measured, with a resolution of ∼ 10 -6 N, by means of a flexure-hinge spring of stiffness 9500 N.m -1 equiped with a capacitive displacement sensor. The distance between mica substrates is measured by white-light multiple-beam interferometry (MBI) [34,35]: the fringes of equal chromatic order produced between the reflective backsides of the substrates are analyzed using the multilayer matrix method [33,36] in order to deduce the thickness and the effective refractive index of the confined medium.

Mica substrates were prepared according to the following protocol. A large mica sheet (10 cm × 10 cm) was cleaved down to a thickness of ∼10-30 µm. One side was thermally evaporated with a 40 nm-thick silver layer, in order to obtain a highly reflective surface for multiple beam interferometry. Samples of ∼ 1 cm 2 were cut from this sheet with surgical scissors, and glued, silver side down, onto glass cylindrical lenses (radius of curvature R ≃ 1 cm), using a UV-curing glue (NOA 81, Norland). Two glued sheets were then re-cleaved using adhesive tape, down to a thickness of 1-4 µm. The cylindrical lenses were then mounted inside the SFA, with their axis crossed at right angle, and the exact thickness of each substrate was determined by MBI, following a procedure described in [37]. Samples were then unmounted and subjected to the various surface treatments described below in order to obtain brushes of PNIPAM covalently grafted on the substrates.

After grafting on both mica sheets, samples were installed back in the SFA, brushes were brought into contact in the absence of solvent, and the total thickness, h tot , of grafted PNIPAM was measured. In what follows, we report the dry thickness of the various brushes as being h dry = h tot /2, assuming that an identical amount of polymer is grafted on each surface. The refractive index for such dry layers was measured to be 1.47±0.02, in good agreement with values reported in other study [23].

The surfaces were then separated, and water injected into the liquid cell until full immersion of the brushes. The temperature inside the SFA was set, to within ±0.01 • C, to a chosen value in the range 20-40 • C, by means of a thermoregulation system described previously [33]. The surfaces were approached by driving the re-mote point of the force-measuring spring at a prescribed velocity V , until a normal load of ∼ 10 -2 N was reached, and then separated at the same driving velocity. All the results presented below have been obtained with V ≃ 4 nm.s -1 . We have checked that lowering the driving velocity by one order of magnitude did not affect the range of measurable repulsive forces by more than a few percents. This indicates that, with the chosen approach velocity, forces mostly arise from steric repulsion between brushes and not from hydrodynamic drag due to solvent flow. Besides, we have found that the effective refraction index (n eff ) for the brush/water system increases from ∼ 1.33 for large (several microns) intersurface separations, up to 1.43±0.02 at the onset of repulsion between brushes. Under stronger brush compressions, n eff stays roughly constant, within experimental resolution, at a value of ≃ 1.45. Since we focus, in what follows, on brush compression, we present results for which the brush thickness has been calculated using a constant n eff = 1.45.

C. Substrates preparation

Scheme 2 summarizes the different steps, following the thickness measurement of a pair of bare mica sheets, which lead to substrates grafted with a given surface density of initiator for Atom Transfer Radical Polymerization (ATRP). (i) The mica samples of known thickness were transfered to a plasma reactor (model Femto from Diener Electronics Germany, operated at 80W), where they were exposed for 10 minutes to a RF-plasma generated in water vapor, at a pressure of 0.4 mbar. Such a treatment, proposed initially by Parker et al. [38], has been shown to produce silanol groups at the mica surface [39], without any significant increase of its roughness. Covalent attachment of silane based molecules on such plasma-activated surfaces is then possible [38,39,40,41].

(ii) Hydroxylated mica surfaces were then immediately immersed in a solution of propyltrimethoxysilane (PTMS) in toluene at 70 • C. The PTMS solution was prepared following the procedure described by Smith et al.

[42]: 0.5 mL of PTMS were mixed with 25 mL of toluene, stirred at 70 • C for 90 minutes, and filtered over a 0.2 µm Teflon membrane before immersion of the substrates. After PTMS grafting, mica surfaces were thouroughly rinsed with toluene, ethanol and water, and dried in an argon flow. This step results in a fraction of the silanol groups at the mica surfaces being replaced by methylterminated molecules. This fraction was varied by adjusting the immersion time between 0 and 1h.

(iii) PTMS grafted samples were subsequently immersed, at room temperature and for a duration ranging from a few seconds to 5 minutes, in an aqueous solution of aminopropyltriethoxysilane (APTES). APTES (100 µL) was diluted in water (100 mL), stirred for 2h at ambient temperature in order to prehydrolyze the ethoxysilane groups, and filtered through a 0.2 µm membrane before use. APTES grafting was followed by copious rinsing with water and ethanol. This yields mica substrates which exhibit a mixed NH 2 /CH 3 functionality at the surface, the ratio of amino to methyl groups being varied by adjusting immersion times in step (ii) and (iii).

(iv) A solution of triethylamine (1.22 mL) in toluene (20 mL) was then prepared and cooled at 0 • C. The silane grafted mica samples were immersed in this solution, to which 255 µL of 2-bromo-2-methylpropionyl bromide were added dropwise. The samples were left in the bath for 2-3 minutes, after which they were rinsed in toluene, washed in ethanol and water and dried in an argon flow. This step results in surface immobilization, on the aminoterminated sites, of the ATRP intiator from which PNI-PAM brushes are grown [25,43].

D. Grafting of PNIPAM brushes

Brushes were grown by ATRP, following a protocole akin to that described in references [22,23,25]. A solution of NIPAM (3.00 g) dissolved in deionized water (15 mL) was deoxygenated by bubbling Argon at room temperature for 30 min. A solution of CuBr (0.044g) in water (3 mL) was prepared and stirred while 210 µL of PMDETA was added for copper complexation. This solution was then mixed to the NIPAM solution under argon bubbling.

A pair of indentically modified mica samples was then placed for 5 minutes in the aqueous solution of NI-PAM and copper catalyst. Next, samples were immersed for 10 min in 10 mL of an aqueous solution of CuBr 2 (0.048g)/PMDETA (126 µL) to quench the polymerisation reaction. Finally, surfaces were rinsed repeatedly with water and dried carefully in an argon stream inside a laminar flow cabinet, where the were installed in the surface forces apparatus.

We have used the same polymerization time for each pair of mica substrates, which we expect to result in brushes formed of chains of comparable molecular weight, the surface density of which shall be controlled by the initial ATRP initiator coverage.

E. Characterization of grafted layers

PNIPAM brushes were first characterized by measuring their dry thickness, using the SFA. In what follows, we present results obtained on brushes of six different thicknesses, which are reported in Table I. Since NIPAM polymerization occured only at the surface of the samples, and not in the bulk solution, we do not know a priori the molecular weight of the grafted chains. However, as presented in the Discussion section below, a scaling analysis of the ratio between swollen and dry thick- Besides, we have performed wetting experiments in order to measure the static contact angle (θ s ) of water as a function of temperature on the various brushes. This was done on flat mica substrates submitted, simultaneously with the SFA samples, to the sequence of chemical modifications described above. Experiments were performed using a custom-built instrument, described in a previous publication [41], which we have adapted in order to regulate the temperature of the substrate and of the water droplets in the range 20-40 • C.

Figure 3 shows the evolution of θ s with temperature for three brush thicknesses. It can be seen that the contact angle stays constant for T 30 • C, increases by approximately 6-7 • while T increases from 30 to 34 • C, and exhibits a plateau at higher temperatures. Such a behavior is in qualitative agreement with previous measurements on similar surfaces [23,25]. It shows that PNIPAM is indeed grafted on the mica substrates, the sharp increase of θ s around 32 • C being the signature of the hydrophobic nature of PNIPAM above its LCST. As seen in Fig. 3, we only observe a very weak influence of brush thickness on θ s . This indicates that wetting is mainly sensitive to the chemical composition of the outermost region of the brush, and not to its detailed structure, as already suggested in previous works [25].

III. RESULTS

We first present the effect of temperature on the measured force-thickness curves during compression of two identical brushes. The results reported in Fig. 4 have obtained with 70nm-thick brushes. The behaviors observed with thicker or thinner brushes displayed the same qualitative features. It can be seen that, as the temperature is increased from 23 to 37 • C:

(i) the onset of repulsive forces occurs at smaller intersurface separations (Fig. 4a and4b), (ii) full compression of the PNIPAM layers (down to their dry thickness) is reached under lower normal forces (Fig. 4a and4b), (iii) upon surface separation, adhesive forces build up for T 30 • C, and are larger at higher temperatures (Fig. 4c).

We also notice that at T 35 • C, in contrast to the monotonic repulsion measured at lower temperatures, attractive interactions between brushes are observed upon approach, before entering the 'hard-wall' repulsion regime. This is illustrated in the inset of Fig. 4a.

The effect of temperature on the range of repulsive forces is further characterized as follows. We measure, as a function of temperature, the thickness h 0 reached under a normal force of 10 -5 N, i.e. under low compression. Note that the choice of such a criterion is arbitrary, it only ensures that the applied force is well above the experimental resolution, and h 0 is therefore lower than the unperturbed brush thickness [44]. We define the ratio α = h 0 /h dry , which we call the "swelling ratio" in the rest of the article. Fig. 5 presents the evolution of α with temperature, for different brush densities. We observe that:

(i) for all densities, α decreases gradually as T increases, displays a stronger temperature sensitivity in the range 30-35 • C, and reaches, for T 35 • C, a constant value close to unity., (ii) the lower the brush density, the larger the value of α at low temepratures.

Finally, we plot on Fig. 6 the measured pull-off forces (the force needed to separate the brushes), as a function of temperature, for various brush densities. It can be seen that adhesion forces between contacting brushes are independent of their grafting density, and increase with temperature above 30 • C.

IV. DISCUSSION

A. Collapse

The temperature dependence of the repulsive range reported in Fig 4a and4b shows that the thickness of the brushes decreases as the temperature is increased: grafted chains evolve from a low temperature swollen state (with α > 1, see Fig. 5) to a high temperature collapsed state (α ≃ 1) where most of the solvent is expelled from the polymer layers. This result is fully consistent with the thermal response of grafted PNIPAM previous observed in experiments using AFM or SFA [20,21,22,23].

Furthermore, the evolution of the swelling ratio with temperature (Fig. 5) clearly shows that the collapse of end-tethered chains, in the range of grafting densities explored here, occurs gradually rather than through a sharp coil-globule transition as observed for isolated chains in solution [12,13]. Such a behavior, which is observed for the first time by means of force measurements, is in good agreement with results obtained from neutron reflectivity [28], surface plasmon resonance [25] or quartz-crystal microbalance [26,27]. It is qualitatively consistent with theoretical predictions by Zhulina et al. [30,31], who have shown that, due to repulsive interactions between densely grafted stretched chains, the collapse of a dense planar brush occurs gradually as the solvent strength is lowered.

Besides, it can be seen on Fig. 5 that the grafting density affects both the magnitude of swelling and the temperature sensitivity (the slope dα/dT ) of the brushes: as T increases from 30 to 35 • C, α exhibits a sixfold decrease for the lowest density, whereas it drops by only a factor of 2.5 for the highest density. Such a result is of importance for applications where large and sharp thickness changes are needed, e.g. for efficient flow control or actuation in brush-grafted nano/microchannels [14,16].

B. Low temperature swelling

The low temperature values of the swelling ratio indicate that under good solvent conditions, less dense brushes swell more than denser ones (Fig. 5). This result can be interpreted as follows, within the Alexander-de Gennes framework for the description of brushes in good solvent [45,46]. The dry thickness h dry of a brush is related to the grafting density by:

h dry d 2 = N a 3 (1)
where d is the average distance between tethering sites (1/d 2 is the grafting density), N is the number of monomer units per chain, and a is the monomer size. Following Alexander and de Gennes, we picture a swollen brush as formed of stretched chains subdivided into blobs of size D b ∼ d (see inset of Fig. 7). The blob size is given by the following expression:

D b = g ν a (2)
with g the number of monomers in a blob. We discuss below the value of the exponent ν.

The swollen brush thickness is:

h swell = n b D b (3)
with n b = N/g the number of blobs per chain. Replacing D b by d in Eq. 2 and 3, one gets for the swollen thickness:

h swell = N a d a 1-1/ν (4)
The ratio of the swollen to the dry thickness thus reads:

α = h swell h dry = d a 3-1/ν (5)
Such a scaling is expected to break down at low enough grafting densities, i.e. at large d, where the swelling ratio most likely saturates at a value comparable to that of an isolated chain.

Under the assumption of semi-dilute brush (i.e. for an average monomer volume fraction φ which stays ≪ 1), the exponent ν shall be close to 3/5, the blob size being governed by excluded volume interactions [46]. At large monomer concentrations, for very dense brushes, it has been shown that ν = 3/5 does not allow to account for the measured swollen thickness [47], and the evolution of h swell with grafting density observed in reference [47] suggests that high density brushes shall be better described with gaussian blobs, i.e. ν ≃ 1/2.

On figure 7, we plot α, measured at 23 • C, as a function of 1/ h dry (which is proportional to d). It can be seen that, at small d, the swelling ratio increases quasi-linearly with the distance between grafting sites, then levels off at large d as expected. The linear increase of α with d observed at small d is in good agreement with expression 5, provided that ν ≃ 1/2. Such a value of ν, in agreement with previous measurements on dense brushes [47], is consistent with the fact that the average monomer concentration in the most dense brushes is large. Indeed, the volume fraction, which can be estimated as φ ≃ 1/α, is found to range from 1/7 to 1/3, values that may be too high for the assumption of semi-dilute brush to strictly hold. In view of the agreement obtained between the measured and predicted evolution of α(d) at small d, we go one step further and use Eq. 5, assuming the monomer size to be a ≃ 5 Å [20,28], in order to evaluate the grafting densities of the brushes for which the linear scaling of α(d) holds. We thus obtain grafting densities, reported in table I, ranging from 9 × 10 -4 to 4 × 10 -3 chains/ Å2 . Next, we check the consistency of our assumption of constant N by estimating the number of monomers per chain using Eq. 1 and the calculated values of d. The polymerization indices are reported in table I, where it can be seen that the average value of N is 4200, with variations of ±10% from brush to brush. This corresponds to an average molecular weight of the grafted chains of 475 kg.mol -1 . Finally, we use the average value of N to calculate, from Eq. 1, the grafting density of the brush for which the above scaling analysis fails.

On Fig. 8, we now plot the swelling ratios, measured at low temperature for the various brushes, as a function of the estimated grafting densities. Such a representation of the data allows for direct comparison of our results with those obtained in previous experimental and theoretical studies. We thus remark that, in a range of density similar to that covered in the work by Plunkett et al. [23], we measure swelling ratios which are much larger than those reported in reference [23], exhibiting moreover a density dependence of opposite sign with respect to that reported by Plunkett et al.. We have, at this stage, no clear explanation such a discrepancy [48]. However, we find that our data compares reasonably well with those of Yim et al. [28], and confirm, on brushes made of longer chains, general trend suggested by these authors. Furthermore, we find a striking qualitative similarity between the decrease of α with 1/d 2 observed in Fig. 8, and the prediction of Mendez et al. [32] for brushes of chains.

FIG. 8: Swelling ratio α as a function of brush density.

C. Compression of swollen brushes

In an attempt to better identify the monomer volume fraction above which the assumption of semi-dilute brushes ceases to hold, we now discuss the validity of the Alexander-de Gennes framework for dense brushes under compression.

To this aim, we compare the low temperature compression curves, F (D)/R, with R the radius of curvature of the surfaces [START_REF]Within the Derjaguin approximation, F (D)/R is proportional to the interaction free energy per unit area[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF], with the following prediction for the free energy per unit area of two compressed brushes [45,[START_REF] Heuberger | [END_REF]52]:

F (D) R = C 7 D 2L -5/4 + 5 D 2L 7/4 -12 (6)
with L the unperturbed thickness of one brush, and the prefactor

C ∝ k B T L/d 3 ,
where k B is the Boltzmann constant and d the distance between grafting sites. For the different brushes studied here, we have computed F (D)/R using for L the brush thickness measured at the onset of repulsion, and for d the value estimated as described above. Doing so, we find a partial agreement between predictions and measurements: the low compression end of the F (D)/R profiles is reasonably well described by Eq. 6, but the prediction is found to systematically underestimate the repulsive forces under strong compression, as illustrated on Fig. 9 [START_REF]Letting L as a free parameter in Eq[END_REF]. Moreover, we observe that the higher the brush density, the narrower the range of agreement between experiments and theory: the calculated F (D)/R fails to reproduce the measurements for values of D/2L lower than ∼ 0.85 for the most dense brushes (Fig. 9a), and for D/2L 0.6 for the brushes of lowest density (Fig. 9b). The existence of a compression range over which agreement is fair may seem surprising, since we have shown in the previous section that α(d) was not well described under the assumption of a semi-dilute layer (ν ∼ 1/2 rather than 3/5). We interprete this apparent paradox as follows. Grafted brushes most likely exhibit, as already discussed in previous studies [23,46,[START_REF] Heuberger | [END_REF], an outer region where the monomer density is lower than that closer to the solid substrate (i.e. the monomer density profile is not step-like but rather decreases smoothly to zero). The above observations thus suggest that during compression of a dense brush, a first step consists in compressing this outer region of lower density, in which the semi-dilute assumption appears to hold, until reaching an average monomer density above which the brush is stiffer than predicted.

These results show that the Alexander-de Gennes model has only a narrow range of validity when dealing with dense brushes under compression. From the measurement of the thickness D f where predictions fail to account for the repulsive forces, one may estimate a monomer volume fraction h dry /D f , which we find in the range 0.2-0.3. We thus obtain, from force-distance measurements, an estimate of the monomer density above which the assumption of semi-dilute brushes ceases to be valid.

D. Wetting, attractive interactions and adhesion

PNIPAM chains bear both hydrophobic (methyl -CH 3 ) and hydrophilic (amide -NH, and carbonyl -C=O) groups along their backbone. As suggested by the temperature dependence of the water contact angle (θ s ) shown on Fig. 3, the outermost layer of a PNIPAM brush is expected to be the seat of local molecular rearrangements, so that, at T ≥ 35 • C, the hydrophobic groups are preferentially exposed at the interface with water. Besides, the range of temperatures over which θ s increases roughly corresponds to the range where brush thicknesses exhibit a steeper decrease with T . This supports the idea that, as T increases and chain dehydration and collapse proceed, a growing fraction of methyl groups is exposed at the surface, until a maximum surface density of CH 3 is reached above 35 • C. Such a picture thus suggests that the attractive interaction between collapsed brushes observed at temperatures above 35 • C (see Fig. 3a and reference [22]) most likely arises from the so-called hydrophobic forces that exist between methyl-terminated surfaces immersed in water [START_REF] Christenson | [END_REF]. Now, the measurements of pull-off forces which do not depend on grafting density (Fig. 5) indicate that adhesion between brushes in contact is, like wetting, mainly controlled by the nature of the chemical groups exposed at the interface. However, we note that while the water contact angle levels off above 35 • C, pull-off forces are found to increase steadily up to 38 • C. This suggests that, once the contact is established between two collapsed PNIPAM layers, some mechanism is at play, which affects interbrush adhesion but not wetting. We believe that this mechanism corresponds to the formation of interchain hydrogen bonds, between -NH and -C=O groups [55] , thus contributing to increase the pull-off force. Such H-bonds are directional and require that the chemical groups involved are favorably oriented along the polymer chains. This should therefore result in a pull-off force that increases with the contact time, due to the dynamics of molecular rearrangements which limits H-bonds formation. We are currently investigating this point, which constitutes a test of the above hypothesis of specific interactions at the interface between two pnipam brushes.

V. CONCLUSIONS

We have performed a systematic study of the effect of grafting density on the collapse, compression and adhesion of PNIPAM brushes. Our study, based on force/distance measurements in a Surface Forces Apparatus, makes use of a plasma activation technique of mica surfaces that allows for covalent grafting of PNIPAM brushes by surface-initiated ATRP. We show that the thermal response of such brushes is strongly affected by the grafting density, in good agreement with recent theoretical predictions [32]: the swelling ratio is found to decrease noticeably at high densities, a result that may have direct implications in e.g. nano/microscale flow control using PNIPAM brushes [14]. Furthermore, the analysis of force-compression curves provides an insight regarding the range of validity of the classical Alexander-de Gennes framework when dealing with dense brushes. Finally, we show that in contrast to the swelling behavior, adhesive forces between brushes do not seem to be affected by the grafting density of the layers. Such an observation leaves completely open the question of how brush thickness may affect cell/substrate adhesive interactions in cell culture applications of PNIPAM [17,19].

VI. AKNOWLEDGEMENTS

We are indebted to Christiane Caroli for her contribution to data interpretation and her precious help with polymer brush theory.

FIG. 1 :

 1 FIG.1: Schematic representation of the experimental configuration. PNIPAM brushes are grafted on curved mica sheets. Interbrush forces during compression/decompression are measured by means of a loading spring. The distance D between the mica surfaces is measured by multiple-beam interferometry. The Fabry-Perot cavity formed by the backsilvered mica sheets is shone with white light. The transmitted light is collected by a microscope objective and sent to a spectrometer for spectral analysis of the fringes of equal chromatic order, from which D is calculated. Experiments are performed over a temperature range of 20-40 • C, which allows to probe thermal response of the brushes when going from the swollen (a) to the collapsed (b) state.

FIG. 2 :

 2 FIG.2: Grafting steps for immobilization of ATRP initiator on mica surfaces.

FIG. 3 :

 3 FIG.3: Static contact angle of water as a function of temperature, measured on PNIPAM brushes of thickness: ( ) 10 nm, (•) 125 nm, (•) 255 nm. Error bars give the resolution on angle determination, comparable to the dispersion of the results observed when measuring at various spots on a surface.

FIG. 4 :

 4 FIG. 4: (a) Force (F)-distance (D) curves measured during approach of two brushes of dry thickness 70 nm, at the temperatures indicated on the figure. Inset: close-up of the F (D) curve at 37.3 • C, showing attractive interaction upon approach. (b) same data as in (a), with forces displayed on a logarithmic scale. (c) Force as a function of time, during separation of two brushes of dry thickness 70 nm, measured at the indicated temperatures. The curves have been horizontally shifted in order to the facilitate comparison.

FIG. 5 :

 5 FIG. 5: Swelling ratio α as a function of temperature, for brushes of thickness: (•) 215 nm, (•) 125 nm, ( ) 70 nm, ( ) 10 nm.

FIG. 7 :

 7 FIG.7: Ratio α as a function of 1/ p h dry (in nm -1/2 ). The straight line is a guide for the eye. Inset: sketch of an Alexander-de Gennes brush made of blobs of size D b ∼ d.

FIG. 9 :

 9 FIG. 9: Normalized force F/R as a function of brush thickness D during compression of brushes of dry thickness: (a) 215 nm, (b) 10 nm. Symbols correspond to measurements, solid lines correspond to F/R calculated with Eq. 6, using 2L = 1550 nm in (a) and 2L = 190 nm in (b).

TABLE I :

 I Molecular parameters of PNIPAM brushes.(a) estimated from the scaling analysis using Eq.2. (b) estimated from Eq. 1 after determination of the grafting density. (c) density calculated using Eq.1, assuming N = 4200, i.e. the average value of chain lengths determined for the thicker brushes.nesses, measured for each brush, allows us to estimate both the grafting densities and the chain length of the grafted layers. These values, reported in TableI, confirm that brushes of different thickness display different grafting densities of chains of roughly constant molecular weight.

	dry	grafting density number of monomer
	thickness (nm) (chain. Å-2 )	units/chain
	215	0.0042 a	4130 b
	155	0.0033 a	3800 b
	125	0.0024 a	4200 b
	70	0.0014 a	4080 b
	50	0.0009 a	4600 b
	10	0.0002 c	4200 c


G Di Natale
email: dinatale@lirmm.fr

M Doulcier

M L Flottes

B Rouzeyre

A Reliable Architecture for Parallel Implementations of the Advanced Encryption Standard

This paper presents an on-line self-test architecture for hardware implementation of the Advanced Encryption Standard (AES). The solution exploits the inherent spatial replications of a parallel architecture for implementing functional redundancy at low cost. We show that the solution is very effective for on-line fault detection while keeping the area overhead very low. Moreover, the architectural modification for on-line test does not weaken the device with respect to side-channel attacks based on power analysis.

Introduction

Standard cryptographic functions such as the Advanced Encryption Standard (AES) are today implemented in a wide range of devices targeting various application domains with security requirements. In addition to the inherent property of these devices, allowing storage and transmission of sensitive information across insecure networks, many applications require high reliability for guarantying a proper digital security. Consequently, as other parts of the system, crypto-cores must be carefully designed in order to provide reliable processing of sensible data. Design for on-line testability of such cores prevents structural failures to cause loss of service and compromise the security.

Fault detection and tolerance schemes for various implementations of cryptographic algorithms have been recently considered. Mainly, two approaches have been developed: based on information redundancy (e.g. the use of codes, [START_REF] Wu | Low Cost Concurrent Error Detection for the Advances Encryption Standard[END_REF] [2] [START_REF] Karri | Concurrent Error Detection Schemes for Fault-Based Side-Channel Cryptanalysis of Symmetric Block Ciphers[END_REF]) or functional redundancy ([3] [4] [START_REF] Bertoni | Error Analysis and Detection Procedures for a Hardware Implementation of the Advanced Encryption Standard[END_REF]).

All the techniques based on codes add some bits to the original data word in order to check its validity. The main issue in these approaches is the prediction of the value of the code on an output, given the input value and the executed operation.

For instance, the prediction of a parity bit is almost straightforward for the ShiftRows, MixColumns and AddRoundKey operations performed in the AES [START_REF]Advanced Encryption Standard (AES)[END_REF] because these transformations are either linear or they just perform some bit permutations (see Section 2 for a detailed description of the AES). Conversely, the prediction of the parity bit is not trivial for the SubBytes operation performed by the so-called S-Boxes. As a consequence, the parity prediction requires larger circuitry. Solutions based on parity codes ( [START_REF] Wu | Low Cost Concurrent Error Detection for the Advances Encryption Standard[END_REF] [START_REF] Di Natale | An On-Line Fault Detection Scheme for SBoxes in Secure Circuits[END_REF]) lead to an overhead of about 20% and high single error detection. However they are not effective in case of multiple faults or single faults that lead to an even number of errors. Other solutions based on the use of more complex codes such as CRC [START_REF] Wu | Low Cost Concurrent Error Detection for the Advances Encryption Standard[END_REF] or systematic nonlinear robust codes [START_REF] Karri | Concurrent Error Detection Schemes for Fault-Based Side-Channel Cryptanalysis of Symmetric Block Ciphers[END_REF] lead to higher fault coverage but at the expense of a significant area overhead (> 60%).

Alternatively, the techniques presented in [START_REF] Karri | Concurrent Error Detection Schemes for Fault-Based Side-Channel Cryptanalysis of Symmetric Block Ciphers[END_REF], [START_REF] Yen | Simple Error Detection Methods for Hardware Implementation of Advanced Encryption Standard[END_REF], and [START_REF] Bertoni | Error Analysis and Detection Procedures for a Hardware Implementation of the Advanced Encryption Standard[END_REF] are based on functional redundancy. They can be used whenever encryption and decryption modules are implemented on the same circuit. Each encoding phase is followed by a decoding and compare phase in order to check if the resulting decoded text matches with the initial plaintext. A similar procedure is employed when the circuit is used for decoding a cipher-text.

Conversely to most of the previously proposed approaches that focus on the S-Boxes only (dominant component, counting up to 75% of the circuit area), we propose a low cost self-test architecture for detecting single and multiple faults in most of the AES hardware. The form of testing is accomplished using duplication and comparison. The main idea is to implement the datapath in such a way that several identical blocks can be defined. With an additional block, online pair wise comparisons of blocks are implemented to check the functionality of the AES hardware. Efficiency and low area overhead are achieved by exploiting the spatial duplication inherent to the parallel implementation of the algorithm. Moreover, since any structural modification on the hardware implementation may jeopardize the digital security, the proposed architecture is also checked with respect to one of the most common attack based on power analysis [START_REF] Kocher | Differential Power Analysis[END_REF].

The paper is organized as follows. Section 2 introduces the basic concepts and the characteristics of the Advanced Encryption Standard algorithm. Section 3 presents the proposed on-line self-test approach, while section 4 discusses the results in terms of area overhead and fault detection capability. Section 5 introduces the problem of side channel attacks based on power analysis, and presents experimental results showing the resistance of the proposed architecture to such an attack. Eventually, Section 6 concludes the paper.

Advanced Encryption Standard

AES [START_REF]Advanced Encryption Standard (AES)[END_REF] is a block cipher adopted as an encryption standard by the U.S. government. AES began immediately to replace the Data Encryption Standard (DES, used since 1976) for the reason that it outperforms in long-term security thanks to, among other things, larger key sizes (128, 192, or 256 key bits). For sake of simplicity, we focus on 128-bit key in the sequel of the paper Another major advantage of AES is its efficient implementation on various platforms. It is suitable for small 8-bit microprocessor platforms, common 32-bit processors, and dedicated hardware implementations that can reach throughput rates in the gigabit range. Several hardware implementations are presented in [START_REF] Zhang | Implementation Approaches for the Advanced Encryption Standard Algorithm[END_REF].

The AES algorithm's internal operations are performed on a two dimensional array of bytes called State. The State consists of 4 rows of 4 bytes. Each byte is denoted by S i,j (0 ≤ i < 4, 0 ≤ j < 4) . The four bytes in each column of the State array form a 32-bit word, with the row number as the index for the four bytes in each word. The initial plain text is a 128-bit block that can be expressed as 16 bytes: in 0 , in 1 , in 2 … in 15 . Encryption and decryption processes are performed on the State, at the end of which the final value is mapped to the output bytes array out 0 , out 1 , out 2 , … out 15 .

The AES is an iterative process composed of 10 rounds. The plain text to cipher is first copied to the State array. After the initial secret key addition (roundkey(0)), the first 9 rounds are identical, with small difference in the 10th round. As illustrated in Figure 1, each of the first 9 rounds consists of 4 transformations: SubBytes, ShiftRows, MixColumns and AddRoundKey. The final round excludes the MixColumns transformation. The encryption scheme in Figure 1 can be inverted to get a straightforward structure for decryption.

SubBytes Transformation

The SubBytes transformation is a non-linear byte substitution that operates independently on each byte of the State using a substitution table (S-Box). This S-Box is constructed by composing two transformations:

1. Take the multiplicative inverse in the finite field GF (2 8 ); the element (00000000) 2 is mapped to itself; 2. Apply the following affine transformation (over GF(2)):

i i i i i i i c b b b b b b           8 mod ) 7 ( 8 mod ) 6 ( 8 mod ) 5 ( 8 mod ) 4 ( ' (1)
for 0 ≤ i < 8, where b i is the i th bit of the byte, and c i is the i th bit of a byte c whose value is fixed and is equal to {01100011}. This transformation can be pre-calculated for each possible input value since it works on a single byte (only 256 values). S-Boxes can be implemented either as a ROM or as combinational logic.

MixColumns Transformation

The MixColumns transformation is performed on the State array column-bycolumn. Each column is considered as a four-term polynomial over GF (2 8 ) and multiplied by a(x) modulo x 4 + 1, where:

a(x) = (00000011) 2 x 3 + (00000001) 2 x 2 + (00000001) 2 x + (00000010) 2 (2)

AddRoundKey Transformation

In AddRoundKey transformation, a roundkey is added to the State array by bitwise XOR operation. Each roundkey consists of 16 bytes generated from the Key Expansion operation described below.

Key Expansion

The key expansion routine, as part of the overall AES algorithm, takes the input secret key of 128 bits and outputs an expanded key of 11*128 bits composed of the input secret key and 10 roundkeys, one for each round. Details of the algorithm for determining the value of each roundkey are given in [START_REF]Advanced Encryption Standard (AES)[END_REF].

Functional redundancy for on-line fault detection

The technique we propose in this paper is designed for all the AES cores (encryption and decryption) that use 16 S-Box repetitions. We do not consider low-area implementations, where there is only one S-Box at the cost of several clock cycles for completing one encryption/decryption round. Typical hardware architecture of the AES with 16 S-Boxes is sketched in Figure 2. Here, sixteen 8-bit registers feed the 16 identical S-Boxes (S). Shiftrows involves only wires for shifting the bytes of the State, it operates on 128 bits. Four identical MixColumns blocks operate on 32 bits each.

S-Boxes

Our goal is to identify a partitioning of the circuit that allows a repetition of identical sub-blocks. These sub-blocks will be compared two-by-two for on-line fault detection thanks to the implementation of an extra sub-block. In the classical architecture depicted in Figure 2, ShiftRows unfortunately prevents such a partitioning since it operates on all the 128 bits.

However by inspecting the AES algorithm, it can be seen that SubBytes and Shiftrows functions can be switched. We thus propose to perform ShiftRows before SubBytes, and even before loading the registers. Figure 3 The same procedure can be applied to the whole circuit and, as a consequence, the datapath can be divided in 4 identical slices that operate on 32 bits each, and that we call RSMA (32-bits Register, 4 S-Boxes, 1 Mixcolumns and 32 xor for the Addroundkey operation). The main idea of the proposed approach is to use one additional RSMA block, and to compare a pair of RSMA blocks at each clock cycle. In particular, at each clock cycle two blocks are fed by the same inputs and the related outputs are compared in order to detect possible faults. Figure 4

ShiftRows RS1 RS2 RS7 RS8 MA 1 RS3 RS4 RS5 RS6 MA 2 RS3 RS1 MA 1 RS7 RS5 ShiftRows RS2 RS8 MA 2 RS6 RS4 RSMA1 RSMA2
= O(1) O(2) O(3) O(4) = K(4) K(3) K(2) K(1)
From the Control Unit um(3) um( 2) um [START_REF] Wu | Low Cost Concurrent Error Detection for the Advances Encryption Standard[END_REF] um( 4)

From the Control Unit lm(0) lm(1)
check( 1) check( 2) check( 3) check( 4)

check(0) u(3) u(2) u(1) u(4) RSMA 4 RSMA 3 RSMA 2 RSMA 1 RSMA 0
UMux (4) 0 1

UMux (3) 0 1

UMux (2) 0 1

UMux (1) The scheduling of the comparisons of a pair of RSMA blocks is a very important issue of the proposed method. One AES encryption lasts 10 clock cycles and there are 5 different configurations. Therefore it's possible to use each of the 5 configurations twice during one encryption. Through the 5 configurations, each RSMA block is compared twice with the left block, once with the right block). Thus, if the 5 configurations are activated twice, each RSMA block is compared 4 times during one encryption. A counter is in charge of the test configuration scheduling.

0 1 0 1 = LMux(4) 0 1 = LMux(3) 0 1 = LMux(2) UMux (1) 1 0 UMux (2) 1 0 UMux (3) 1 0 UMux (4)

Results and Fault Detection Analysis

This section provides results related to the area overhead and the fault detection analysis of the proposed approach. The proposed architecture has been described in VHDL and synthesized using Synopsys Design Compiler [9] using a 130nm

CMOS library by STM [10]. We considered that all the keys used in the AddRoundKey step (see Section 2) are pre-computed and stored in the circuit.

The area of the original circuit is 52961 µm 2 (corresponding to 9660 logic cells)

while the area of the proposed architecture is 71357 µm 2 (corresponding to 13084 logic cells and 34.7% of area overhead). The speed penalty is only 6% due to the insertion of the additional multiplexers.

Concerning the efficiency of the proposed architecture with respect to the fault detection, our functional redundancy strategy differs from the classical Double Modular Redundancy (DMR) scheme.

A classical DMR architecture allows detecting all the faults (single and multiple) that lead to an error (i.e., a difference at the output of one of the duplicated modules). Starting from the moment of appearance of the fault, the fault latency depends on the inputs applied to the circuit, only. In other words, the fault is detected as soon as the input vector can sensitize the fault and propagate it up to the output of the module (i.e., the input of the comparator between the two modules). Anyway, a system based on classical DMR scheme does not deliver faulty responses without noticing it (unless in case of equivalent faults in the two modules).

Our technique is able to detect any single or multiple fault leading to a wrong RSMA output value (as for the classical DMR) but only when the affected RSMA is compared with another one. Conversely to DMR, the dynamic reconfiguration of the modules leads to a comparison of each module twice every 5 clock

Therefore it can happen that the system produces erroneous responses without noticing it even in presence of a single stuck-at.

We question here the probability to get an error on the AES output and to not detect it. This probability can be analyzed by computing the probability P err (f) of not detecting an error on the circuit's outputs while a given fault f affects the circuits. Here, only non-redundant faults are of interest, i.e. we focus on testable faults. For this analysis we focus on single stuck-at faults only, because the number of multiple faults is too high to be analyzed. However, unless the extremely very low probable case of multiple faults composed of 5 equivalent faults in the 5 RSMA modules, all multiple faults are covered by our technique.

With regard to the proposed architecture, P err (f) is the probability that the fault f is activated (i.e. sensitized and propagated in such a way that it leads to an error) during at least one of the 6 clock-cycles during which the faulty RSMA is not compared, and it is not activated during the 4 clock cycles when the RSMA is compared.

Let denote p f the probability of activation of a fault f into an RSMA module, i.e., the probability that for a random input pattern the fault is sensitized and the error is propagated to its output. In the hypothesis to have several distinct functional inputs, we can consider that the device is fed by a random source. In addition, as demonstrated in [START_REF] Hellekalek | Empirical evidence concerning AES[END_REF], the inherent properties of the AES makes that the sequence of input values that are applied to consecutive rounds of the same encryption can be considered as random. Therefore, the probability p f is equal to the ratio of input vectors that test f over the number of possible input vectors. The number of possible input vectors for the RSMA, is 2 32 . Since fault simulation cannot be applied in exhaustive way, we split the problem in two parts. From one side, S-Boxes have 8 input bits only, consequently exhaustive analysis is possible and p f can be obtained for each fault through simulation. Since MixColumns and AddRoundKey are invertible functions, all the errors appearing on the output of the S-Boxes propagate through the functions to the comparator. From the other side, p f of the MixColumns has been calculated thanks to its modular structure involving 4 identical 8-bit inputs sub-functions (2 8 combinations). Finally, the AddRoundKey operations involve only xor operations and are very easily tested.

For both MixColumns and AddRoundKey, each fault is tested with p f =50%. Since Sboxes are bijective functions, randomness properties are still kept at the input of MixColumns and AddRoundKey.

For the proposed architecture, the probability that f is not activated during the clock-cycles of comparison is equal to   4 1 f p  while the probability that f is activated during at least one of the clock-cycles without comparison is equal to

  6 1 1 f p  
. Finally, it comes: In order to calculate the overall error probability, we simulated all the faults in the Sboxes to determine the distribution of probabilities of activation of the faults. An overall of 3860 stuck-at faults are present in our implementation. Basically, we calculated how many faults are activated by one test pattern (p f = 1/256), how many faults are activated by 2 patterns (p f = 2/256), and so on. Figure 6 summarizes, for each probability p f , the number FD(p f ) of faults with that activation probability.

      6 4 1 1 1 ) ( f f err p p f P      (3)
                                                      i Sbox Sbox ERR i i i FD Faults P (4)
Concerning the MixColumn and AddRoundKey, there are #Faults MCAK =750 faults, all of them with p f =50%. Therefore the error probability P ERR-Sbox of

MixColumns and AddRoundKey is calculated based on equation (3).

It comes that the overall probability P ERR of the RSMA is:

% 99 . 9 # # 4 #         RSMA MCAK ERR MCAK Sbox ERR Sbox ERR Faults P Faults P Faults P (5)
The architecture has thus a probability of 90.01% to detect any fault in the RSMA during a single encryption (10 clock cycles).

Let's now analyze the evolution of this probability based on the number of encryptions. When we perform E encryptions, an RSMA block is compared during 4E clock cycles, while it is not compared during 6E clock cycles.

The error probability can therefore be rewritten as follows:

        E f E f err p p f E P        6 4 1 1 1 , (6)
Considering the fault distribution FD given in Figure 6 and the probability of error detection in the MixColumns and AddRoundKey, we can re-calculate the overall error probability P ERR of the RSMA block in function of the number of encryptions (Figure 7). As it can be seen, the error probability slightly increases up to 14% for 5 encryptions, while for higher encryption numbers it tends to 0.

The error probability augmentation from 0 to 5 encryptions can be explained by the fact that at the beginning the probability to exercise the faulty module with random test patterns increases more quickly than the probability to compare the faulty module with a good one while exciting the fault. Since we focus on permanent fault, after a while (i.e. 5 encryptions) the probability to detect the fault (from comparison) is predominant. Namely, for 300 encryptions, the fault detection probability is equal to 99.9% Equivalent curves would be obtained in case of 192-bit or a 256-bit key, since the datapath of the device does not change. The only difference in case of longer keys is a higher number of executed rounds. Therefore, we expect lower number of encryptions for achieving the same error probability.

Differential Power Analysis

An important issue when dealing with cryptographic cores is the sensitivity of the architecture implementation to side-channel attacks, in particular against Differential Power Analysis (DPA). We focused on this attack because among all the known possible attacks, this is one of the cheapest and easiest to perform.

Basically, the DPA attack is a statistical technique relying on the correlation that exists between the current consumed by the device and the processed data.

We now introduce some theoretical issues that allow the reader to understand the principle underlying the DPA attack.

Let's consider the output of a gate whose state depends on both the plain text under ciphering (primary inputs) and the secret key. It is called the target node.

Let's consider now a sequence of input patterns P 0 , P 1 , …, P n that generate the transitions T 1 (P 0 P 1 ), T 2 (P 1 P 2 ), ..., T n (P n-1 P n ) on the circuit primary inputs.

A logic simulation of the circuit while monitoring the target node allows classifying these input transitions in two sets, according to a guess on the key:

 PA, composed by the transitions that make the target node to commute from 0 to 1 and therefore that make the target gate to consume;

 PB, composed by the transitions that do not lead the target gate to participate to the power consumed by the circuit (i.e., transitions from 0 to 0, 1 to 1, and 1 to 0 on the target node).

Figure 8 represents the power consumption of the device when stimulated by numerous input vectors. We assume in this example that the guess on the secret key is correct. In other word, the simulation is performed with the key actually used in the circuit from which power consumptions are collected. Each rectangle represents the total power consumed by the circuit when a new vector is applied to the inputs. In this figure, and just for clarity of explanation, the power consumption is represented by a rectangle corresponding to the average of the consumption over the transition time. In the following, this issue will be redefined in a more precise way. The set of transitions on the circuit inputs is split in the two sets: in the left part there are the PA transitions and the related consumptions while in the right part there are the PB transitions and their corresponding consumptions. A part of the power consumption related to the transitions belonging to PA is due to the power consumed by the target gate (shaded rectangles). Obviously, the commutation from 0 to 1 of non-target nodes also contribute to the power consumption of the circuit, however the input transitions that lead to such commutations are assumed to be evenly distributed to sets PA and PB. If a large number of transitions are considered, mean consumptions related to sets PA and PB are almost equal, except for the contribution of the target node.

In other words, since the two sets are classified in such a way that the set PA always leads to a component of power consumption that is not present in the set PB, the difference between the two mean powers computed from set PA and set PB must show a noticeable difference. During a DPA attack, the target node is chosen in such a way that it depends on a small part of the key only, so that all the key suppositions can be considered. For example, when DPA is conducted against the AES, the target node is chosen as the output of an S-Box that depends on only 8 (of 128) secret key bits. Thus, only 256 key guesses are needed, instead of 2 128 . The process is iterated on every S-Box.

For each key guess, the two sets PA and PB are created according to the results of the logic simulation and the key guess under evaluation. The power mean values are calculated for each set using the simulated power traces of the circuit under attack for each transition. Finally, the differences of the mean values of the two sets are calculated. When the key guess is correct (and only in this case), PA actually includes the input transitions that lead to a transition 0 to 1 on the target node while PB does not include any of these transitions. The difference between the mean power obtained from PA and PB can be observed in this case. On the contrary, when the curves are classed in PA or PB independently from the actual value of the secret key, the two average curves do not present any noticeable difference. The classification process is illustrated in Figure 9 where K x is assumed to be the correct key, the one actually used during ciphering.

Conclusions

In this paper we proposed a low cost architecture for detecting single and multiple faults in the hardware implementation of the Advanced Encryption Standard (AES) during its mission mode. The solution, based on spatial redundancy, reorders the AES algorithm subtasks. This modification does not influence the actual encryption function and it allows the implementation of 4 identical blocks working on 32-bits each. Thanks to this parallel and duplicated architecture, only one additional 32-bits block is added in the circuit leading to 4 tests per encryption cycle for every block. The solution is very effective in terms of fault latency and fault coverage while keeping the area overhead very low (about 34.7%). Finally, it is shown how the proposed implementation reduces the correlation between the power consumption and the processed data, therefore it does not make easier attacks based on Differential Power Analysis.

Figure 1 :

 1 Figure 1: AES Algorithm (encryption) ShiftRows Transformation In this transformation, the bytes in the first row of the State do not change. The second, third, and fourth rows shift cyclically to the left one byte, two bytes, and three bytes, respectively.

Figure 2 :

 2 Figure 2: Typical AES Implementation

  depicts the proposed operation switching, where an RS block represents a Register/S-Boxes pair, whereas MA represents a MixColumns/AddRoundKey pair. In this figure we described the proposed method for a smaller part of the whole circuit. In particular, the figure depicts the reordering on 8 RS blocks and 2 MA blocks instead of the actual 16 and 4 ones, respectively.

Figure 3 :

 3 Figure 3: Switch between Shiftrows and SubBytes

1 0Figure 4 :

 14 Figure 4: AES Architecture with RSMA duplication and comparison

Figure 5 Figure 5 :

 55 Figure 5 represents P err (f) in function of p f . It must be noticed that the hard-to-test faults   0  f p

Figure 6 :

 6 Figure 6: Fault Distribution FD(p f )Assuming that each fault has the same probability to appear in the circuit, the overall error probability P ERR-Sbox of each Sbox is calculated as the weighted average of the values P err (f) according to the distribution FD(p):

Figure 7 :

 7 Figure 7: Error probability w.r.t. the number of encryptions

Figure 8 :

 8 Figure 8: Power consumption after pattern partitioning

3 Figure 9 :

 39 Figure 9: Pattern classification for several key guesses It's important to note that the actual attack is performed by measuring and analyzing the instantaneous power consumptions over the whole transition period, and not using the time-averaged value as shown in Figures 8 and 9.

Figure 10

 10 Figure10shows the appearance of the result of an attack over a period of 1ns and 8 key guesses. Each of the 8 curves represents the DPA curve, i.e. the difference between the mean powers issued from the transitions classified in sets PA and PB, in function of time. The curve that shows the higher peak (bold line in Figure10) corresponds to the correct key guess.

Figure 10 :

 10 Figure 10: Real DPA curve

Figure 11 :

 11 Figure 11: DPA Curves

  details the behavior of a part of the circuit where one extra RSMA block has been added. In this figure,

	LMux(2), LMux(3) and LMux(4) are multiplexers with an additional output
	that is asserted whenever the two inputs are equal (i.e., a multiplexer with a
	comparator).
	Table 1 details the signals controlled and observed by the control unit. For
	instance, when RSMA4 and RSMA3 work together, the UMux(3) let the input
	I(4)	I(3)	I(2)		I(1)
	Reg	Reg	Reg	Reg	Reg
	Mix Columns	Mix Columns	Mix Columns	Mix Columns	Mix Columns

I(4) go into the RSMA3. Among the five signals coming from the comparators, only one at a time is considered by the control unit. For example, in the above case, the check(4) signal is verified, i.e., the two related RSMA blocks are checked.

Table 1 :

 1 Signals controlled and observed by the Control Unit

	Compared	Um	Lm	To check
	RSMA4, RSMA3	1000	11	check(4)
	RSMA3, RSMA2	1100	01	check(3)
	RSMA2, RSMA1	1110	00	check(2)
	RSMA1, RSMA0	1111	00	check(1)
	RSMA0, RSMA4	0000	11	check(0)