Adrien Cutivet

Flavien Cozette

Meriem Bouchilaoun

Ahmed Chakroun

Osvaldo Arenas

Marie Lesecq

Jean-Claude De Jaeger

Abdelatif Jaouad

François Boone

Hassan Maher

Characterization of Dynamic Self-Heating in GaN HEMTs Using Gate Resistance Measurement

Keywords: GaN, temperature measurement, dynamic selfheating, gate metal thermometry, HEMT

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

I. INTRODUCTION

EMTs based on GaN are attractive to RF and power electronics designers due to the outstanding GaN properties (wide band gap, high saturation velocity and high current density) allowing tremendous performance in terms of power density and cut-off frequency. Cost-wise, GaN based heterostructure on silicon substrate is of major interest for several telecommunication and power conversion markets. However, this technology suffers from inherent limitation in terms of heat dissipation affecting devices overall performance and reliability. Extracting accurately the device temperature is thus of prime importance for modeling electrical characteristics and predicting device lifetime.

Literature reports on an electrical method (G-RTD) using gate resistance temperature dependence to determine device temperature in a straightforward manner [START_REF] Estreich | A DC technique for determining GaAs MESFET thermal resistance[END_REF][START_REF] Fattorini | Channel temperature estimation in GaAs FET devices[END_REF][START_REF] Schwitter | Study of Gate Junction Temperature in GaAs pHEMTs Using Gate Metal Resistance Thermometry[END_REF]. A comparison of this method with micro-Raman spectroscopy has proved this method to be reliable with a net uncertainty of ±5% [START_REF] Paine | Measurement of Temperature in GaN HEMTs by Gate End-to-End Resistance[END_REF]. G-RTD method is however restricted to transistors featuring This paragraph of the first footnote will contain the date on which you submitted your paper for review. This work was partly supported by Fonds de Recherches du Québec -Nature et Technologies (FRQNT), Conseil de recherches en sciences naturelles et en Génie du Canada (CRSNG), Direction Générale de l'Armement (DGA) and the labex GaNeX.

A. Cutivet, F. Cozette, M. Bouchilaoun, A. Chakroun, O. Arenas, A. Jaouad, F. Boone and H. Maher are with Université de Sherbrooke, Sherbrooke QC J1K2R1 Canada (e-mail: adrien.cutivet@usherbrooke.ca).

F. Cozette, M. Lesecq and J.-C. De Jaeger are with Univ. Lille, CNRS, Centrale Lille, ISEN, Univ. Valenciennes, UMR 8520 -IEMN, F-59000 Lille, France (e-mail: flavien.cozette@ed.univ-lille1.fr).

double-ended gate where separated pads enable direct measurement of gate resistance. For transistors with a single gate access, the use of a dedicated metal sensor upon a passivation layer has been successfully demonstrated [START_REF] Arenas | Electrothermal Mapping of AlGaN/GaN HEMTs Using Microresistance Thermometer Detectors[END_REF].

While many other studies detail successful approaches to measure on-wafer HEMT steady-state temperature, fewer studies are related on the matter of self-heating for timedependent dissipated power. However, it is crucial to model the temperature rise for RF applications with time-dependent power level (such as radars and data transmission using amplitude modulation) and power electronics where transistors are used in switching operations.

In this frame, Micro-Raman spectroscopy has demonstrated the possibility to provide excellent accuracy with an approximate 1 µm spatial resolution and time resolutions down to 10 ns [START_REF] Kuball | Time-Resolved Temperature Measurement of AlGaN/GaN Electronic Devices Using Micro-Raman Spectroscopy[END_REF][START_REF] Riedel | Nanosecond Timescale Thermal Dynamics of AlGaN/GaN Electronic Devices[END_REF][START_REF] Bagnall | Transient thermal dynamics of GaN HEMTs[END_REF]. Thermoreflectance imaging has been proved to provide useful information in a complementary way with Micro-Raman [START_REF] Martin-Horcajo | Transient Thermoreflectance for Gate Temperature Assessment in Pulse Operated GaN-Based HEMTs[END_REF]. Regarding methods based on electrical measurement, [START_REF] Mustafa | Influence of parasitic effects of the "3ω" measurement setup to improve the determination of GaN HEMTs thermal impedance[END_REF] reports on a method based on 3ω measurement using HEMT ON-resistance to extract thermal dynamic characteristic, but only in the linear operation mode of 𝐼 characteristic (ohmic area). In this paper, a new and robust electrical method for doubleended gate HEMTs is introduced, it makes use of gate resistance frequency measurement to monitor dynamic temperature variations at any bias point. This new method enables a complete and accurate thermal impedance determination suitable for further modeling and simulation works useful for power circuit designers.

II. EXPERIMENTAL PROCEDURE

A. Device Under Test

The method is carried out on a coplanar common-gate AlGaN/GaN MOS-HEMT fabricated on commercial epiwafers (EPIGaN) grown by Metal-Organic Chemical Vapor Deposition (MOCVD) on Si substrate. The epitaxial structure consists of a 1.8 µm Al0.18Ga0.82N back barrier layer, a 150 nm undoped GaN channel layer, a thin 4 nm Al0.45Ga0.55N barrier layer and a 50 nm SiN in-situ dielectric layer. A sheet carrier density of 1.08.10 13 cm -2 and an electron mobility of 1810 cm 2 .V -1 •s -1 are determined by Hall Effect measurement. The 50 nm SiN dielectric layer is dry-etched before e-beam evaporation of the ohmic contact metal stack (Ti/Al/Ni/Au). After annealing, a nitrogen ion implantation isolation is performed and the contact resistance determined by TLM [START_REF] Chakroun | Optimized Pre-Treatment Process for MOS-GaN Devices Passivation[END_REF]. A Ni/Au (40/600 nm) metal deposition is performed featuring photolithography and e-beam evaporation to make a 1.5 µm long meander gate electrode with 0.5 µm and 2.0 µm field-plate extensions toward source and drain respectively. The deposited Au resistivity is evaluated to be 9.1×10 -8 Ω.m. Gate leakage current is lower than 10 nA for all biases used in this study thanks to the high quality of the gate oxide layer. Gate-todrain and gate-to-source distances are 6.0 µm and 1.5 µm respectively for a total gate width of 318 µm. A SEM image of the device is shown in figure 1.

B. Measurement Setup

The setup is described in figure 1. A DC current source injects a current (𝐼 ) along the gate to measure gate resistance (𝑅 ). Gate-source voltage is controlled with a bias while a Keithley 6221 source applies a DC and AC external current (𝐼 𝑥 ) with an external resistance (𝑅 𝑥 ) permitting to control . Considering a static drain current 𝐼 and a drain-source voltage with alternative variations 𝐼 𝑓 and 𝑓 , the HEMT dissipated power (𝑃) in frequency domain is as follows:

𝑃 𝑓 = 𝑓 . 𝐼 + . 𝐼 𝑓 (1)
𝑃 𝑓 = . 𝑓 . 𝐼 𝑓

A SR850 DSP lock-in amplifier from Stanford Research Systems permits the measurement of ′ 𝑓 and 𝐼 𝑓 for a frequency range of 1 mHz to 102.4 kHz. By considering the following assumptions:

 The device temperature (𝑇) is assumed to be the highest value along the gate length axis averaged along the gate width axis.  The gate is isotherm and its temperature is equal to 𝑇 [START_REF] Sodan | Experimental Benchmarking of Electrical Methods and µ-Raman Spectroscopy for Channel Temperature Detection in AlGaN/GaN HEMTs[END_REF].  The dependence of 𝑅 with temperature is linear with respect to 𝑇 which in turn is linear with respect to 𝑃 for the range of variation.  Values of 𝐼 are chosen sufficiently small so that ′ magnitude is low enough to consider 𝐼 constant along the gate.

The device thermal impulse response (𝑍 𝑇 ), also known as the thermal impedance, is introduced and yields:

𝑍 𝑇 𝑓 = 𝑇 𝑓 𝑃 𝑓 ⁄ = 𝑅 𝑓 ( 𝜕𝑅 𝜕𝑇 . 𝑃 𝑓 ) ⁄ (3)
The device thermal step response (𝑍 𝑇 ′ ), also known as transient thermal impedance, can be approximated if supposed constant over the range of temperature investigated by a simple variable change [START_REF] Vermeersch | A Shortcut to Inverse Fourier Transforms: Approximate Reconstruction of Transient Heating Curves from Sparse Frequency Domain Data[END_REF]:

𝑍 𝑇 ′ 𝑡 = |𝑍 𝑇 𝜋𝑓 → 𝑡 ⁄ | (4)
While gate leakage current (~10 -8 A) can be neglected in comparison with 𝐼 (~10 -4 A), a displacement current arising from gate-to-drain electrical coupling may account for a significant parasitic contribution on ′ 𝑓 such as:

′ 𝑓 = 𝑅 𝑓 . 𝐼 + 𝑗𝛼. 𝑓. 𝑓 ( 5
)
Where ∝ is a real constant accounting for the total gate-todrain capacitance and distributed gate resistance. A deembedding technique was introduced in [START_REF] Paine | Measurement of Temperature in GaN HEMTs by Gate End-to-End Resistance[END_REF] to cancel the effect of a parasitic gate conduction current. A similar approach is hereby introduced to cancel the effect of this displacement current using complex values:

𝑅 𝑓 = ( ′ 𝑟 + 𝑓 - ′ 𝑟 -𝑓 ) ( 𝐼 ) ⁄ ( 6
)
Where ′ 𝑟 + 𝑓 and ′ 𝑟 -𝑓 are ′ 𝑓 measurement obtained with a positive and negative value of 𝐼 respectively.

III. MEASUREMENT AND RESULTS

A. Steady-state results

The dependence of 𝑅 on 𝑇 is investigated through on-wafer measurement using 4-wire sensing and a thermally controlled chuck while the device is unbiased. The gate access pad resistance is estimated to be less than 1 ohm and is therefore neglected. Figure 2.a shows the results for a chuck temperature ranging from 25 to 100°C A linear dependency is acknowledged with 𝜕𝑅 𝜕𝑇 = . 𝑚Ω. 𝐾 -and a correlation factor of 0.9996.

B. Extensive results and analysis

To assess the reproducibility and validity of this thermal impedance extraction method, measurements are conducted under different experimental conditions described in table 1. Figure 3 shows |𝑍 𝑇 𝑓 | extraction from 1 Hz to 100 kHz. For standard and 1-4 conditions, corresponding to the transistor saturation region, (eq.1), (eq.3) and (eq.6) are used. Condition 5 consists in using the transistor in ohmic region. In that case the second harmonic term 𝑃 𝑓 is typically higher than 𝑃 𝑓 and therefore (eq.2) and (eq.3) are used. As expected, all the experimental condition variations have no effect on the extraction giving the same values in a ±3% range, except for the results obtained in ohmic region for which |𝑍 𝑇 𝑓 | values are typically inferior. This last result is attributed to the fact that a lower proportion of electric field is located near the drain-side gate edge at low which enables a better local heat dissipation [START_REF] Choi | The Impact of Bias Conditions on Self-Heating in AlGaN/GaN HEMTs[END_REF].

The results are coherent with literature [START_REF] Mustafa | Influence of parasitic effects of the "3ω" measurement setup to improve the determination of GaN HEMTs thermal impedance[END_REF] with |𝑍 𝑇 𝑓 | dropping down significantly for frequencies above 500 Hz and maintaining a quasi-constant value for frequencies below. |𝑍 𝑇 𝑓 | is 40 K.W -1 at 100 kHz, meaning that a higher frequency is needed to attain thermal impedance cut-off frequency, as stated in literature [START_REF] Mustafa | Influence of parasitic effects of the "3ω" measurement setup to improve the determination of GaN HEMTs thermal impedance[END_REF]. A slight increase of |𝑍 𝑇 𝑓 | is observed below 10 Hz, attributed to an imperfect contact of the sample backside to an ideal thermal mass. The values obtained at the lowest frequencies are coherent with 𝜕𝑇 𝜕𝑃 ⁄ steady-state value evaluated from III.A results (≈90 K.W -1 ). Furthermore, the phase of 𝑍 𝑇 𝑓 is extracted using (eq.3) and (eq.6), and is shown in figure 3 inset for frequencies above 1 kHz. The phase value is observed to decrease to attain -° at 100 kHz as it is expected to converge toward a -° value at cut-off frequency where the temperature variation is completely out of phase with the dissipated power variation.

In a final step, the device transient thermal impedance 𝑍 𝑇 ′ is extracted using (eq.4). The results are shown in figure 3 from 1.6 µs to 160 ms. At 1.6 µs, temperature has already risen to nearly half of the steady-state value which is coherent with literature to the authors knowledge [START_REF] Bertoluzza | Three-dimensional finiteelement thermal simulation of GaN-based HEMTs[END_REF][START_REF] Jardel | An Electrothermal Model for AlGaN/GaN Power HEMTs Including Trapping Effects to Improve Large-Signal Simulation Results on High VSWR[END_REF].

Conditions Parameters Standard = . , = . , 𝑓 = . and 𝐼 = . 𝑚𝐴 1 Lower 𝑅 measurement current: 𝐼 = . 𝑚𝐴 2 Different bias: = . , = .

3

Lower signal amplitude: 𝑓 = . 4

Another HEMT from the same process batch 5

Ohmic region: = , 𝑓 = .

Tab. 1. Sets of measurement parameters used for 𝑍 𝑇 𝑓 extraction.

IV. CONCLUSION

This paper introduces an original and straightforward thermal impedance extraction method relying entirely on electrical characterization for transistors suited for direct gate resistance measurement. The theory and the experimental setup are described and tested on GaN HEMT devices. Thermal impedance extraction is conducted under various experimental conditions to assess validity and overall reliability of this new method. The results are shown to be independent of power signal amplitude, gate current resistance measurement and gate bias conditions as well as identical for another similar HEMT. This highlights the accuracy of this technique as well as its adaptability and potential for application on transistors from other technologies. Furthermore, an experimental setup capable of a maximum operation frequency up to 20 MHz would allow extracting the transient thermal impedance with a time resolution under 10 ns, which is equivalent to state of the art Micro-Raman spectroscopy performance.

Fig. 1 .

 1 Fig. 1. SEM image of a common-gate GaN HEMT under test with configuration of voltage and current sources. Annotated contacts: G (Gate access n°1), G' (Gate access n°2), D (Drain access) and S (Source access).

Figure 2 .

 2 Fig. 2. (a) Gate resistance versus temperature obtained with on-wafer 4-wire sensing and thermally controlled chuck. (b) Device under test 𝐼 characteristic and associated steady-state temperatures extracted for and ranging from -0.5 to 1.5 V by steps of 0.5 V and 0 to 10 V respectively.

Fig. 3 .

 3 Fig. 3. HEMT thermal impedance in frequency domain and transient thermal impedance in time domain obtained under various experimental conditions. Inset: phase extraction of thermal impedance.

  Characterization of dynamic self-heating in GaN HEMTs using gate resistance measurement Adrien Cutivet, Flavien Cozette, Meriem Bouchilaoun, Ahmed Chakroun, Osvaldo Arenas, Marie Lesecq, Jean-Claude De Jaeger, Abdelatif Jaouad, François Boone, and Hassan Maher

H measurement is 0.35 Ω.mm. The 50 nm SiN dielectric layer is dry-etched in the gate electrode area and a 5 nm SiOx gate dielectric is deposited by PECVD


Member, IEEE Rémi Cogranne
email: remi.cogranne@utt.fr

Member, IEEE Vahid Sedighi
email: vsedigh1@binghamton.edu

IEEE Jessica Fridrich Fellow

PRACTICAL STRATEGIES FOR CONTENT-ADAPTIVE BATCH STEGANOGRAPHY AND POOLED STEGANALYSIS

Keywords: Batch steganography, pooled steganalysis, adaptive embedding, optimal detection. 1

This paper investigates practical strategies for distributing payload across images with content-adaptive steganography and for pooling outputs of a single-image detector for steganalysis. Adopting a statistical model for the detector's output, the steganographer minimizes the power of the most powerful detector of an omniscient Warden, while the Warden, informed by the payload spreading strategy, detects with the likelihood ratio test in the form of a matched filter. Experimental results with state-of-the-art content-adaptive additive embedding schemes and rich models are included to show the relevance of the results.

steganography in digital images, arguably the most popular and most studied cover objects.

Recent years have seen a remarkable progress in steganography and steganalysis. Syndrome trellis codes [START_REF] Filler | Minimizing additive distortion in steganography using syndrome-trellis codes[END_REF] gave birth to numerous modern, content-adaptive data hiding algorithms [START_REF] Penný | Using high-dimensional image models to perform highly undetectable steganography[END_REF][START_REF] Holub | Universal distortion design for steganography in an arbitrary domain[END_REF][START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF]. The science of detection of hidden data called steganalysis has also remarkably improved over the past few years with the introduction of rich media models [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF][START_REF] Denemark | Selection-channel-aware rich model for steganalysis of digital images[END_REF] and new machine learning tools [START_REF] Kodovský | Ensemble classifiers for steganalysis of digital media[END_REF][START_REF] Cogranne | Is ensemble classifiers needed for steganalysis in highdimensional feature spaces?[END_REF].

In batch steganography, the payload is spread over multiple covers while pooled steganalysis jointly analyzes multiple objects for detection. Introduced in [START_REF] Ker | Batch steganography and pooled steganalysis[END_REF], these two topics are among the most pressing open problems today [START_REF] Ker | Moving steganography and steganalysis from the laboratory into the real world[END_REF]. Payload spreading strategies for non-adaptive hiding schemes and targeted detectors were studied in [START_REF] Ker | Batch steganography and the threshold game[END_REF][START_REF] Ker | Perturbations hiding and the batch steganography problem[END_REF][START_REF] Ker | Batch steganography in the real world[END_REF] with the conclusion that the payload should either be concentrated in as few covers as possible or spread evenly.

Pooled steganalysis was studied in [START_REF] Ker | Perturbations hiding and the batch steganography problem[END_REF] under the assumption that the Warden knows the payload sizes but not their assignment to individual images. In a different setup, the authors of [START_REF] Ker | The steganographer is the outlier: Realistic large-scale steganalysis[END_REF] used a local outlier factor to identify the steganographer in the wild. The topic of learning optimal pooling functions appeared in [START_REF] Penny | Optimizing pooling function for pooled steganalysis[END_REF], and the problem of sequential detection was investigated in [START_REF] Cogranne | A sequential method for online steganalysis[END_REF].

In this paper, we work in spatial domain because of the availability of good and tractable models that recently lead to important advances in steganography and steganalysis [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF][START_REF] Ker | Rethinking optimal embedding[END_REF]. By adopting a statistical model of Warden's detector, the problems of batch steganography and pooled steganalysis are formalized within a unified theoretical framework that allows us to design and study realistic payload spreading strategies that can be viewed as approximations of optimal but practically infeasible approaches.

In Section 2, we adopt a model of Warden's detector and formalize optimal pooled steganalysis in Section 3 and optimal content-adaptive batch steganography in Section 4. Several practical alternatives for optimal batch embedding strategies are introduced in Section 4. Experiments with contentadaptive embedding schemes and steganalysis detectors built with rich models appear in Section 5. The paper is concluded in Section 6.

PROBLEM STATEMENT

Abstracting away from specific details, we assume that the actors are aware of the state-of-the-art in the corresponding fields, namely that the steganographer uses content-adaptive embedding and the steganalyst a classifier trained on possibly high-dimensional features from R d . For now, we will disregard the details of exactly how the classifier is trained and simply assume that, when applied to a feature from a single image, the classifier θ returns a scalar output: θ : R d → R that is subsequently compared with a threshold to reach the decision.

Denoting the ith image with

x (i) = (x (i)
kl ) and its representation in the feature space as z (i) ∈ R d , the steganographers generate a source of I images x (i) , i = 1, . . . , I, that are either all cover or all stego embedded with payloads R i . The number of images is assumed to be arbitrarily large. On the other hand, the Warden inspects a set of B images x (i) , i = 1, . . . , B, with a classifier trained with a high-dimensional feature set. Due to the way the features are built and the fact that the test statistic is a projection of high-dimensional features, the Warden's single-image detector output, denoted θ (i) = θ(z (i) ), is a sample from a Gaussian distribution N (0, σ 2 ) [START_REF] Cogranne | A sequential method for online steganalysis[END_REF][START_REF] Cogranne | Modelling and extending the ensemble classifier for steganalysis of digital images using hypothesis testing theory[END_REF].

Given B ≥ 1 images x (i) , i = 1, . . . , B, in Warden's pooling bag, the Warden faces the following hypothesis testing problem:

H 0 : θ (i) ∼ N (0, σ 2 ), ∀i (1)
H 1 : θ (i) ∼ N µ i (R i ), σ 2 , ∀i, (2)
where µ i (R i ) is the expected shift of the detection statistic (over messages) when embedding payload size R i in x (i) . In (2), we adopted the so-called shift hypothesis [START_REF] Ker | Batch steganography and pooled steganalysis[END_REF], meaning that the embedding affects only the mean of the detector output but not its distribution. Note that we allow this shift to be a different function of the payload size for each image, hence the subscript i of µ. In addition, because we did not impose any assumption on the steganographers' payload spreading strategy, R i can also be different for each image.

OPTIMAL POOLED STEGANALYSIS AND PRACTICAL APPROXIMATIONS

From the formulation of the hypotheses in (2), the Warden's problem consists of maximizing the detection accuracy given a set of B inspected images

x (i) , i = 1, . . . , B.
The case in which the Warden does not have any information about the payload strategy used by the steganographers, R = (R 1 , . . . , R B ), has been addressed in [START_REF] Cogranne | A sequential method for online steganalysis[END_REF]. The conclusion of this prior work is that the steganalyst should simply average all outputs of the single-image detector θ (i) . In this paper, we consider an "omniscient" Warden who knows the spreading strategy R = (R 1 , . . . , R B ) and the expectations µ i (R i ). In this case, (2) reduces to a test between simple hypotheses for which the Most Powerful (MP) test that maximizes the detection power for a given false-alarm probability is the Likelihood Ratio (LR) test, the matched filter [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]:

Λ B = 1 σ µ B 2 B i=1 µ i (R i )θ (i) , (3)
where

µ B = (µ 1 (R 1 ), . . . , µ B (R B )
) denotes the outputs of Warden's detector on all images embedded with payloads R and x 2 is the Euclidean norm of x. The term

(σ µ B 2 ) -1
is just a normalization factor. From Eq. ( 3) and from the distribution of the detector's output (2), it is immediate that Λ B follows

Λ B ∼ N (0, 1) under H 0 N √ B µ B 2 , 1 under H 1 . (4)
Note that the power of this LRT only depends on µ B 2 .

Since in practice the steganalyst does not know the expectations µ i (R i ), they need to be estimated. In this paper, we chose a polynomial regression of second degree in three variables: the payload R i , the change rate r i caused by embedding payload R i , and i , the deflection coefficient based on MiPOD's [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF] cover model:

i = Np kl=1 (r (i) kl ) 2 (σ (i) kl ) -4 , (5)
where (σ

(i) kl ) 2 is the variance of pixel x (i)
kl , estimated from its neighborhood as in [4, Section V], and r

(i) kl is the change rate of pixel x (i)
kl . The regressor coefficients were estimated from the training part of the image database (images available to the Warden) using the least square estimator. Replacing the detector output expectation in the optimal LR under H 1 by the estimates μi gives us the "estimated LR":

Λ B = 1 σ μB 2 B i=1 μi θ (i) . (6)

BATCH EMBEDDING BY MINIMIZING STATISTICAL DETECTABILITY (AND EMPIRICAL SOLUTIONS)

Once the steganographers have chosen an embedding scheme and have been granted access to a source of digital images x (i) , i = 1, . . . , I, their task is to select a payload spreading strategy R = (R 1 , . . . , R B ) to minimize detectability. An omniscient steganographer with perfect knowledge of the expectation of Warden's detector output µ i (R i ) for all i = 1, . . . , I can find optimal combination of payloads, R , by minimizing the statistical detectability against the MP adversary or the expectation of the LR Λ B (4), µ B 2 :

R = arg min R B i=1 µ i (R i ) 2 , (7)
s.t. R = B i=1 R i , (8)
where Eq. ( 8) is the steganographer's payload constraint.

Evaluating for each image the expectation of the detector's output µ i (R i ) can be very cumbersome. Additionally, the steganographers will likely be ignorant of the detector used by the Warden and especially how it has been trained. In this paper, we thus consider the following alternatives to optimal spreading that are feasible to implement in practice: The steganographer adopts a cover model and spreads payload over images so that each image from the bag contributes with the same value as the KL divergence (deflection coefficient) i (5).

3.

Trust the distortion [Distortion Limited Sender (DiLS)]. The steganographer spreads payload over images so that each image from the bag contributes with the same value of distortion.

Among the above strategies, the only one that minimizes a distortion (detectability) over all pixels from all images is the IMS. Here, the unknown expectation of the detectors' output is replaced with the distortion function on which the (adaptive) embedding scheme is based. The DeLS strategy is the only one that operates with statistical detectability. Ideally, the steganographer should minimize the total KL divergence B i=1 i . However, this would be rather expensive to implement especially for large B and also computationally infeasible to test (see below). Similar complexity issues arise when minimizing the sum of distortions over all B images. Spreading the payload by finding a fixed value of the deflection coefficient (distortion) for all B images that communicates the required payload can be implemented much more efficiently.

It is important to note that experimental evaluation of each strategy in practice would be very time consuming as the Warden needs to determine the vector of payloads R and extract features for each bag, which makes the training very expensive. We hence adopt two more simplifying assumptions that will allow us to execute the experiments with a significantly lower computational burden. We will assume that over time the steganographers maintain an average communicated payload R. Having access to a large number of images I, for each spreading strategy the steganographer determines the payload that would be embedded in the ith image if all I B images were in the embedding bag. This payload is essentially a tag attached to each image with the tag value depending on the spreading strategy. For large enough bags B, the actual payloads determined for each bag will be approximately the same as the "asymptotic" tags. This will allow us to execute the experiments by pre-computing the feature vectors for all images in the training (and testing) set and then performing the pooled steganalysis by randomly drawing B features from the testing set. Note that under this simplification, the embedding is "bagless" as the tags are determined only by R and the spreading strategy.

NUMERICAL RESULTS

We use three additive state-of-the-art embedding schemes: S-UNIWARD [START_REF] Holub | Universal distortion design for steganography in an arbitrary domain[END_REF], HILL [START_REF] Li | A new cost function for spatial image steganography[END_REF], and MiPOD [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF]. All three algorithms were modified to avoid making embedding changes in saturated pixels (grayscale with value 255) to prevent weak- ening their security as documented in [START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF][START_REF] Sedighi | Effect of saturated pixels on security of steganography schemes for digital images[END_REF]. The image source in all experiments is BOSSbase 1.01 [START_REF] Bas | Break our steganography system -the ins and outs of organizing BOSS[END_REF] split 10 times randomly into 5,000 images for training (Warden's database) and 5,000 for testing (Alice's database), unless mentioned otherwise. All experiments were conducted with a single-image LSMR-based linear classifier described in [START_REF] Cogranne | Is ensemble classifiers needed for steganalysis in highdimensional feature spaces?[END_REF] trained on SRM features [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF]. Since the Warden is omniscient, she trains on the same stego source produced by Alice, which depends on the embedding scheme, average secret communication rate R, and spreading strategy.

Due to space limitations, we only include the most interesting results regarding the proposed batch steganography strategies and pooling methods. Table 1 shows the accuracy of three pooling methods as a function of the bag size B used by Eve for steganalysis. Note that the DiLS is by far the worst, though for larger bag size the difference becomes rather small, while the DeLS performs almost as well as the IMS. The ranking of steganographic algorithms, batch strategies, and pooling methods also almost always remain consistent over the bag size, which is in agreement with the proposed statistical model presented in Sections 2-3. Regarding the accuracy of the pooling methods, as one can expect the optimal LR test always has the best performance while the proposed "estimated LR" reaches a comparable performance only for DiLS. For the most secure DeLS and IMS batch strategies, the performance of the estimated LR drops significantly in comparison with the optimal test, indicating thus a possible detection improvement with a more accurate predictor of the detector output. The average test proposed in [START_REF] Cogranne | A sequential method for online steganalysis[END_REF] for an ignorant Warden does not know the batch strategy and performs much worse across all batch strategies.

The P E for bag size B = 1 in the table corresponds to the performance of the single-image detector. It should be contrasted with the single-image detector trained to detect the most commonly considered uniform spreading strategy or payload-limited sender (PLS) that embeds the mean payload R = 0.2 bpp in every image. The detector of the PLS achieves P E ≈ 0.35 for both HILL and MiPOD while the DeLS and IMS senders are detected by the single-image detector at P E ≈ 0.46, which testifies to the rather large gain in security due to payload spreading. It is also worth pointing out that, while MiPOD has been evaluated as the most secure additive embedding scheme [START_REF] Cogranne | Is ensemble classifiers needed for steganalysis in highdimensional feature spaces?[END_REF][START_REF] Sedighi | Contentadaptive steganography by minimizing statistical detectability[END_REF][START_REF] Sedighi | Effect of saturated pixels on security of steganography schemes for digital images[END_REF], it appears slightly less secure than HILL in the batch mode as HILL seems to spread the payload more efficiently across multiple images.

The ROC curves in Fig. 1 show the probability of correct detection as a function of the false-alarm probability for S-UNIWARD [START_REF] Holub | Universal distortion design for steganography in an arbitrary domain[END_REF] with mean payload R = 0.1 bpp and pooling bag size B = 100. We note that, in agreement with the statistical model described in Sec. 2, the ranking of batch strategies and pooling methods is consistent for almost all false-alarm probabilities. Also note the small differences between the accuracy of pooling methods for DiLS (lightest curves) compared to other batch strategies and how the proposed "estimated LR" fails to match the performance of the optimal LR. Finally, note how close the IMS and DeLS spreading strategies are in terms of security. This, of course, depends on the statistical model of the cover Alice uses and on the distortion function on which the embedding is based.

Last, but not least, Fig. 2 provides a useful insight about the batch strategies. This figure shows the empirical distribution of payloads among all 10,000 BOSSbase images for all batch strategies for S-UNIWARD at mean payload R = 0.2 bpp. We note that the IMS and DeLS strategies are similar in the sense that both tend to put small payloads in the vast majority of images and allocate most of the hidden data in a limited number of images for which the detection is the most challenging, such as very textured images. On the other hand, the DiLS often embeds a payload that is close to the mean payload R and thus adapts to image content to a much smaller degree. Qualitatively similar results have been observed for other embedding schemes and all tested mean payloads.

CONCLUSIONS

In this paper, we study the problem of content-adaptive batch steganography and pooled steganalysis for an omniscient Warden aware of the payload-spreading strategy and equipped with a single-image detector trained as a classifier between the cover and stego sources. By adopting a statistical model for the output of the single-image detector, we derive the optimal pooling function as a likelihood ratio in the form of a matched filter and its approximations realizable in practice. We also consider several batch strategies that can be efficiently implemented in practice and test them, together with pooling strategies on state-of-the-art steganography, drawing numerous interesting conclusions for practitioners of steganography as well as the steganalyst.

1 . 2 .

 12 Trust the steganography [Image Merging Sender (IMS)]. The steganographer merges all B images into one and lets the embedding algorithm spread the payload. Trust the cover model [Detectability Limited Sender (DeLS)].

Fig. 1 .Fig. 2 .

 12 Fig.1. Batch strategies and pooling methods evaluated using ROC curves for S-UNIWARD at R = 0.2 bpp.

Table 1 .

 1 Evolution of minimal total probability of error under equal prior P E as a function of pooling bag size B, R = 0.2 bpp.

	Embedding scheme Batch strategy Pooling method Bag size: 1	2	5	10	20	50	100	200
			Average	0.4224	0.3930 0.3372 0.2811 0.2137 0.1134 0.0487 0.0175
		DiLS	Optimal LR (3)	0.4224	0.3778 0.2936 0.2224 0.1513 0.0720 0.0380 0.0112
			Estimated LR (6)	0.4224	0.3852 0.3128 0.2431 0.1685 0.0725 0.0210 0.0120
			Average	0.4521	0.4368 0.4077 0.3739 0.3291 0.2525 0.1782 0.1039
	HILL	DeLS	Optimal LR (3)	0.4521	0.4272 0.3782 0.3304 0.2812 0.2043 0.1452 0.0903
			Estimated LR (6)	0.4521	0.4352 0.4030 0.3666 0.3225 0.2446 0.1737 0.1005
			Average	0.4663	0.4506 0.4277 0.4026 0.3663 0.3046 0.2369 0.1594
		IMS	Optimal LR (3)	0.4663	0.4444 0.3988 0.3515 0.2930 0.2150 0.1528 0.1014
			Estimated LR (6)	0.4663	0.4483 0.4208 0.3923 0.3516 0.2838 0.2171 0.1408
			Average	0.4444	0.4260 0.3924 0.3528 0.3004 0.2138 0.1375 0.0665
		DeLS/DiLS	Optimal LR (3)	0.4444	0.4178 0.3638 0.3129 0.2581 0.1852 0.1334 0.0586
	MiPOD		Estimated LR (6) Average	0.4444 0.4610	0.4258 0.3885 0.3482 0.2948 0.2074 0.1341 0.0670 0.4477 0.4204 0.3905 0.3527 0.2802 0.2129 0.1312
		IMS	Optimal LR (3)	0.4610	0.4368 0.3863 0.3200 0.2522 0.1590 0.1080 0.0691
			Estimated LR (6)	0.4610	0.4419 0.4095 0.3680 0.3206 0.2391 0.1693 0.0967

This work focuses on

This work has been done while Rémi Cogranne was a visiting scholar at Binghamton University. The work on this paper was supported by Air Force Office of Scientific Research under the research grants number FA9950-12-1-0124. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation there on. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied of AFOSR or the U.S. Government.