Liang Zhou

Baoyu Zheng

Anne Wei

Benoît Geller

Jingwu Cui

A Scalable Information Security Technique: Joint Authentication-Coding Mechanism for Multimedia over Heterogeneous Wireless Networks

Keywords: multimedia security, authentication, joint source-channel coding, wireless networks

As multimedia is expected to be a major traffic source in the next-generation wireless networks, there have been increasing concerns about the security issues of wireless transmission of multimedia in recent years. Wireless networks, by their natures, are more vulnerable to external intrusions than wired ones. Therefore, many applications demand authenticating the integrity of multimedia content delivered wirelessly. In this work, we propose a framework for jointly authenticating and coding multimedia to be transmitted over heterogeneous wireless networks. We firstly provide a novel graphbased authentication scheme which can not only construct the authentication graph flexibly but also trade-off well among some practical requirements such as overhead, robustness and delay. And then, a rate-distortion optimized joint source-channel coding (JSCC) approach for error-resilient scalable encoded video is presented, in which the video is encoded into multiple independent streams and each stream is assigned forward error correction (FEC) codes to avoid error propagation. Furthermore, we consider integrating authentication with the specific JSCC scheme to achieve a satisfactory authentication results and end-to-end reconstruction quality by optimally applying the appropriate authentication and coding rate. Simulation results show the effectiveness of the proposed authentication-coding scheme for multimedia over wireless networks.

I. INTRODUCTION

As multimedia is expected to be a major traffic source on the next-generation wireless networks, the demand for transmitting the multimedia content over wireless networks has increased.

In contrast to the abundance of methods have been proposed to design robust and efficient schemes for delivering multimedia content over error-prone wireless networks, there are only very few works paying attention to the security aspect of such transmission. In fact, as more and more applications require authenticated multimedia streams, it is important to protect the authenticity of the streams in the aspects of integrity and non-repudiation. In order to design a satisfactory authentication scheme for a wireless multimedia transmission system, it would be essential to take into account the following practical requirements:

• Low communication overhead: It refers to the additional bytes transmitted along with the stream packets. These additional bytes include i.e. MAC (Message Authentication Code), Crypto Hash values or digital signatures.

• Robust against packet loss: The packets of the stream should be able to be authenticated with high probability under varied channel conditions with different packet loss rate. This requirement is particularly useful for multimedia streams which can tolerate some packet loss.

• Less receiver delay: It refers to the delay from the time the packet is received to the time when it is authenticated by the receiver. When consuming streaming media, each packet usually has its deadline after which it becomes useless. As a result, a large receiver delay could cause a packet to miss its deadline.

A. Related Works

The authentication problem has been attempted mainly using two approaches: a naive solution of authenticating a potential long stream is to sign each network packet using digital signature.

However the problem is that signing algorithms nowadays are computationally expensive, and it is not worthy to compute and verify one signature for each packet [START_REF] Zhang | An Optimal Content-Aware Authentication Scheme for Streaming JPEG-2000 Images over Lossy Networks[END_REF]; since it is too expensive to sign every packet of the stream, we can organize packets into groups and sign only one packet within each group [START_REF] Wu | Scalable Authentication of MPEG-4 Streams[END_REF]. This approach can be further classified into graph-based approach [START_REF] Gennaro | How to Sign Digital Streams[END_REF][START_REF] Peffig | Efficient Authentication and Signing of Multicast Streams over Lossy Channels[END_REF][START_REF] Song | Expander Graphs for Digital Stream Authentication and Robust Overlay Networks[END_REF][START_REF] Miner | Graph-Based Authentication of Digital Streams[END_REF] and erasure-code-based approach [START_REF] Park | Efficient Multicast Stream Authentication using Erasure Codes[END_REF]. [START_REF] Gennaro | How to Sign Digital Streams[END_REF] proposed an authentication scheme using a simple hash chain. It has low overhead and low receiver delay, but it cannot tolerate any packet loss; [START_REF] Peffig | Efficient Authentication and Signing of Multicast Streams over Lossy Channels[END_REF] provided EMSS, which uses a hash chain where each packet contains the hashes of previous packets and the signing is on the last packet. Obviously, it easily leads to a high receiver delay; [START_REF] Song | Expander Graphs for Digital Stream Authentication and Robust Overlay Networks[END_REF] presentd an authentication scheme based on the expander graph and theoretically derived the lower bound of authentication probability (AP). However, it has a very large communication overhead which is unacceptable for real applications; [START_REF] Miner | Graph-Based Authentication of Digital Streams[END_REF] was based on the random graph. The signing is on the first packet, and each packet contains the hashes of every subsequent packet with certain probability. Therefore, it also has high communication overhead; [START_REF] Park | Efficient Multicast Stream Authentication using Erasure Codes[END_REF] was proposed to use erasure code for stream authentication. For each block, the digital signature is coded with erasure code and then scattered into the packets. As long as the number of loss packets is less than a threshold, all received packets can be authenticated. This scheme has a high computation overhead due to the erasure coding. In addition, it also suffers from a high receiver delay, because the receiver has to wait for a minimum number of the received packets before authentication.

B. Main Contributions

The main contributions of this paper are as follows: firstly, we present a novel graph-based authentication (NGBA) approach which can not only construct the authentication graph flexibly but also trade-off well between the aforementioned practical requirements. Secondly, we propose an analytical joint source-channel coding (JSCC) approach for error-resilient scalable encoded video for lossy transmission, in which the video is encoded into multiple independent substreams based on 3-D SPIHT (3-D set partitioning in hierarchical trees) algorithm to avoid error propagation. Furthermore, the final realization of joint authentication-coding (JAC) system is the highlight of the proposed scheme because the ultimate goal of such scheme is to achieve an optimal end-to-end multimedia quality under the overall limited resource budget.

C. Outline

The rest of paper is organized as follows. In section II, we provide some technical preliminaries used in this work. Section III introduces the novel graph-based authentication scheme and joint source-channel coding, respectively. In section IV, we optimize the proposed joint authenticationcoding scheme. Finally, we present some selected simulation results and give some concluding remarks.

II. PRELIMINARIES

A. Definitions and Notations

Considering a sender transmitting consecutive packets {P 1 , ..., P n } in a broadcast data stream, we construct an authentication graph (AG) to authenticate received packets. In particular, we construct a directed acyclic graph of n vertices where a vertex i corresponds to the packet P i .

Let e(i, j) denotes a directed edge starting from i and ending at j. An edge e(i, j) in the graph indicates the authentication relationship between packet P i and P j : upon receiving packet P i and P j , if a receiver can authenticate both the contents and the source of P i , then it can authenticate the contents and source of P j . One of the packets, denoted by P sig , is signed with a public key signature algorithm. Hence, packet P i can be authenticated if and only if there is a path from P i to the signature packet that only includes nodes corresponding to the received packets [START_REF] Zhang | Rate-Distortion-Authentication Optimized Streaming of Authenticated Video[END_REF].

We denote the probability that P i is linked to P sig via such a path by P r[P i → P sig ].

For every stream, we are interested in the value of P r i = P r[P i → P sig |P i is received] for i ∈ 1, ..., n. In particular, we allow the sender to input desired values for these authentication probabilities. It is useful to allow a different AP for each packet, because the packets in the stream may actually vary in priority. Consequently, packets deemed more important will be more tolerant to loss (because redundant authentication information will be included), and the less important packets will be less tolerant of loss, in order to avoid unnecessary overhead.

B. Hash-Based Authentication Tool

A public hash function may be used to link the packets in a multicast stream to a signature.

Recall from subsection II-A that e(i, j) is representing in a graph, then in the corresponding authentication scheme, the ability to authenticate P j implies the ability to authenticate P i . P i may have a positive in-degree itself, indicating that hashes of other packets are included within P i . In this case, the hash of P i is taken after all other hashes it requires are included in it.

We require strictly hash-based authentication graphs to be acyclic, so as to avoid dependencies between packets which can not be fulfilled [START_REF] Lysyanskaya | Authentication without Identification[END_REF].

One major advantage of this signature-based approach is that it can protect data integrity while ensuring non-repudiation. Therefore, it is useful for general authentication applications when digital evidence is concerned. Other merits of this approach include achieving both low computation and communication overhead, and resisting to packet loss [START_REF] Li | Joint Source-Channel-Authentication Resource Allocation and Unequal Authenticity Protection for Multimedia over Wireless Networks[END_REF]. We consider adopting this approach as the underlying authentication algorithm in this research to take advantage of these desirable merits.

III. SYSTEM DESCRIPTION

The proposed joint authentication and coding system is shown in Fig. 1. At the sender, the multimedia content is firstly passed to the JAC control unit, where it runs the JAC scheme (which will be described in section IV) and outputs the optimal source code rate, channel code rate, and authentication rate. The JSCC unit encodes the multimedia according to both the source rate and outputs the compressed code stream. In the packet signing unit, AG is constructed using the proposed NGBA approach (which will be described in subsection III-A). Therefore, the main task at the sender is to sign and protect the code stream by joint authentication and coding before transmission. At the receiver, error correction is firstly performed on the received stream in the channel decoding unit. Residue errors may still exist in the output stream which passes to the source decoder. We assume that the source decoder is error-resilient, where techniques such as synchronization mark and CRC (cyclic redundancy check) are applied to the code stream. Note that bit errors would trigger verification false alarms, and thus it is important to skip packets with bit errors during authentication. The verifiability information passes to the source decoding unit, so that during multimedia decoding, those non-verifiable packets are skipped.

A. Novel Graph-Based Authentication

In order to obtain lower overhead and higher AP while maintain the same level of delays and robustness against packet loss, we propose a novel graph-based authentication approach where one signature is amortized among a group of packets connected with some regular graphs.

1) Authentication Graph Construction:

Assume the stream is divided into a number of blocks and each block contains M + 1 (M ≥ 0) packets, where only one signature is generated for each block, and the M packets and the signature packet P sig are connected using the regular graph.

Assuming M = n × m + t (n, m, t ≥ 0), the definition of the graph is given below.

The M data packets are divided into m stages, and each stage has n packets, and the t is the remaining packets. The packet is denoted as P (u, v), where u ∈ {0, 1, ..., m -1} indicates the stage and v ∈ {0, 1, ..., n -1} indicates the packet in a stage. In this graph, there exists a directed edge e(P (u 1 , v 1 ), P (u 2 , v 2 )) from packet P (u 1 , v 1 ) to packet P (u 2 , v 2 ), if either of the following conditions is met: [START_REF] Zhang | An Optimal Content-Aware Authentication Scheme for Streaming JPEG-2000 Images over Lossy Networks[END_REF].

u 1 = u 2 + 1 and v 1 = v 2 ; (2). u 1 = u 2 + 1 and v 1 = v u 2 2 , where v u 2
2 is different from v 2 only at the bit position u 2 . In addition, there also exists a directed edge from all packets in stage 0 to the signature packet P sig .

AG CONSTRUCTION

• If there exists m = log 2 n + 1 and t = 0. Each directed edge e(P (u 1 , v 1 ), P (u 2 , v 2 )) is realized by appending the hash of the packet P (u 1 , v 1 ) to P (u 2 , v 2 ). Fig. 2(a) gives an example of the authentication graph, with 4 stages and 8 data packets in each stage. The signature packet P sig contains the signature and hashes of all packets in stage 0 to log 2 n -1 have two hashes, and the packets in the last stage do not have any hash.

• If there exists m = log 2 n + 1 but t = 0, the remaining packets t are constructed using the following units (shown in Fig. 3). Note that all packets in stage 0 to m -1 have two hashes, and the packets in the last stage (just for the t) do not have any hash. 2) Lower Bound of AP: For all pairs of nodes (i, j), we include a directed edge from node i to node j with probability p (0 < p ≤ 1), we call a graph constructed in this way a p-random graph. For notational convenience, we note P sig = P 1 . Lemma 1: With a p-random authentication approach and no packet loss, a packet P i (i ≥ 2), can be authenticated with at least the probability:

P r[P i → P sig |P i is received] ≥ 1 -(1 -p)(1 -p 2 ) i-2 (1)
Proof: We calculate the probability that node i connects to signature node (node 1) in the corresponding p-random graph as follows. First, with probability p, e(i, 1) exists and so, node i connects to the signature node. With probability (1 -p)p, e(i, 1) does not exists but e(i, i -1)

does, so i can connect to signature node via a path from i -1 to signature node. Proceeding in this way, we get the following expression:

P r[P i → P sig |P i ] ≥ p + (1 -p)pP r[P i-1 → P sig |P i-1 ] + ... + (1 -p) i-2 pP r[P 2 → P sig |P 2 ] (2)
Apply the induction assumption for 1, ..., i -1, to the right hand side of the inequality above, we have:

p + (1 -p)p(1 -(1 -p)(1 -p 2 ) i-3 ) + ... + (1 -p) i-2 p(1 -(1 -p)) (3)
We simplify this expression by factoring out terms of the form (1 -p). As a first step, we have:

1-(1 -p)[1 -p + (1 -p)p(1 -p 2 ) i-3 -(1 -p)p + (1 -p) 2 p(1 -p 2 ) i-4 -... -(1 -p) i-2 p + (1 -p) i-3 p(1 -p 2 ) -(1 -p) i-3 p + (1 -p) i-2 p] (4)
Continuing to factor in this way, we eventually get:

1 -(1 -p) i-1 [p(1 + p) i-3 + p(1 + p) i-4 + p(1 + p) i-5 + ... + p(1 + p) + 1 + p] = 1 -(1 -p) i-1 [p( 1 -(1 + p) i-2 1 -(1 + p) -1) + 1 + p] (5)
This simplifies to:

1 -(1 -p)(1 -p 2 ) i-2 .
Theorem 1: With a p-random authentication approach in a lossy network, in which each packet is lost independently at random with probability q, packet P i , i ≥ 2, can be authenticated with probability:

P r[P i → P sig |P i is received] ≥ 1 -(1 -p)(1 -(p(1 -q)) 2 ) i-2 (6)
Proof: We assume that P 1 is always received (this may be accomplished with high probability by transmitting it multiple times, or empowering receivers to request re-transmission if it is not received), when we follow the same type of argument as used in the proof of Lemma 1, we get: Let

P r[P i → P 1 |P i ] ≥ p + (1 -p)p(1 -q)P r[P i-1 → P 1 |P i-1 ] + ... + (1 -p(1 -q)) i-3 p(1 -q)P r[P 2 → P 1 |P 2 ] (7)
a i (p) = 1 -(1 -p)(1 -p 2 ) i-2
, the authentication probability found in Lemma 1. From the equality above, it follows that

P r[P i → P 1 |P i ] ≥ ( a i (p(1 -q)) -p 1 -p )(1 -p(1 -q)) + p(1 -q) (8)
The statement of the theorem follows from substituting in the expression for a i (p(1 -q)).

In the case of the proposed NGBA, a packet P (u, v) can not be authenticated unless there is a path to the signature packet at the receiver. The authentication probability P r[P (u, v)] is equivalent to probability that such path exists

P r[P (u, v)] ≥ 1 -(1 -p)(1 -(p(1 -q)) 2 ) u , u ≥ 0 (9)
We can see that P r[P (u, v)] depends only on u and q, and all packets in the same stage have the same P r[P (u, v)]. As we travel from stage 0 to stage m -1, the authentication probability decreases, because a packet in the later stage has more independency than that in the earlier stage. However, this trend is slowed down by the proposed graph where a packet in the later stage has more paths to the signature packet. Therefore, the minimum authentication probability P r min under random packet loss can be achieved as follows

P r min = 1 -(1 -p)(1 -(p(1 -q)) 2 ) m (10)

B. Joint Source Channel Coding

In this subsection, we first introduce the architecture of JSCC, and then describe the packet-loss pattern approximation employed in this paper to represent the channel packet-loss process. 1) The Architecture of JSCC: The proposed coding architecture contains two parts as shown in Fig. 4. Unit-1 uses the 3-D SPIHT codec that generates independent embedded streams, while Unit-2 uses the coding constraints and channel condition to pack the bit-streams into packstreams of quality layers. This two-units structure collects incremental contributions from the various streams into SNR scalable quality layers in a way similar to that of embedded block coding with optimized truncation. The streams and rate-distortion functions generated by Unit-1 can be processed independently to channel conditions. The source and channel allocation algorithm in Unit-2 must be efficient to cope with the time varying channel conditions.

Unit-1 uses an embedded coding technique that generates multiple independent embedded streams. A video is divided into several independently encoded for additional functionality in

Unit-1. The video coder divides the 3-D wavelet coefficients into multiple blocks according to their spatial and temporal relationships, and then encodes each block independently using the 3-D SPIHT algorithm. Fig. 5 shows an example the separation of the 3-D wavelet transform coefficients into four independent blocks, each of which retains the spatio-temporal structure of 3-D SPIHT. The proposed method allocates source bits to each embedded bitstream to minimize the total distortion of a video clip. Moreover, the video scalability is imparted by the layering concept and the scalable stream is organized into quality layers.

2) Packet-Loss Pattern Approximation:

We use Reed-Solomon (RS) code as the channel coding strategy because it is effective for recovering erased symbols when their locations are known [START_REF] Kim | Distrubuted Video Streaming Using Multiple Description Coding and Unequal Error Protection[END_REF]. For a (n, k) systematic RS code with a block length n, the source symbol is k. The first k encoded symbols are source symbols correctly when the number of loss symbols is less than the minimum distance d min = n -k + 1 of the code. The performance of an RS decoder can be characterized by the code correct probability

P c (n, k) = n-k m=0 P (n, m) (11)
where P (n, m) is the probability of m erasure within a block of n symbols. In a binary symmetric channel without memory, we have

P (n, m) =   n m   P m B (1 -P B ) n-m ( 12
)
where P B is the mean packet loss rate [START_REF] Fu | A Joint Source and Channel Coding Algorithm for Error Resilient SPIHT-Coded Video Bitstreams[END_REF]. In general, for channels with memory, it is more complicated to calculate . Here, we use a two-state Markov model (i.e. Gilbert model) to simulate the bursty packet loss behavior [START_REF] Qu | Cross-layer QoS Control for Video Communications over Wireless Ad Hoc Networks[END_REF]. The two states of this model are denoted as G (good) and B (bad). In state G, packets are received correctly and timely, whereas, in state B, packets are assumed to be lost. This model can be described by the transition probabilities P GB from state G to B and P BG from state B to G. The then the average P B is given by

P B = P GB P GB + P BG (13)
The Markov model is a renewal model, and such models are determined by the distribution of error-free intervals, known as gap. Let gap of length σ be the event that after a lost packet, σ -1 packets are received and then again a packet is lost. The gap density function g(σ) gives the probability of a gap length σ. The gap distribution function G(σ) gives the probability of the gap length greater than σ -1. These functions can be derived as [START_REF] Fu | A Joint Source and Channel Coding Algorithm for Error Resilient SPIHT-Coded Video Bitstreams[END_REF] g

(σ) =    1 -P BG , σ = 1 P BG (1 -P GB ) σ-2 P GB , σ > 1 (14) G(σ) =    1 -P BG , σ = 1 P BG (1 -P GB ) σ-2 , σ > 1 (15)
Let R(n, m) be the probability of m -1 erroneous symbols within the next n -1 symbols following an erroneous symbol. It can be calculated using the recurrence

R(n, m) =      G(n), m = 1 n-m+1 σ=1 g(σ)R(n -σ, m -1), 2 ≤ m ≤ n (16)
Then the probability of errors within m a block of n symbols is

P (n, m) =        n-m+1 σ=1 P B G(σ)R(n -σ + 1, m), 1 ≤ m ≤ n 1 - n σ=1 P (n, m), m = 0 (17)

IV. OPTIMIZATION FOR JOINT AUTHENTICATION AND CODING

The purpose of joint authentication and coding is to achieve two objectives: (1) optimize the source and channel coding bits for minimizing the end-to-end distortion, and (2) optimize the authentication bits for achieving satisfactory AP. Notice that AP determines the probability that a packet is non-verifiable, which should be skipped during reconstruction. Since the skip will result in distortions to the multimedia content, we may find that it is possible to unify the two objectives into one single form, i.e., maximizing the end-to-end PSNR (Peak Signal-to-Noise Ratio) at the receiver relative to the original sequence. It can be defined as

P SN R(dB) = 10 log 10 ( 255 2 M SE ) ( 18
)
where M SE is the mean-square error between the original and the decoded luminance frame.

Fig. 6 shows how multiple encoded sequences of different quality levels are protected based on systematic RS codes. For notational convenience, we define the bit-plane 1 as the highest bit plane and the bit-plane I s as the lowest bit plane to be sent for sub-stream-s [START_REF] Sungdae | A full-featured, error-resilient, scalable wavelet video codec based on the set partitioning in hierarchical trees (SPIHT) algorithm[END_REF]. Let N s be the number of packets that are used to send the combined source data and redundancy for sub-stream-s in a GOP (Group of Pictures) and L be the packet size in bytes. In this scheme, the bits belonging to bit-plane i (1 ≤ i ≤ I s ) are filled into k s,i packets and the remaining c s,i = N s -k s,i packets are filled with channel coding redundancy. In other words, the source data for bit-plane i is protected by RS code (N s , k s,i ).

We propose the JAC scheme which is performed on GOP basis. We define the total number of packets to be sent from all sources for a GOP period N as Fig. 6. Multiple-substream bit-plane with error protection.

N ≤ R × N GOP F × L ( 19
)
where R is the total coding rate in bytes/s for the combination of source coding (r s ), channel coding (r c ) and authentication (r a ) for all sources, N GOP is the number of frames in a GOP, F is the frame rate in frames/s. In this framework, we assume that are n s sources and source-s transmits sub-stream-s to the receiver for s = 1, 2, ..., n s (n s ≥ 1). Then the proposed algorithm divides N into N 1 (t i ), N 2 (t i ),..., N n s (t i ) so as to maximize the expected quality at the receiver, where N s (t i ) represents the total number of packets transmitted by source-s at GOP period t i for s = 1, 2, ..., n s (n s ≥ 1). Taking account into the effective rate of source s, N s (t i ) should satisfy the following condition:

N s (t i ) ≤ R s (t i ) × N GOP F × L ( 20
)
where R s (t i ) is the total rate of source-s at the GOP period t i . In typical transform coding, each coefficients is quantized independently. The overall distortion is exactly the summation of the distortion at each source. The probability for an authentic packet P i to be decodable and verifiable is P r i (1 -P B ). In this case, the distortion is merely due to source coding. If the packet is either non-decodable or non-verifiable, the distortion depends on the specific errorconcealment scheme. Here, we consider setting the values to zeros when a packet is either non-decodable or non-verifiable. Therefore, we can state our source and channel allocation algorithm as follows: Given N , R s (t i ) and the tolerated minimum authentication probability P r thr , the proposed algorithm finds N s (t i ) and K s (t i ) = (k s,1 (t i ), k s,2 (t i ), ..., k s,I s (t i )) for s = 1, 2, ..., n s , that maximize the expected quality at the receiver given by

P SN R(t i ) = ns s=1 (P r s (1 -P B )) Is l=1 ( N s -k s,l-1 j=Ns-k s,l +1 P (j, N s ) Is i=l P SN R s (i)) (21) subject to n s s=1 N s = N, N s ≤ Rs(t i )×N GOP F ×L P r s ≥ P r thr , s = 1, 2, ..., n s
where P r s is the average AP of source-s; P (j, N s ) is the probability that j packets are lost out of N s packets sent by source-s; P SN R s (i) is the expected quality at the receiver when the receiver decodes up to the ith bit-plane for sub-stream-s; I s is the last bit plane to be sent for source-s.

Each source independently runs the proposed rate allocation algorithm to get its optimal number of packets to transmit for a GOP period, using the information contained in the control packets that the receiver sends to all sources. The proposed algorithm tries all possible combinations of (N s , K s ) that satisfy the constraints in (21) and choose one that maximizes the expected quality. Once the optimal (r s , r c ) value is found, the source code rate, channel code rate and authentication rate are determined.

V. SIMULATION RESULTS AND DISCUSSION

We conducted simulation experiments to test the performance of the proposed streaming framework. First of all, we describe the simulation environment. Secondly, we present the main simulation results where we show the objective and subjective results of the performance under different scenarios. Finally, we conclude this section based on the selected simulation results described.

A. Simulation Environment

For these experiments, we use the QCIF Weather Forecast test sequence at F = 30f rames/s, N GOP = 16 and n s = 2. A three-level wavelet decomposition is applied to a group of 16 frames

[3] M, 1 M b-1 Erasure Code [6] M, 1, 2 [m , M] M-m NGBA M, 1 1 n/2 m+1
and the 3-D wavelet coefficients are divided into two groups using the method proposed in [START_REF] Kim | Distrubuted Video Streaming Using Multiple Description Coding and Unequal Error Protection[END_REF].

In order to provide a representative evaluation of system performance, for each simulation run second intervals, the environments are kept constant. It should be noted that all the simulation results in this section have been obtained using averaged 300 runs in order to obtain statistically meaningful results.

B. Selected Simulation Results and Discussions

At first, we compare the proposed NGBA approach with other existing approaches. Table I summarizes the 4 authentication approaches based on aforementioned requirements. (Note: M is the block size; b is the maximum edge length; m is the minimum number of received packets to recover the hashes and the signature in a block; m is the number of the stage; n is the number of packets each stage contains)

In most approaches, the authentication probability and communication overhead conflict with each other, that is, increasing the overhead will increase the authentication probability, and vice versa. Fig. 7 shows the authentication probabilities under different communication overheads.

Assuming the loss probability is 30%, the total number of packet is 1024, each hash has 16bytes and each signature has 128 bytes. For EMSS approach, the length of each edge is uniformly distributed in the interval [START_REF] Zhang | An Optimal Content-Aware Authentication Scheme for Streaming JPEG-2000 Images over Lossy Networks[END_REF]128]; Fig. 8 shows that our proposed approach outperforms other approaches except the Erasure Code in terms of overhead and authentication probability. From the above figures, we can see that the NGBA outperforms existing approaches in terms of integrating overhead, robustness, authentication probability and receiver delay.

And then, to demonstrate the effectiveness of our proposed joint scheme, we plot the endto-end rate-distortion curves for the test sequence at packet loss rate equal to 5% and 15%, respectively. The proposed resource allocation scheme (JAC+NGBA) is benchmarked against other two schemes:1) JAC+EMSS, in which the overall resource allocation is performed between source channel coding and authentication, but the resource within authentication is equally allocated using the basic EMSS scheme. 2) JSCC+EMSS, in which the resource for source and channel coding is jointly allocated whereas that for authentication is fixed, and the basic EMSS is applied. Fig. 9 shows the performance comparison between our proposed scheme and the competing schemes. The proposed JAC+NGBA scheme can be seen to achieve a much higher performance in terms of end-to-end PSNR compared to the competing schemes. When the packet loss rate is 5% and the overall rate ranges from 0.5 to 3, the average PSNR using the proposed scheme is 37.85 dB while it is 34.60 dB and 34.26dB for the case of JAC+EMSS and JSCC+EMSS, thus, around 3.2-3.6 dB performance gain can be achieved on average using the proposed scheme. Similarly, when the packet loss rate is 15%, around 2.6-5.6dB performance gain can be achieved on the average. It should be noted that JAC+EMSS also outperforms JSCC+EMSS, especially when the packet loss rate is high. For example, when the packet loss rate is 5%, the average performance gap is only 0.34dB; while packet loss rate is 15%, the gap increases to 3.05dB.

Moreover, to examine how the JAC is affected by the channel condition, we fix the overall code rate and examine how r s , r c and r a vary, as the packet loss rate increases from 5% to 15%. Table II illustrates the unitary results for the test sequence. From the table, we observe that when the channel condition is good, most of the bits are allocated for source coding and authentication. When the channel condition is poor, the large portion of bits is allocated for channel coding. As expected, the PSNR of reconstructed image decreases as packet loss rate increases.

The above objective results are based on a quantitative assessment of reconstructed PSNR values. In Fig. 10, we also show some subjective results based on the reconstructed frames taken from the decoded test sequence of the simulation run when P B = 15% and the overall rate is 4bpp. From Fig. 10, we can see that the proposed JAC+NGBA scheme can provide improved subjective performance compared to the other competing schemes. These results again support the preceding objective assessments.

In order to provide a more comprehensive evaluation of the proposed joint scheme, in Table III, we repeat the results for other QCIF video sequences under the same simulation configuration as the previous experiments (Note: r a = 0.3 for JSCC+EMSS). From Table III, it can be observed that the proposed JAC+NGBA method has considerable performance advantage comparing to the other competitive methods, which is due to the proposed scheme has the characteristics of network-adaptive and error-resilient.

VI. CONCLUDING REMARKS

In this paper, we have been focusing on designing a joint authentication and coding system in order to achieve satisfactory authentication results and end-to-end reconstruction quality under the overall limited resource budget. Firstly, we provide a novel graph-based authentication approach which can not only construct the authentication graph flexibly but also trade-off well between some practical requirements. Secondly, we propose an analytical joint source-channel coding approach for error-resilient scalable encoded video for lossy transmission. Furthermore, we integrate authentication with coding to achieve an optimal end-to-end multimedia quality under

Fig. 1 .

 1 Fig. 1. Architecture of the joint authentication and coding scheme.

Fig. 2 (

 2 b) gives an example of the constructed AG when M = 34 = 8 × 4 + 2.

Fig. 2 .Fig. 3 .

 23 Fig. 2. The examples of authentication graph

Fig. 4 .

 4 Fig. 4. The architecture of JSCC.

Fig. 5 .

 5 Fig. 5. Separate the 3-D wavelet transform coefficients into four independent sub-streams.

  we generate a random topology on the disc of unit area as a 2D Poisson point process with total number of nodes equal to 25. The transmission range r for each node is kept constant during the simulation at the value of r = 0.2 × (1/ √ π) such that the sum of the transmission regions for all the 25 nodes (i.e., 25 × πr 2 = 1) almost completely covers the unit disc, thus ensuring a high degree of connectivity. Each node is assigned the fixed transmission rate W i = 2M bps, which is a basic rate available in the IEEE 802.11b standard. During transmission, the environments are updated every 1 second which can cause changes in the channel condition. During successive 1

Fig. 7 .

 7 Fig. 7. Authentication probabilities at different overheads. (Packet loss probability is fixed at 30%)

Fig. 8 .

 8 Fig. 8. Authentication probabilities at different loss probabilities. (The overheads is 30 bytes per packet)

Fig. 9 .

 9 Fig. 9. End-to-end rate-distortion curves. (a) packet loss rate is 5% (r a = 0.4 for JSCC+EMSS). (b) packet loss rate is 15% (r a = 0.25 for JSCC+EMSS).

Fig. 10 .

 10 Fig. 10. Subjective comparisons of decoded frames for Weather Forecast sequence. (a) Joint JAC and NGBA; (b) Joint JAC and EMSS; (c) Joint JSCC and EMSS; (PSNR=36.7 dB, 34.2 dB, 30.8 dB, respectively).

TABLE II JAC

 II RATE UNDER DIFFERENT P B (OVERALL RATE=3BPP)

	PB	5%	6%	7%	8%	9%	10% 11% 12% 13% 14% 15%
	r s	0.54 0.52 0.51 0.48 0.46 0.41 0.38 0.35 0.33 0.29 0.25
	rc	0.04 0.13 0.16 0.21 0.24 0.31 0.38 0.42 0.48 0.53 0.59
	r a	0.42 0.35 0.33 0.31 0.30 0.28 0.24 0.23 0.19 0.18 0.16
	PSNR(dB) 41.7 41.1 40.6 40.1 39.3 38.8 37.8 36.9 36.0 35.4 34.6

VII. ACKNOWLEDGMENTS

This work is supported by the International Project PRA-SI (financed by France and China government) under Grant No. SI04-03, the Key Project of Nature Science Foundation of Jiangsu (China) under Grant BK2007729 and the Climbing Plan in NJUPT under Grant NY207061.

Moreover, we also thank the anonymous reviewers for insightful comments and suggestions.

the overall limited resource budget. The simulation results show the effectiveness of our joint authentication-coding scheme for multimedia over wireless networks.


Thierry Petit

Andrew C Trapp

Assessing Both Solution Diversity and Solution Quality in Constraint Programming

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A Tonnoir

Dirichlet-to-Neumann operator for diffraction problems in stratified anisotropic acoustic waveguides

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Dirichlet-to-Neumann operator for diffraction problems in stratified anisotropic acoustic waveguides

Version française abrégée

Nous nous intéressons ici à l'étude et la résolution numérique d'un problème de diffraction dans un guide d'ondes bi-dimensionnel infini acoustique anisotrope et stratifié (représenté Figure 1). Autrement dit, on cherche à calculer la solution sortante des équations [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF], où la matrice A est définie par [START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF]. Afin de réduire les calculs à une zone bornée, il existe dans la littérature plusieurs approches (voir par exemple [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF][START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF][START_REF] Bonnet-Ben Dhia | An alternative to the Dirichlet-to-Neumann maps for waveguides[END_REF]). L'une d'elle, développée dans le cas isotrope (i.e. A = Id), consiste à écrire des conditions dites transparentes sur des frontières artificielles Σ ± a bornant le domaine de calculs (voir [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF][START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF]). Pour cela, on

Email address: antoine.tonnoir@gmail.fr (A. Tonnoir).

Preprint submitted to the Académie des sciences exploite la représentation modale de la solution dans les parties non perturbées Ω ± a du guide. En effet, la géométrie séparable des demi-guides Ω ± a permet de montrer par une technique de séparation de variables que la solution p se décompose sous la forme d'une série modale [START_REF] Bonnet-Ben Dhia | An alternative to the Dirichlet-to-Neumann maps for waveguides[END_REF]. À l'aide de cette forme explicite de la solution, on peut obtenir un opérateur de Dirichlet-to-Neumann (DtN) qui relie la trace p| Σ ± a à la dérivée normale ∂ ν p| Σ ± a de la solution sortante associée dans Ω ± a .

Dans le cas où A = Id, les champs transverses des modes ne sont plus les éléments propres d'un problème auto-adjoint, mais sont solutions du problème aux valeurs propres quadratique (4). On montre qu'ils ne sont alors plus orthogonaux entre eux (on peut même montrer qu'ils ne forment plus une base de Riesz de L 2 (Σ ± a )), ce qui pose problème pour justifier la décomposition modale (3) dans les demi-guide Ω ± a . L'objectif de cette note est de montrer que la solution se décompose en fait sur des modes dans des demi-guides déformés Ω ± a,α , et de prouver qu'on peut obtenir via la décomposition modale (12) un opérateur DtN. Pour cela, l'idée est d'exploiter un changement de coordonnées (X, Y ) = (x + α(y), y) qui préserve la géométrie du guide, où α(y) est choisi en fonction de A, voir l'équation (7). Il est connu que changer le système de coordonnées revient à changer l'anisotropie du milieu, voir [START_REF] Chen | Acoustic cloaking and transformation acoustics[END_REF]. Ainsi, on peut réécrire les équations dans les parties non perturbées du guide d'ondes sous la forme (8). Dans ces nouvelles coordonnées, le calcul des modes revient alors à la résolution d'un problème aux valeurs propres linéaires auto-adjoint, ce qui permet de retrouver des propriétés de complétude et d'orthogonalité, non plus sur les sections droites Σ ± a du guide, mais sur des sections déformées Σ ± a,α , voir Figure 2. On peut alors obtenir des conditions transparentes, réécrire le problème en domaine borné, et montrer comme dans [START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF] un résultat d'existence et unicité, voir théorème 3.1.

Introduction

In this note, we are interested in the study and the numerical resolution of the diffraction problem in a stratified anisotropic acoustic bi-dimensional waveguide. In other words, we want to solve the following equations:

   -div(A(x, y)∇p) -ω 2 p = f in Ω, A(x, y)∇p • ν = 0 on ∂Ω, (1)
where p denotes the pressure field, ω > 0 the frequency and ν the outward unitary normal. The geometry Ω of the waveguide is represented on Figure 1, and can be decomposed into three parts

Ω = Ω - a ∪ Ω a ∪ Ω + a
where Ω ± a = {±x ≥ a} × [-h, h], Ω a = {|x| ≤ a} ∩ Ω, and a, h > 0. h represents the height of the waveguide, and 2a the length (in the x -direction) of the bounded subdomain Ω a . This subdomain Ω a may be perturbed and may contain defects, as represented on Figure 1. Moreover, we suppose that Supp(f ) ⊂ Ω a . We also denote by Σ ± a = {x = a} × [-h, h] the interfaces between Ω a and Ω ± a . Besides, A(x, y) is a symmetric positive definite matrix that verifies A(x, y) = A(y) if (x, y) ∈ Ω a and:

⌦ = ⌦ a ⌦ + a ⌦ a ⌃ + a ⌃ a x y
A(y) =   c 1 (y) c 3 (y) c 3 (y) c 2 (y)   , where      c 1 (y), c 2 (y) > 0, c 1 (y)c 2 (y) -[c 3 (y)] 2 > 0.
(2)

In the sequel, for simplicity reasons, we will not recall the dependency of A(y) (and its coefficients) with respect to y, and we will simply denote A(y) by A.

Classically, to define the diffraction problem, we must impose a radiation condition at infinity (see for instance [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF]). This condition ensures that the solution is outgoing (that is to say "propagates" toward infinity), and is defined thanks to the modal decomposition of the solution in the half-guides Ω ± a . Modes are particular solutions of the homogeneous equations (1) (i.e. f = 0) of the form e iβx ψ(y) in a perfect waveguide of separable geometry R × [-h, h]. In the isotropic case (i.e. A = Id), simple calculations show that there are two families of modes {p ± n } n∈N , which are given by:

p ± n (x, y) = e ±iβnx ψ n (y), where ψ n (y) = N n cos nπy 2h and β n = i ω 2 - πn 2h 2 , ∀n ∈ N.
We consider the complex square-root with branch-cut on R -. N n is a normalisation coefficient chosen so that the family {ψ n } n∈N is an orthonormal basis of L 2 ([-h, h]). Supposing that ω is not a cut-off frequency (that is β n = 0, ∀n ∈ N), the modes p + n (resp. p - n ) are rightgoing (resp. leftgoing). Therefore, we say that p is outgoing if it expands in Ω + a (resp. Ω - a ) as a series of modes p + n (resp. p - n ). Thus, except at the cut-off frequency, we define the diffraction problem by equations ( 1) and the radiation condition.

Moreover, given a Dirichlet data g ± on Σ ± a , we can express the outgoing solution p in the half-guide Ω ± a verifying p = g ± on Σ ± a . Indeed, thanks to the orthogonality of the ψ n , it comes that

p(x, y) = n≥0 g ± , ψ n L 2 ([-h,h]) e ±iβn(x-a) ψ n (y), in Ω ± a . (3)
Using this modal decomposition, we derive the so-called Dirichlet-to-Neumann (DtN) operators T ± that map the Dirichlet data g ± to the normal derivatives ∂ ν p(g ± ), where p(g ± ) denotes the above function.

Thanks to the DtN operators T ± , we can formulate the diffraction problem in the bounded domain Ω a by writing "transparent" boundary conditions (TBC) ∂ ν p = T (p| Σ ± a ) on Σ ± a (see [START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF]).

Now, what about the anisotropic case (A = Id) ? We can still compute the modes, at least numerically. Indeed, injecting the form of the solution e iβx ψ(y) into the equations (1) leads to solve:

     -(c 2 ψ ) -iβ c 3 ψ + (c 3 ψ) -ω 2 -c 1 β 2 ψ = 0 in [-h, h], c 2 ψ + ic 3 βψ = 0 on {-h, h}. (4)
In the case where c 3 = 0, this problem is a linear self-adjoint eigenvalue problem where β 2 represents the eigenvalue and ψ the eigenfunction. We can then extend the classical results of the isotropic case (that is to say proving orthogonality and completeness of the transverse part of the modes). The difficulty arises when c 3 = 0. Indeed, (4) is then a quadratic eigenvalue problem where β is the eigenvalue, and the major point is that the transverse part of the modes are no longer orthogonal.

The purpose of this note is to propose a method that enables to compute the modes by solving a linear self-adjoint eigenvalue problem, and that enables to recover the completeness and the orthogonality relations of the transverse parts of the modes, not on the straight section [-h, h] of the waveguide, but on "deformed" section of the waveguide that will be precise in the sequel. Thus, we will be able to derive DtN operators and reformulate the diffraction problem in a bounded domain. A numerical validation is presented at the end of the note.

Modes computation

The key idea consists in using an appropriate change of coordinates. Indeed, changing the coordinates amounts to change the anisotropy of the medium. This idea has been intensively used in acoustic cloaking, see for instance [START_REF] Chen | Acoustic cloaking and transformation acoustics[END_REF]. Another application that we can mention is the construction of water-wave guide shifter, see [START_REF] Berraquero | Pagneux Experimental realization of a water-wave metamaterial shifter[END_REF]. Let us consider a straight waveguide of geometry R × [-h, h]. In order to compute the modes, we propose a particular change of coordinates of this form:

(X, Y ) = (x + α(y), y), (5)
where α(y) is a function of the transverse coordinate y. Let us point out that this change of coordinates ( 5) is not the usual one associated to the anisotropy axes. The advantage of this mapping is that it "eliminates" the extra-diagonal term c 3 while preserving the geometry of the waveguide (because Y = y). Indeed, in these new coordinates, the gradient operator ∇ rewrites :

∇ = R ∇, R =   1 0 α 1   , where α = dα dy and ∇ =   ∂ X ∂ Y   . Noting that div = ∇ t , it comes div (A∇•) = ∇ t R t AR ∇ (•) .
After simple computations, we get that

R t AR =   c c 3 + α c 2 c 3 + α c 2 c 2   , where c = c 1 + 2α c 3 + (α ) 2 c 2 > 0. ( 6
)
To check that c > 0, we simply consider the polynomial P (z) = c 1 + 2c 3 z + c 2 z 2 . Its discriminant 4 (c 3 ) 2 -c 1 c 2 is strictly negative because of (2), and therefore the sign of P (z) for all z is the same as c 2 (or c 1 ).

Given (6), in order to eliminate the extra-diagonal terms in R t AR, the "smart" choice of α is α = -c3 c2 , so that

α = y -h -c 3 (s) c 2 (s) ds + C ste . ( 7
)
The constant term C ste simply corresponds to a shift in the x-direction and can be chosen arbitrarily. In the rest, we will suppose C ste = 0. With this choice of α, the equations (1) simplify to

-c∂ 2 XX p -∂ Y (c 2 ∂ Y p) -ω 2 p = 0 in R × [-h, h], (8)
where p(X, Y ) = p(x, y). We recall that the coefficients c and c 2 depend on the transverse coordinate Y = y. Let us remark that, since c > 0 and c 2 > 0, we are still led to solve a Helmholtz type equation.

Concerning the free boundary conditions A∇p • ν = 0 on R × {-h, h}, we have:

A∇p = AR ∇ p =   c 1 + α c 3 c 3 0 c 2   ∇ p =   c∂ X p + c 3 ∂ Y p c 2 ∂ Y p   (9)
Let us underline that, for our choice of α , c 1 + α c 3 = c. Therefore, the boundary conditions become c 2 ∂ Y p = 0, or equivalently ∂ Y p = 0. To sum up, this change of coordinates (X, Y ) = (x + α(y), y) with α given by (7) enables to (i) eliminate the cross-derivative in the operator div(A∇•),

(ii) get boundary conditions that only involve derivative with respect to Y , (iii) preserve the separable geometry R × [-h, h] of the waveguide.

Then, we can look for particular solutions the form e iβX ψ(Y ), which leads to solve the following generalized linear and self-adjoint eigenvalue problem:

   -c 2 ψ = ω 2 -cβ 2 ψ in [-h, h], ψ = 0 on {±h}, (10)
where β 2 is the eigenvalue and ψ the associated eigenvector. Because the operator -(c 2 (•) ) is symmetric and positive, we get by classical spectral theory results the following proposition: Proposition 2.1 There exists a sequence of positive eigenvalues {λ n = ω 2 -cβ 2 n } n∈N and associated eigenfunctions { ψ n } n∈N solutions to (10). Moreover, the eigenvalues λ n tend to +∞ with respect to n, and the eigenfunctions ψ n form a basis of L 2 ([-h, h]) that verifies the orthogonality relations:

ψ n , ψ m c := h -h c(s) ψ n (s) ψ m (s)ds = 0, if λ n = λ m .
In what follows, we consider that the ψ n are normalized with respect to the scalar product (•, •) c . Then, the two families of modes p ± n are given by (using the coordinates (x, y)) p ± n (x, y) = ψ n (y)e ±iβn(x+α(y)) , where

β n = (ω 2 -λ n )/c. ( 11
)
Supposing that ω is not a cut-off frequency, the family p + n (resp. p - n ) corresponds to the rightgoing (resp. leftgoing) modes. Therefore, except at the cut-off frequencies that form a countable set, we can define the radiation condition for the diffraction problem [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF]. Let us remark that at a given x, let say x = 0, the evanescent modes (for which β n = i|β n |) are exponentially growing (or decaying, depending on the sign of α) in the cross-section y of the waveguide because of the term e iαβny = e -α|βn|y . Due to this exponential behavior, we can prove that the transverse parts of the modes on straight cross-sections Σ ± a are not Riesz basis of L 2 (Σ ± a ). Now, when considering "deformed" cross-sections, everything goes fine since we recover the orthogonality and completeness of the transverse parts of the modes.

Derivation of the Dirichlet-to-Neumann operator

Let us now come back to our initial problem [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF] where Ω is a locally perturbed waveguide. We denote by Σ ± a,α = Ω ∩ {±X = a} two deformed cross-sections of the waveguide. Let us remark that if A is constant, then Σ ± a,α are oblique straight lines because α(y) = -c 3 y/c 2 , and therefore X = a ⇔ xc 3 y/c 2 = a. Given the traces p| ± Σa,α on Σ ± a,α and recalling that { ψ n } n∈N defined in proposition 2.1 is an orthonormal

Using these explicit expressions of the solution in Ω ± a,α and noticing that A∇p(x, y)•ν| Σ ± a,α = ±c∂ X p(X, Y ), we can derive the DtN operators that map the traces p| ± Σa,α to the normal derivatives A∇p • ν| Σ ± a,α on the boundaries Σ ± a,α :

T ± α : p| Σ ± a,α ∈ H 1/2 (Σ ± a,α ) → T ± α p| Σ ± a,α = n≥0 iβ n p| Σ ± a,α , ψ n c ψ n ∈ H -1/2 (Σ ± a,α ). (13)
With this formulation, we can prove as in [START_REF] Goldstein | A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[END_REF] the following result: Theorem 3.1 Let us suppose that ω is not a cut-off frequency. The problem (14) (or equivalently [START_REF] Givoli | Numerical method for problems in infinite Domains[END_REF] with the radiation condition) admits a unique solution in H 1 (Ω a,α ), except for at most a discrete set of frequencies ω.

Let us emphasize that in some perturbed waveguides, there exist at some frequencies (which are not the cut-off frequencies) trapped modes solutions to the homogeneous problem (14). Thanks to formulation (14), we can numerically compute the solution in Ω ± a,α . To illustrate these "new" TBC, we have considered a particular case of stratified medium: On Figure 3, we have represented the real part of the computed solution for two sizes of Ω a,α . As we can see, the conditions on Σ ± a,α are "transparent" because they have no influence on the computed solution p a .

Ω = R × [-h,

Figure 1 .

 1 Figure 1. Geometry of the waveguide and notations. The black dot represents a defect.

Figure 2 .

 2 Figure 2. Geometry of the waveguide and new notations basis of L 2 ([-h, h]), we can get the associated outgoing modal decomposition of the solution in Ω ± a,α : p(x, y) =

  Thus, we can derive TBCs and reformulate the diffraction problem (1) with the radiation condition in a bounded domain as follows: Find p a ∈ H 1 (Ω a,α ) solution to   -div (A∇p a ) -ω 2 p a = f in Ω a,α , A∇p a • ν = T ± α

1 

 1 , where c 3 (y) = y + h 2h(1 + 0.001) , h = 1, and ω = 10.

Figure 3 .

 3 Figure 3. Solution p a of the problem (14) for two sizes of Ω ± a . The dotted circle represented the support of f .

Acknowledgements

The author would like to thank Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss and Vincent Pagneux for their useful advices and discussions.