Abstract:
Described herein is a method and system for hierarchical wireless video with network coding which limits encryption operations to a critical set of network coding coefficients in combination with multi-resolution video coding. Such a method and system achieves hierarchical fidelity levels, robustness against wireless packet loss and efficient security by exploiting the algebraic structure of network coding.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority from Provisional application No. 61/317,532 filed Mar. 25, 2010 under 35 U.S.C. §119(e) which application is hereby incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     This invention was made with government support under Grant No. FA9550-06-1-0155 awarded by the U.S. Air Force and under Contract No. N66001-08-C-2013 awarded by the Space and Naval Warfare Systems Command. The government has certain rights in this invention. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     Not Applicable. 
     FIELD OF THE INVENTION 
     The concepts described herein generally relate to network coding schemes and more particularly to secure network coding for multi-resolution wireless video streaming. 
     BACKGROUND OF THE INVENTION 
     As is known in the art, there has been abundant research aiming at ensuring a reasonable quality of video experience for wireless users. 
     As is also known, the task of providing video streaming of variable quality to a heterogeneous set of receivers with different subscription levels is still an open issue. One challenge is to serve wireless users with video streams that: (i) are of different quality, depending on subscription level; and (ii) provide security guarantees to ensure that only authorized users will access the protected video streams. 
     In order to illustrate this problem, one can consider the scenario illustrated in  FIG. 1 , in which nodes A, B and C are interested in a video stream served by node S, but they have paid for different video qualities, for example different layers of a multi-resolution video stream. Node S can connect to the receivers through three relay nodes R 1 , R 2 , R 3  in wireless range, but with poor channel quality. Due at least in part to the noisy nature of the wireless medium, at least some packets transmitted by node S are lost. Reliable video transmission, however, requires node S to retransmit the lost packets using feedback received from nodes A, B and C. Moreover, relays R 1 , R 2 , R 3  need to synchronize and schedule transmissions to ensure that each node A, B, C receives all packets without duplicates. Under this scenario, video quality can decrease, because some video frames are not delivered in a timely fashion and are therefore skipped. 
     Moreover, given the broadcast property of the wireless medium, nodes that did not have subscription access to certain layers can potentially overhear the transmitted packets. In  FIG. 1 , for example, node B could overhear layer  3  frames. Preventing unauthorized access to certain layers in the presence of relay nodes thus imposes a challenging security problem, in particular because encryption of the complete video stream is often deemed unfeasible in resource-limited mobile terminals. 
     Furthermore, real time decoding of high-quality video already consumes a great deal of processing power, and can become overwhelming in conjunction with resources required for the decryption of large files. Moreover, a lossy wireless medium imposes additional requirements to the security mechanisms, such as robustness to losses and limited synchronization to prevent scheduling problems. 
     To reduce the amount of processing power required, one can reduce the complexity of the decoding by partially encrypting the video data. However, it is relatively difficult to evaluate the degree of security provided by partial encrypting schemes. The use of layered coding in wireless scenarios was seen as promising, but it is likely to yield prioritization and scheduling problems. For instance, some prior art work has shown that even a relatively simple prioritization of a base layer is not a trivial task. 
     SUMMARY OF THE INVENTION 
     In order to address the above and other problems, a technique known as network coding can be used. The network coding approach allows nodes in a network to combine different information flows by means of algebraic operations. This principle leads to an unconventional way of increasing throughput and robustness of highly volatile networks, such as wireless networks, sensor networks and peer-to-peer systems. Network coding is known to have benefits for wireless communications. It is also known that network coding can also reduce, or in some cases even minimize, decoding delay with feedback, making it suitable for multimedia streaming. 
     Described herein is a method and system for hierarchical wireless video with network coding which limits encryption operations to a predetermined set of network coding coefficients in combination with multi-resolution video coding. Such a method and system achieves hierarchical fidelity levels, robustness against wireless packet loss and efficient security by exploiting the algebraic structure of network coding. 
     Protection of a wireless video stream, while increasing the overall robustness to losses and failures, reducing scheduling problems and adding resilience, is also possible using network coding. By viewing the network code as a cipher, it is possible to create a lightweight cryptographic scheme that reduces the overall computational complexity. Thus, network coding inspires a reformulation of the typical separation between encryption and coding for error resilience. 
     It is unnecessary to perform security operations twice, since one can take advantage of the inherent security of this paradigm. As described herein, it is possible to take advantage of the above benefits of network coding to develop and analyze a novel secure network coding architecture for wireless video. Described herein is a multicast setting in which several devices, which are in general heterogeneous and have limited processing capabilities, subscribe to multi-resolution streaming video in a lossy wireless network. 
     Also described are security operations performed at the network coding layer which allow: (i) a reduction in the number of encryption operations while meeting the prescribed security guarantees, (ii) the resulting lightweight security scheme to be combined with efficient layered codes and streaming protocols for wireless video and (iii) matching network coding with scalable video streams, relying on network coding&#39;s asynchronous operation and inherent robustness to link failures and packet loss. Contributions described herein are as follows: (1) a secure scalable network coded method for video streaming designed for delay-sensitive applications that exploits the robustness of network coding with manageable complexity and quantifiable security levels; (2) demonstration of how hierarchical codes for scalable video based on successive refinement can be combined with network coding in scenarios where not all the nodes are authorized to receive the best quality; (3) analytical evaluation of the security properties of the novel scheme described herein, and discussion of performance and implementation in a wireless streaming service; (4) a description of insights and system considerations regarding implementation in real scenarios; and (5) preliminary proof-of-concept for a network coded video architecture in several wireless scenarios via simulation. 
     It has been found that by exploiting the algebraic structure of network coding, the triple goal of hierarchical fidelity levels, robustness against wireless packet loss and efficient security can be achieved and a secure scalable network coded method and system for video streaming designed for delay-sensitive applications that exploits the robustness of network coding with manageable complexity and quantifiable security levels is provided. 
     In accordance with the concepts, systems and techniques described herein, encryption operations are limited to a critical set of network coding coefficients provided by a source node in combination with multi-resolution video coding. The source node utilizes an n×n lower-triangular matrix A, in which n is the number of layers in a group of pictures (GoP). Matrix A is used for encoding at the source only and each non-zero entry of matrix A is an element a ij  chosen uniformly at random from all non-zero elements of the field F q \{0}. The GoP is divided into a plurality of vectors  b   (1)  . . .  b   (w) , each of the vectors having K symbols S 1 -S K  in which the k th  symbol of each vector belongs to a corresponding one of the n layers in the GoP and wherein the number of vectors created is computed as size of GoP/n. At least one symbol of each vector  b   (i)  is encrypted for each use of the encoding matrix wherein the output of the operation of a stream cypher is denoted as a symbol P with a random key  K  as E(P, K ). The encoding matrix A is successively applied to the information symbols to be sent to provide encoded information symbols which comprise a payload of one or more packets. Each of the one or more packets comprise a header and the payload and the header comprises locked and unlocked coefficients. Each line of a first matrix A is encrypted with a corresponding layer key wherein the first matrix A corresponds to a locked coefficients matrix. An n×n identity matrix corresponding to the unlocked coefficients is provided. The one or more packets are encoded in relay nodes in accordance with a random linear network coding (RLNC) protocol wherein algebraic coding is performed on unlocked coefficients, locked coefficients and the payload. The relay nodes identify the layer of a packet by looking at the first non-zero position in the unlocked coefficients, and packets are mixed with packets of the same or lower layers only. 
     In accordance with a further aspect of the concepts, systems and techniques described, a method for streaming video data in a network including a server node, a plurality of relay nodes and one or more receiver nodes, comprises performing a one-time key distribution between the source node and each of the one or more receiver nodes and dividing the video data into more than one group of pictures (GoP), each of the more than one group of pictures having a predetermined time of duration. For each group of pictures (GoP), generating at the source node an n×n lower-triangular matrix A, in which n is the number of layers in the GoP and using matrix A for encoding at the source only with each non-zero entry of matrix A being an element chosen uniformly at random from all non-zero elements of the field F q \{0}. The method further includes dividing the GoP into a plurality of vectors  b   (1)  . . .  b   (w) , each of the vectors having K symbols S 1 -S K  in which the k th  symbol of each vector belongs to a corresponding one of the n layers in the GoP and wherein the number of vectors created is computed as size of GoP/n. The method further comprises encrypting at least one symbol of each vector  b   (i)  for each use of the encoding matrix wherein the output of the operation of a stream cypher is denoted as a symbol P with a random key  K  as E(P, K ) and applying the encoding matrix A successively to the information symbols to be sent to provide encoded information symbols which comprise a payload of one or more packets with each of the one or more packets comprising a header and a payload. The method further comprises encrypting each line of a first matrix A with a corresponding layer key wherein the first matrix A corresponds to a locked coefficients matrix and generating an n×n identity matrix corresponding to the unlocked coefficients wherein the header of the packet comprises the locked and unlocked coefficients. The method further comprises encoding the one or more packets in relay nodes in accordance with a random linear network coding (RLNC) protocol wherein algebraic coding is performed on unlocked coefficients, locked coefficients and payload and the relay nodes identify the layer of a packet by looking at the first non-zero position in the unlocked coefficients, and packets are mixed with packets of the same or lower layers only. 
     In one embodiment, dividing the video data into more than one group of pictures (GoP), comprises dividing the video data into more than one GoPs having a time of duration of one (1) second. 
     In one embodiment, performing algebraic coding on unlocked coefficients, locked coefficients and the payload comprises performing algebraic coding indistinguishably on unlocked coefficients, locked coefficients and the payload. 
     In one embodiment, the method further comprises applying, via the receivers, Gaussian elimination following standard RLNC over the unlocked coefficients and recovering the locked coefficients by decrypting each line of the matrix with the corresponding key and obtaining plaintext by a substitution process. 
     In one embodiment, the protected symbols are encrypted with the key for the lowest level in the network such that all legitimate participants in the protocol can decrypt the locked symbols. 
     In one embodiment, the method further comprises sending a first line of the matrix unencrypted and starting the encryption of symbols at symbol  2  so that layer  1  is accessible by all nodes in the network. 
     In one embodiment, only a single key per layer is used for multi-resolution encryption and wherein the single key is shared among all receivers. 
     In one embodiment, encrypting comprises encrypting the base layer of the GoP in order to achieve maximum security. 
     In one embodiment, composing a payload of the packets includes forming the payload by concatenating all the vectors A(E(b 1 , K ), b 2 , . . . , b x ) T . 
     In one embodiment, encrypting each line of matrix A with a corresponding layer key comprises encrypting each line of matrix A with a corresponding layer key via the source. 
     In one embodiment, a packet from an nth layer corresponds to the nth line of matrix A such that that each packet of layer x includes packets from layers  1 , . . . , x−1, x. 
     In one embodiment, the method further comprises sending a first line of the matrix unencrypted and starting the encryption of symbols at symbol  2  so that layer  1  is accessible by all nodes in the network. 
     In one embodiment, when performing a linear combination of one packet of layer x with a packet of layer y&gt;x, the resulting packet belongs to layer y. 
     In accordance with a still further aspect of the concepts, systems and techniques described herein a method of generating packets for transmission on a network comprises generating an n×n lower triangular matrix in which each non-zero element is chosen uniformly at random out of all non-zero elements of a finite field, dividing plaintext into vectors of elements wherein a first position of each vector is encrypted using a stream cipher and multiplying the matrix by each of the vectors to generate a payload. 
     In one embodiment, coefficients of the matrix are locked using one different key for each line of the matrix and placed in a header of the packets. 
     In one embodiment, the method further includes generating one line of an identity matrix for each line of the locked coefficients and sending the packets out to the network. 
     In one embodiment, generating an n×n lower triangular matrix comprises generating a 3×3 lower triangular matrix. 
     In one embodiment, dividing plaintext into vectors of elements comprises dividing plaintext into vectors of 3 elements. 
     In accordance with a further aspect of the concepts, systems and techniques described, a system for streaming video data in a network comprises a server node, a plurality of relay nodes and one or more receiver nodes. 
     With this particular arrangement, a secure scalable network coded system for streaming video data in a network including a server node, a plurality of relay nodes and one or more receiver nodes is provided. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing features of this invention, as well as the invention itself, may be more fully understood from the following description of the drawings in which: 
         FIG. 1  is a block diagram of a source S which streams video to three sink nodes A, B and C through relay nodes R 1 , R 2  and R 3  in a wireless setting. 
         FIG. 2  is a block diagram of a coding system in which a source node generates multilayer video which is provided to a network encoder and is transmitted via a wireless transmission. 
         FIG. 3  is a layer model in which video data is divided into groups of pictures (GoP) with the duration of 1 second. GoPs are then subdivided into layers. 
         FIG. 4  is a diagrammatical illustration of operations performed at a source node. 
         FIG. 5  is an illustration of the encryption of the locked coefficients. 
         FIG. 6  is a block diagram illustrating modules of one exemplary system implementation (entities that are external to the system, i.e. key distribution and generation of a multiresolution stream, are in dashed. 
         FIG. 7  is plot of size of data to be encrypted for the scheme described herein versus traditional encryption (encryption of the whole data). 
         FIG. 8  is a plot of played rate as a function of loss probability P loss , for the scheme described herein (NC 1 ), three streams with network coding (NC 2 ) and without network coding (WoNC). 
         FIG. 9  is a plot of the load on the server as a function of the loss probability P loss . 
         FIG. 10  is a plot of CDF versus decoding time for loss probability R loss =0.4, for layer  3 . 0   0 . 2   0 . 4   0 . 6   0 . 8   1   
         FIG. 11  is a plot of percentage of skipped segments versus probability of loss, P loss , for layer  3 . 
         FIG. 12  is a plot of percentage of segments played in lower quality as a function of the probability of loss P loss.    
         FIG. 13  is a plot of initial buffering delay as a function of loss probability P loss , for layer  3 . 
         FIG. 14  is a plot of played quality as a function of segment ID for P loss =0.4. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring now to  FIG. 1 , a source or server node S streams video to three sink or receiver nodes A, B and C (or more simply “sinks” or “receivers”) through relay nodes R 1 , R 2  and R 3  in a wireless setting. The probability of dropping a packet in each link (in dashed) is denoted as P loss . The sinks A, B, C subscribed for different video quality, thus one must devise mechanisms to ensure reliable delivery over the wireless medium, and protection against unauthorized access. The operation of source node S is described in detail below (and particularly in conjunction with  FIG. 4  below). 
     Referring now to  FIG. 2 , a source node S generates multilayer video and provides the multilayer video to a network encoder. Network encoder encodes the video (i.e. the video is fed to the network encoder) and is subsequently transmitted through a wireless network having relay nodes R 1 , R 2 , R 3  (e.g. as shown in  FIG. 1 ) to one or more destination or receiving nodes (e.g. nodes A, B, C as shown in  FIG. 1 ). The video is fed to a network encoder and then undergoes the transmission in a wireless network. 
     One concept described herein is directed toward how to generate a secure, scalable stream by matching the multilayer video generated by source node S with the network encoder. 
     In considering a network model and abstractions, one may consider an abstraction of a wireless network where the source S and relay nodes R 1 , R 2 , R 3  only have access to the identifiers of the sinks (e.g. the IP addresses). Thus, there is no centralized knowledge of the network topology or of the encoding functions. 
     Referring briefly to  FIG. 3 , a layer model is shown. One can adopt a model of video layers as described in Z. Liu, Y. Shen, S. S. Panwar, K. W. Ross, and Y. Wang, “Using layered video to provide incentives in p2p live streaming,” in  P 2 P - TV &#39; 07 : Proceedings of the  2007  workshop on Peer - to - peer streaming and IP - TV , New York, N.Y., USA, 2007, pp. 311-316, ACM. It should, however, be appreciated that other layer models may also be used in accordance with the concepts, systems and techniques described herein. 
     As illustrated in  FIG. 3 , video data is divided into “groups of pictures” or GoPs (also interchangeably referred to herein as “video segments”) with a constant time duration. In the exemplary embodiment described, herein the GoPs have a duration of one (1) second. Other durations, may of course, also be used. The data is then encoded into L layers (with four (4) layers being shown in  FIG. 3 ); each layer is divided into a fixed number of packets. It should be noted that each layer is dependent upon all previous layers. That is, layer  1  is necessary to decode layer  2 , layer  2  is necessary to decode layer  3 , etc. 
     Consider a threat posed by a passive attacker with the following characteristics: (1) the attacker can observe every transmission in the network; (2) the attacker has full access to information about the encoding and decoding schemes; (3) the attacker is computationally bounded, and thus unable to break hard cryptographic primitives. 
     The goal of the attacker is to recover the multicast video stream at the highest possible quality. 
     Network coding and security can be accomplished via random linear network coding (RLNC). RLNC is a completely distributed scheme to implement network coding protocols, whereby nodes draw several coefficients at random and use them to form linear combinations of incoming packets. The resulting packet is sent along with the global encoding vector, which records the cumulative effect of the linear transformations suffered by the original packet while on its path from the source to the destination. The global encoding vector enables the receivers to decode by means of Gaussian elimination. 
     Next described are concepts, techniques and systems related to secure network coding for video streaming. 
     Referring now to  FIG. 4 , the operations at a source node (e.g. source node S in  FIGS. 1 and 2 ) are illustrated in  FIG. 4 . In general overview with reference to the exemplary embodiment illustrated in  FIG. 4 , a source node generates a 3×3 lower triangular matrix in which each non-zero element is chosen uniformly at random out of all non-zero elements of a finite field. The plaintext is divided into vectors of 3 elements and the first position of each vector is encrypted using a stream cypher. The matrix is multiplied by each of the vectors to generate the payload. The coefficients of matrix A are locked using one different key for each line of the matrix and placed in the header of the packets. One line of the identity matrix is generated for each line of the locked coefficients. The packets are then sent out to the network. 
     Proceeding now in more detail, the scheme starts with a one-time key distribution between the source node and the receiver nodes (aka sink nodes). As keys can be reused, only one key per layer is needed for multi-resolution encryption (a single key for the single resolution video case), that would be shared among all receiver nodes. Then, for each GoP, the source node generates an n×n lower-triangular matrix A, in which n is the number of layers in the GoP. Matrix A is used for encoding at the source node only. Each non-zero entry of A is an element a ij  chosen uniformly at random from all non-zero elements of the field F q \{0}. 
     The GoP is then divided into vectors  b   (1)  . . .  b   (w) , in which the first symbol of each vector belongs to layer  1 , the next symbol belongs to layer  2 , etc. The number of vectors created is [size of GoP/n] (it should be appreciated that, for clarity, inconsistencies regarding the proportion between the number of symbols in the layers are ignored). Then, at least one symbol of each vector  b   (i)  is encrypted for each use of the encoding matrix. As layers are dependent—layer i is needed to decode layer i+1—a preferred approach is to encrypt the more informative base layer of the GoP in order to achieve maximum security (in this case, b 1  for each vector  b   (i) )). The output of the operation of a stream cypher is denoted as a symbol P with a random key  K  as E(P, K ). Finally, the payload of the packets is composed by applying the encoding matrix A successively to the information symbols to be sent, i.e., the payload is formed by concatenating all the vectors A(E(b 1 , K ), b 2 , . . . , b x ) T . 
     Next, the source encrypts each line of matrix A with the corresponding layer key. Matrix A is the locked coefficients matrix. The source then generates a n×n identity matrix I, which corresponds to the unlocked coefficients. The packets are comprised of the header and the payload. The header includes the locked and unlocked. Note that, because of the nested structure of coding, determined by the triangular matrix, a packet from layer  1  corresponds to the first line of matrix A, a packet from layer  2  corresponds to the second line of matrix A, etc, so that each packet of layer x includes packets from layers  1 , . . . , x−1, x (i.e. a packet from an nth layer corresponds to the nth line of matrix A such that that each packet of layer x includes packets from layers  1 , . . . , x−1, x). Note also that when performing a linear combination of one packet of layer x with a packet of layer y&gt;x, the resulting packet belongs to layer y. 
     The relays encode packets according to the rules of standard RLNC protocols. The algebraic coding is performed indistinguishably on unlocked coefficients, locked coefficients and payload. Relays identify the layer of a packet by looking at the first non-zero position in the unlocked coefficients, and packets are mixed with packets of the same or lower layers only. 
     The receiver nodes apply Gaussian elimination following standard RNC over the unlocked coefficients. The locked coefficients are recovered by decrypting each line of the matrix with the corresponding key. The plaintext is then obtained by forward substitution. Note that the protected symbols should be encrypted with the key for the lowest level in the network (that is,  K   1 ), so that all legitimate participants in the protocol can decrypt the locked symbols. If layer  1  is to be accessible by all nodes in the network, the first line of the matrix should be sent unencrypted and the encryption of symbols should start at symbol  2 . 
     Table I summarizes the scheme operation. What follows is an elaboration on the matching of multiresolution video and security, prioritization and scheduling issues as well as a security analysis. 
     
       
         
               
             
           
               
                 TABLE I 
               
               
                   
               
             
             
               
                 Initialization (source nodes): 
               
               
                 A key management mechanism is used to exchange n shared keys with the sink 
               
               
                 nodes (one for each layer); 
               
               
                 The source node generates a n × n lower triangular matrix A in which each of the 
               
               
                 non-zero entries is an element from the multiplicative group of the finite field, α ε F q \ 
               
               
                 {0}; 
               
               
                 The coefficients corresponding to a distinct line of the n × n identity matrix are 
               
               
                 added to the header of each coded packet. These correspond to the unlocked 
               
               
                 coefficients. 
               
               
                 Each line / of the matrix A is encrypted with shared key K 1  and placed in the header 
               
               
                 of each packet. These coefficients correspond to the locked coefficients; 
               
               
                 The source node applies the matrix A to the packets to be sent, and places them in 
               
               
                 its memory. 
               
               
                 Initialization (relay nodes): 
               
               
                 Each node initializes n buffers, one for each layer in the network. 
               
               
                 Operation at relay nodes: 
               
               
                 When a packet of layer I is received by a node, the node stores the packet in the 
               
               
                 corresponding buffer; 
               
               
                 To transmit a packet of layer I on an outgoing link, the node produces a packet by 
               
               
                 forming a random linear combination of the packets in buffers 1, . . . , /, modifying 
               
               
                 both the unlocked and locked coefficients without distinction, according to the rules 
               
               
                 of standard RLNC based protocols. 
               
               
                 Decoding (sink nodes): 
               
               
                 When sufficient packets are received: 
               
               
                 The sink nodes perform Gaussian elimination on the matrix of unlocked coefficients, 
               
               
                 applying the same operations to the remainder of the packet, thus obtaining the 
               
               
                 original locked coefficients and coded packets; 
               
               
                 The receiver then decrypts the locked coefficients using the corresponding keys Ki 
               
               
                 for level i; 
               
               
                 The receiver performs forward substitution on the packets using the locked 
               
               
                 coefficients to recover the original packets; 
               
               
                 The receiver decrypts the encrypted symbols to form the original plaintext. 
               
               
                   
               
             
          
         
       
     
     Bringing security to multiresolution video may be accomplished via a triangular encoding matrix. As seen, upon generating a new GoP, the source divides it into vectors  b   (1)  . . .  b   (w) , mixing all layers, and applies the matrix A to each of them to obtain the payload, that is: c (i) =A b   (i) . 
     Referring now to  FIG. 5 , a plurality of different key layers are used to encrypt a corresponding plurality of different lines of a matrix A. As illustrated in  FIG. 5 , the encryption of the locked coefficients includes a first layer which corresponds to the first line of the matrix and is encrypted with the key for layer  1 . The remaining locked coefficients are encrypted line by line according to a similar mechanism. This concept achieves security since only the recipients with the corresponding keys can decode the encrypted line, and consequently the layer. 
     It should be appreciated that standard network coding operations can be employed over the unlocked coefficients also when the layers are encrypted with different keys. Furthermore, even if packets from different layers are combined, reverting the operations through the use of unlocked coefficients subsequently reverts all combinations of different layers, so that the original information can be recovered (for simplicity of the discussion, and without loss of generality, one considers matrix A to have one row per layer  3 ). 
     It should be noted that traditional RLNC mixes all packets by using a full square matrix. This, however, is not suitable for layered coding, since it is not possible to extract individual layers unless one matrix is used for each layer. The triangular matrix coding described herein effectively mixes the layers, allowing for differentiated recovery of successive layers by nodes with different access levels, while relying on the dissemination of lower-level packets to achieve the resilience necessary for higher-level packets to be delivered in a timely fashion. Moreover, the triangular matrix form provides priority to the base layer, as all upper layer packets contain the base layer. Thus, the common prioritization and scheduling of the base layer is solved in a natural way. Below is provided a comparison of the concept and scheme described herein with traditional RLNC addressing scheduling and prioritization issues. 
     The choice of a triangular matrix further meets two important requirements. First, it allows removal of the arbitrary delay introduced by a typical RLNC full-matrix at the source, since the source can code packets as soon as they are generated and does not have to wait for the end of the generation to send them. Furthermore, the use of a triangular matrix also allows for a unique mapping between the unlocked and locked coefficients that does not compromise security: a non-zero unlocked coefficient in column i corresponds to the combination of packets p 1 , . . . , p i  inside the corresponding packet. This is a way of determining the layer of a packet at relay nodes and allow the use of the feedback strategies for minimizing the decoding delay mentioned above. 
     Next described is a model used to perform a security analysis. Let A=(α ij ) be the n×n lower triangular encoding matrix used for performing coding at the source. Each of the non-zero coefficients a ij , i≧j is uniformly distributed over all non-zero elements of a finite field F q , q=2 u , and mutually independent. 
     Let the original data, or plaintext, be a sequence of w vectors  b   (1)  . . .  b   (w) , in which  b   (x) =(b 1   (x) , b 2   (x) , . . . , b n   (x) ) T , 1≦x≦w. All vectors  b   (x)  are independent of A. It is assumed that the successive refinement algorithm used to generate scalable video is optimal. Thus, P(B i =b i )=(q−1) −1 , ∀b i εF q \{0}. For simplicity in the proofs, it is assumed that the plaintext is pre-coded to remove zeros. This can be achieved by mapping elements of F q  into F q−1 , thus incurring a negligible rate penalty of (q−1)/q. 
     The proofs are generalized to include more than one encrypted symbol per use of the encoding matrix. Also, m represents the number of encrypted symbols per reuse of the encoding symbols. We abstract from the particular cypher used for locking the coefficients. For the plaintext, the use of a stream cypher is assumed such that the probability of the output of the encoding operation E(P,  K ) is independent of the plaintext P and the distribution of the output is uniform among all non-zero elements of F q \{0}, that is, P(E(P,  K ))=(q−1) −1 . The parameters of the cypher should be adjusted to approximate these criteria. In the proofs, to obtain these properties, one considers the use of a one time pad in which one symbol of the key is used for each symbol of the plaintext that is encrypted. The key is represented by w random vectors K (1)  . . . K (w) , each with m positions (that is, with wm symbols of key in total). Furthermore, P(K i =k i )=(q−1) −1 , ∀k i εF q \{0}. 
     The vector to which the matrix is applied, that is, the vector (E(b 1 ,K 1   (1) , . . . , E(b m   (x) , K m   (x) , b m+1   (x) , . . . , b n   (x) ) T , is denoted  e   (x) . Each payload vector is represented by  c   (x) =(c 1   (x) ) T , where x corresponds to reuse x of A and
 
 C   i   (x) =( min(1,)/Σ/   j=1 ) a   ij   E ( b   j   (x)   ·K   j   (x) +( i /Σ/ l=m+1 ) a   il   b   l   (x) .
 
     In the description herein, random variables are described in capital letters and instances of random variables are represented in lowercase letters. Vectors are represented by underlined letters and matrices are represented in boldface. Without loss of generality, one can abstract from the network structure and consider the payload of all packets together in the security proofs. Characterized below is the mutual information (denoted by I(•; •)) between the encoded data and the two elements that can lead to information disclosure: the encoding matrix and the original data itself. Theorem 1 evaluates the mutual information between the payload and the encoding matrix, and Theorem 2 evaluates the mutual information between the payload and the original data. 
     Theorem 1: The mutual information between A and AE(1), AE(2), . . . , AE(w) is zero:
 
 I ( A;A E     (1)   ,A E     (2)   , . . . ,A E     (w) )=0.
 
     Theorem 1 is a generalization of the result in Equation 24 and shows that the cost of a statistical attack on the encoding matrix is the cost of a brute-force attack on all entries of the matrix, independently of the number of reuses. 
     Theorem 2: The mutual information between  B   (1) , . . . , B   (w)  and A E   (1) , . . . , A E   (w)  is given by the expression:
 
 I (   B     (1)   , . . . , B     (w)  and  A E     (1)   , . . . , A E     (w) )=log( q− 1)max( f ( w,n,m ), 0),
 
where f (w,n,m)=w(n−m)−( n(n+1) / 2 ).
 
     The equation in Theorem 2 shows that the cost of attacking the plaintext is the cost of discovering the encoding matrix. Thus, one gets a threshold at which there is a reduction of the search space needed to attack the plaintext due to multiple reuses of the matrix A. Notice that there is no disclosure of the plaintext with a single use of the encoding matrix. Below the number of uses in the threshold, the mutual information is 0 and thus, it is not possible to perform a statistical attack on the payload. When the number of uses of the encoding matrix surpasses the threshold, the mutual information grows with w. In the extreme case in which the number of encrypted symbols is equal to the number of symbols in the matrix, the mutual information is always zero (however, in this case, one would not require the encoding matrix to be hidden). 
     The triangular matrix grants unequal protection to the layers of the plaintext. One can easily see that the search space for discovering layer i+1 is larger than the search space to discover layer i. Take, for instance, the case in which m=0—then, for layers i and i+1, an attacker needs to guess, respectively, i and i+1 entries of the matrix. 
     It is believed that the expression in Theorem 2 allows fine tuning the trade-off between complexity and security by varying n (the size of the matrix), m (the number of encrypted symbols) and the size of the field. 
     Referring now to  FIG. 6 , an exemplary system includes a source node S, a relay node R and a receiver which comprises a decoder D. Also shown in  FIG. 6  are a multi-resolution stream and a key distribution system K which are illustrated in phantom since they are external to the system. Consider a scenario such as the one in  FIG. 1 , with a system architecture as depicted in  FIG. 6 , the different components of the system and their practical implications are next described. 
     The technique described herein requires shared keys between source nodes and destination nodes. While the specifics of a particular key distribution mechanism are not relevant to the concepts described herein, exemplary key distribution techniques include, but are not limited to, offline pre-distribution of keys or authentication protocols such as Kerberos or a Public Key Infrastructure (PKI). It should be noted that the need for keys to be shared among several legitimate nodes in a network arises frequently in multicast scenarios and is commonly denominated as broadcast encryption or multicast key distribution. Layer I nodes should keep I keys (one for each layer), and thus, the number of keys exchanged is equal to Σ( L / I=1 ) It I , in which t I  represents the number of recipients of layer I in the network and L the total number of layers in the stream. 
     With respect to multiresolution encoder encoding and security the main requirements of security protocols for multimedia streams are: (i) to work with low complexity and high encryption efficiency, (ii) to keep the file format and synchronization information and (iii) to maintain the original data size and compression ratio. As can be seen from the description provided herein, the scheme described herein has been designed to meet criterion (i). Criterion (ii) is codec-dependent, but in general the scheme described herein is able to meet it. Taking, for example, the MJPEG video codec4, one can use the JPEG2000 option of placing all headers from all blocks of the image on the main header of the file and satisfy criterion (ii). Finally, network coding does not change the size or compression ratio of the stream, so the scheme described herein satisfies criterion (iii). 
     As also shown herein, the maximum level of security is obtained when the compression is optimal and yields a result that is nearly uniform. Thus, the scheme described herein imposes a set of parameters for the codec in order to maximize the entropy of the file. In the MJPEG codec, two such coding decisions would be to choose larger tile sizes and maximum compression rate on the arithmetic coding step. Another approach would be to perform an extra data protection step together with compression. The size of the base layer can be seen as another parameter to increase the compression ratio. As an example, in JPEG2000, each encoded symbol increases the resolution of the stream, therefore it is possible to vary the size of each layer taking the constraints of the security mechanism into consideration. 
     The source encoder node S includes security, loss recovery and network coding modules. The security module and its interoperation with network coding are described herein e.g. in conjunction with  FIG. 4  above. 
     However, it should be appreciated that more than one row of the matrix for each layer is used. In that case, the mapping between the unlocked and locked coefficients suffers a shift: if 2 packets per layer are used, a packet with unlocked coefficients vector (1, 1, 0, . . . 0) belongs to layer  1  and a packet with vector (1, 1, 1, 0, . . . 0) belongs to layer  2 . The division of the payload into vectors should also accommodate this shift. Codecs in which each new symbol (decoded in order) contributes to increased resolution of the output video (such as the MJPEG2000) might benefit from an approach with a finer granularity. This granularity can be fine-tuned by the number of lines of the encoding matrix that belong to each layer. Another important system requirement is to use an encryption mechanism for which the ciphertext is of the same size of the plaintext (e.g. AES in stream cipher mode) in order to keep the size of the symbols constant. 
     An important aspect of the encoder is the rate at which intermediate nodes generate and send linear combinations to the receiver. If a relay generates and forwards a linear combination every time an innovative packet from the server is received, then many redundant packets may arrive at destinations. To solve this issue, the server generates a credit for each coded packet, which is further assigned to one of the intermediate relays. Next, only the relay who receives also the credit associated with the packet is allowed to send a linear combination. 
     After transmitting a complete generation, and before streaming the next one, the server starts the loss recovery process. To recover lost packets, the server sends redundant linear combinations for each layer, mixing all packets of the layer. This process continues until all the receivers for that layer can decode or the server has another segment to stream. 
     The network encoder is a component of the wireless relays of the network and includes layer classification and network coding. As described above, packets of layer I should only be combined with packets of lower layers, i.e., I, I−1, . . . 1. This is done in order to maintain the diversity of layers in the network, because when combining a packet of layer I with layer I+1, the layer of the resulting packet is I+1. After classifying the packet, a relay generates and forwards a linear combination if he received the credit assigned to that packet. 
     The decoder is a component of the receiver that includes security, decoding and buffering and feedback. When enough packets are received, the receiver performs Gaussian elimination to decode packets using the unlocked coefficients. The security process corresponds to the recovery of the locked coefficients and encrypted symbols of the payload and is explained above. 
     Since in the scheme described herein relay nodes perform coding on the packets of the same (and lower) layers, the shape of the triangular matrix sent by the source is not kept through the network. Thus, a received packet, even if innovative in terms of rank, might not be decodable immediately. Hence, the system described herein requires a decoding buffer at the receivers. This decoding buffer takes into account the maximum allowable delay of the video stream, similar to the play buffer at the receivers, and will preemptively flush the current undecoded packets if the delay requirement is not met. Once a full layer is decoded, it is stored in the playback buffer. 
     A node starts the playback once it decodes a number of segments in the lowest quality. If a frame is not received until the time of playback, then it is discarded and the subsequent frame is played instead. Likewise, if the frame is available in a lower quality, it is played in a lower quality than the one the node has access to. At time step k the node plays segment k in the quality in which it is available. If the segment was not decoded (not even in the lowest quality), then the node stops the playback process and starts buffering. If after some buffering timeout, the node decodes segment k, then it plays it in the quality in which it is available; otherwise, the node skips segment k and plays the next one. 
     Considering a system with minimal feedback, in order to free the wireless channels from unnecessary transmissions, the receivers send positive feedback to the server whenever they decode a segment in the desired quality. For example, a layer  3  receiver sends a unique feedback packet when it has decoded layers  1 ,  2  and  3 . 
     Next described is an evaluation of the system described herein in terms of security complexity as well as an evaluation of system performance in a lossy wireless scenario. 
     Referring now to  FIG. 7 , a volume of data to be encrypted according to the size of the plaintext for the scheme described herein is compared with traditional encryption, for typical packet sizes of 500 bytes (for video packets in cellular networks), 1000 bytes (for example, for video over wifi networks) and 1500 bytes (the typical IP packet size). In this example, one encrypted symbol per generation is assumed. For traditional encryption mechanisms, which perform end-to-end encryption of the entire payload, the volume of data that must be encrypted increases linearly with the size of the protected payload. It is not difficult to see that the scheme described herein substantially reduces the size of information to be encrypted. The gains get higher as the maximum size of the packet increases, since the number of matrices to be generated is smaller, and more data can be sent in each packet containing the same matrix of coefficients. 
     Naturally, the required number of cryptographic operations is directly related to the volume of data to be encrypted. If one considers a stream cipher, the number of encryption operations increases linearly with that volume, and therefore, the computational complexity is greatly reduced by the novel scheme described herein as shown in  FIG. 7 . Note that these values are indicative only, and correspond to the theoretical gains when the size of the packet is the only parameter determining the number of reuses of the encoding matrix. The security penalty, which is quantified in above, is not considered for the purposes of this analysis. Note as well that the end values depend on the design of the codec, as well as on the size chosen for each layer. 
     Communication and Computational overhead are next discussed. 
     The ability to reduce the volume of data to be encrypted comes at the cost of including locked coefficients in the data packet. 
     Table II shows the overhead introduced by the novel scheme described herein for each packet and for coefficients with size of 8 and 16 bits, for some values of reference for wireless networks with nodes with several processing capabilities. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 VOLUME OVERHEAD OF LOCKED COEFFICIENTS 
               
               
                 (PER PACKET). 
               
             
          
           
               
                   
                   
                   
                 OVERHEAD 
                   
               
               
                   
                 MAXIMUM IP 
                 #CODED 
                 IN F q   
               
             
          
           
               
                   
                 PACKET SIZE 
                 PACKETS h 
                 q = 2 8   
                 q = 2 16   
               
               
                   
                   
               
             
          
           
               
                   
                 500 
                 4 
                 0.80% 
                 1.60% 
               
               
                   
                   
                 8 
                 1.60% 
                 3.20% 
               
               
                   
                   
                 12 
                 3.20% 
                 6.40% 
               
               
                   
                 1000 
                 4 
                 0.40% 
                 0.80% 
               
               
                   
                   
                 8 
                 0.80% 
                 1.60% 
               
               
                   
                   
                 12 
                 2.40% 
                 4.80% 
               
               
                   
                 1500 
                 4 
                 0.27% 
                 0.53% 
               
               
                   
                   
                 8 
                 0.53% 
                 1.07% 
               
               
                   
                   
                 12 
                 0.80% 
                 1.60% 
               
               
                   
                   
               
             
          
         
       
     
     Note that the inclusion of locked and unlocked coefficients allows avoidance of the use of homomorphic hash functions, which are very expensive in terms of computation. 
     Due to the inclusion of an extra set of coefficients (the locked coefficients), the novel scheme described herein requires additional operations, which are shown in Table III. For the purpose of the analysis described herein, it is considered that, in comparison to the multiplication, the sum operation yields negligible complexity. 
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE III 
               
             
             
               
                   
               
               
                 COMPUTATIONAL COST OF INCLUDING THE LOCKED 
               
               
                 COEFFICIENTS 
               
             
          
           
               
                   
                   
                 DETAILED 
                 TOTAL 
               
               
                 NODE 
                 OPERATION 
                 COST 
                 COST 
               
               
                   
               
               
                 Source 
                 Generation of vectors of 
                 negligible 
                 — 
               
               
                 Node 
                 identity matrix 
               
               
                   
                 Encryption of locked 
                 See Section V-A1 
               
               
                   
                 coefficients 
               
               
                 Relay 
                 Performing extra random 
                 nh multiplication 
                 O(nt) 
               
               
                 Node 
                 linear operations on locked 
                 operations and (n − 
               
               
                   
                 coefficients (combining t 
                 1)h sum 
               
               
                   
                 packets) 
                 operations 
               
               
                 Sink 
                 Decrypt locked coefficients 
                 See Section V-A1 
                 O(n2) 
               
               
                 Node 
                 to obtain the matrix ML of 
               
               
                   
                 plain-text locked coefficients 
               
               
                   
                 Forward-substitution using 
                 O(n2) 
               
               
                   
                 recovered locked 
               
               
                   
                 coefficients 
               
               
                   
                 Decrypt one encrypted 
                 See Section V-A1 
               
               
                   
                 symbol per use of the 
               
               
                   
                 encoding matrix 
               
               
                   
               
             
          
         
       
     
     Next described is wireless video performance. An evaluation is provided of the performance of the protocol described above in the multi-hop multi-path scenario from  FIG. 1 , in which the server S sends video to three (3) heterogeneous receivers A, B and C, through relays R 1 , R 2  and R 3 , over lossy wireless links. In the description hereinbelow, the focus is solely on the performance of the scheme in terms of throughput and robustness to losses, and its ability to deliver quality video to a heterogeneous set of receivers. The novel layered network coding model (scheme NC 1 ) described herein is compared to a standard RLNC (scheme NC 2 ) and also to an implementation without network coding (scheme WoNC). In scheme NC 2  the server sends a different stream for every layer. Each segment is encoded in different qualities, using a full coefficient matrix for each layer. Relay nodes perform RLNC operations on the received packets that belong to the same generation and to the same or lower layers. In this case, since a sink of layer L needs to receive a full-rank matrix for layers  1 ,  2 , . . . L, sinks acknowledge each layer that they decode. Error recovery is similar to scheme NC 1 . In scheme WoNC, the server sends the native packets without coding them. In this case, the intermediate nodes just forward uncoded packets normally. The sinks send as feedback the ids of the packets they received. If some packets are lost, the server retransmits them. 
     A simulation setup is next described. The ns-2 simulator 2.33 described in S. Mccanne, S. Floyd, and K. Fall, “ns2 (network simulator 2),” http://www-nrg.ee.lbl.gov/ns/ with the default random number generator is used for this version. The network coding libraries are independently programmed. The video stream is a constant bit rate traffic over UDP, where the server is streaming at 480 kbps during 100 seconds. Each layer has a fixed size of 20 packets and three (3) layers for the system are considered. This yields a generation of 60 packets, corresponding to 1 second of video. The packet size is 1000 bytes. As a propagation model, two-ray ground is used and the loss probability P loss  is taken as a simulation parameter. Since it was shown that RTS/CTS has a negative impact on the performance, it was disabled for all experiments. In order to simulate heavy loss conditions, MAC layer retransmissions were also disabled. The rate at the MAC layer is 11 Mbps. 
     The receivers start to playback the video stream once they have decoded at least five (5) segments of the lowest quality. The buffering timeout for a segment that has not been decoded until its playback deadline arrives is set to one (1) second. Furthermore, a perfect feedback channel is assumed (that is, no feedback packets are lost). In order to take full advantage of the broadcast nature of the wireless medium, the relays listen to transmitted packets in promiscuous mode. 
     The following metrics: (i) played rate at the receivers, (ii) initial buffering delay, the time interval from receiving the first packet to the beginning of the playback, (iii) decoding delay, the time elapsed from receiving the first packet of a segment until that segment is decoded, (iv) skipped segments, percentage of segments skipped at playback, (v) lower quality segments, percentage of segments played in lower quality than the one requested, (vi) playback quality, average quality in which each segment is played and (vii) load on the server, defined as the ratio between the total rate sent by the server and the streaming rate. In all plots, each point is the average of 10 runs and the vertical lines show the standard deviation. 
       FIGS. 8-14  illustrate results achieved via the concepts, techniques and systems described herein. 
     Referring now to  FIG. 8 , the rate played by each receiver vs. loss probability is shown. Played rate as a function of loss probability P loss , for the technique described herein (NC 1 ), three streams with network coding (NC 2 ) and without network coding (WoNC) as shown. As can be seen from examination of  FIG. 8 , scheme NC 1  and scheme NC 2  are less affected by losses, due to the inherent reliability of network coding in volatile environments, with the scheme described herein performing consistently better. Scheme WoNC, as expected, performs poorly as the medium becomes unreliable. 
     Referring now to  FIG. 9 , the load on the server in function of the loss probability P loss  is shown. One can see in  FIG. 9  that the load on the server grows exponentially as the loss increases. In general, the network coding approaches need to send less coded packets to recover losses. At P loss =0.9, the load is slightly higher for network coding since the server preemptively sends redundant packets until it receives the feedback from the receiver that the segment is decoded, while for scheme WoNC the server retransmits packets only when it receives feedback from the receivers. Since most of the packets are dropped, scheme WoNC never retransmits. 
     Referring now to  FIG. 10  CDF of decoding delay for loss probability P loss =0.4, for layer  3  is shown.  FIG. 10  shows that the network coding approaches are able to decode segments within a second as the server sends redundant linear combinations in a feed-forward manner. Scheme WoNC needs a longer decoding time, because the server waits for the feedback before retransmitting. The plot shown corresponds to a layer  3  receiver and the behavior for other layers is similar. 
     Referring now to  FIGS. 11 and 12 , these figures show the percentage of segments that are skipped and played in lower quality, respectively. Note that with network coding, no segments are skipped for any layers, and, as expected, more segments are played in lower quality as the losses increase. On the other hand, without network coding, there are fewer segments played in lower quality, but at the same time the percentage of skips grows significantly with ploss, because the packets retransmitted by the server do not arrive at the receivers in due time. This effect is exacerbated at higher losses, where no segment is ever played (and hence never skipped either). 
     Referring now to  FIG. 13 , Initial buffering delay in function of loss probability ploss, for layer  3  is shown. One can see in  FIG. 13  that for our scheme, the receivers buffer for a shorter time before starting the playback. The initial buffering delay grows slowly with the probability of loss, because a single network coded packet can recover multiple losses. For scheme WoNC, when losses are high, the receivers are not able to decode anything, thus they never start to play the file. 
     The plots shown in  FIGS. 11 and 13  correspond to layer  3 . The behavior for other layers is similar and slightly better, since layer  3  receivers need to receive more packets than lower layer nodes. 
     Referring now to  FIG. 14 , a plot of Played quality for P loss =0.4 is shown.  FIG. 14  shows the average quality in which every segment is played, when P loss =0.4. A skipped segment accounts as played in a quality equal to 0. Note that the network coding approaches show a high resilience to errors and the video file is constantly played in the desired quality by each receiver compared to scheme WoNC, again with our scheme showing better performance. 
     Finally, it should be noted that the scheme described herein outperforms scheme NC 2  due to the triangular encoding matrix used for coding and to the nested structure of the video layers. These characteristics result in a higher robustness to losses ( FIG. 8 ), better video quality with fewer skips and fewer segments played in lower quality ( FIG. 12 ) and shorter buffering delay ( FIG. 13 ). 
     Described herein is a practical scheme for scalable video streaming that exploits the algebraic characteristics of random linear network coding (RLNC). 
     On the one hand, the concepts, systems and schemes described herein ensure differentiated levels of security for distinct users. On the other hand, the properties of the network coding paradigm assure the resilience to packet losses over wireless channels. The security evaluation proves that it is possible to reduce significantly the number of encryption operations (or, equivalently, the complexity requirements) while quantifying the security levels. 
     It should be noted that the system and techniques described herein were focused on eavesdropping attacks. Network pollution attacks can be dealt with using conventional techniques in albeit some conventional techniques have added in terms of delay and complexity. 
     As part of our ongoing work we are looking at ways to mitigate the effects of such Byzantine attacks under the real-time constraints of streaming services. 
     Having described preferred embodiments of the invention it will now become apparent to those of ordinary skill in the art that other embodiments incorporating these concepts may be used. Accordingly, it is submitted that that the invention should not be limited to the described embodiments but rather should be limited only by the spirit and scope of the appended claims.