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Network Coding for Error Correction and Security Raymond W. Yeung The Chinese University of Hong Kong Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Outline • • • • • • Introduction Network Coding vs Algebraic Coding Network Error Correction Secure Network Coding Applications of Random Network Coding in P2P Concluding Remarks Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Introduction Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Network Coding Example The Butterfly Network Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 b1 b1 b2 b2 b1 b1 b2 b2 b1 b1 b2 b2 b1 b2 b1 b1+b2 b2 b1+b2 b1+b2 Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Network Coding Example with Two Sources Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 b1 b2 b1 b2 b2 b1+b2 b2 b1 b2 b1 b2 b1 b1 b1+b2 b1+b2 Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Wireless/Satellite Application b1 b1 b2 b1+b2 b1+b2 b2 t=1 t=2 t=3 50% saving for downlink bandwidth! Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Two Themes of Network Coding • When there is 1 source to be multicast in a network, store-and-forward may fail to optimize bandwidth. • When there are 2 or more independent sources to be transmitted in a network (even for unicast), store-and-forward may fail to optimize bandwidth. In short, Information is NOT a commodity! Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Model of a Point-to-Point Network • A network is represented by a graph G = (V,E) with node set V and edge (channel) set E. • A symbol from an alphabet F can be transmitted on each channel. • There can be multiple edges between a pair of nodes. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Single-Source Network Coding • The source node s generates an information vector x = (x1 x2 … xk) Fk. • What is the condition for a node t to be able to receive the information vector x? • Max-Flow Bound. If maxflow(t) < k, then node t cannot possibly receive x. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 The Basic Results • If network coding is allowed, a node t can receive the information vector x iff maxflow(t) ≥ k i.e., the max-flow bound can be achieved simultaneously by all such nodes t. (Ahlswede et al. 00) • Moreover, this can be achieved by linear network coding for a sufficiently large base field. (Li, Y and Cai, Koetter and Medard, 03) Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Global Encoding Kernels of a Linear Network Code • Recall that x = (x1 x2 … xk) is the multicast message. • For each channel e, assign a column vector fe such that the symbol sent on channel e is x fe. The vector fe is called the global encoding kernel of channel e. • The global encoding kernel of a channel is analogous to a column in the generator matrix of a classical block code. • The global encoding kernel of an output channel at a node must be a linear combination of the global encoding kernels of the input channels. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 An Example k = 2, let x = (b1, b2) Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 1 0 1 0 0 1 0 1 b1 b1 b2 b2 1 0 1 1 1 1 0 1 b1 b1+b2 b2 1 1 b1+b2 b1+b2 Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Network Coding vs Algebraic Coding Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Linear Multicast • A message of k symbols from a base field F is generated at the source node s. • A k-dimensional linear multicast has the following property: A non-source node t can decode the message correctly if and only if maxflow(t) k. • By the Max-flow bound, this is also a necessary condition for a node t to decode (for any given network code). • Thus the tightness of the Max-flow bound is achieved by a linear multicast, which always exists for sufficiently large base fields. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 An (n,k) Code with dmin = d • Consider a (n,k) classical block code with minimum distance d. n • Regard it as a network code on an n d 1 combination network. • Since the (n,k) code can correct d-1 erasures, all the nodes at the bottom can decode. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 The n n d 1 Combination Network s n n-d+1 n-d+1 Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 • For the nodes at the bottom, maxflow(t) = n-d+1. • By the Max-flow bound, k maxflow(t) = n-d+1 or d n-k+1, the Singleton bound. • Therefore, the Singleton bound is a special case of the Max-flow bound for network coding. • An MDS code is a classical block code that achieves tightness of the Singleton bound. • Since a linear multicast achieves tightness of the Maxflow bound, it is formally a network generalization of an MDS code. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Two Ramifications of Single-Source Network Coding • The starting point of classical coding theory and information-theoretic cryptography is the existence of a conduit through which we can transmit information from Point A to Point B without error. • Single-source network coding provides a new such conduit. • Therefore, we expect that both classical coding theory and information-theoretic cryptography can be extended to networks. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Network Error Correction Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Point-to-Point Error Correction in a Network • Classical error-correcting codes are devised for point-to-point communications. • Such codes are applied to networks on a link-bylink basis. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Channel Decoder Channel Decoder Network Encoder Channel Encoder Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Motivation for Network Error Correction • Observation Only the receiving nodes have to know the message transmitted; the immediate nodes don’t. • In general, channel coding and network coding do not need to be separated Network Error Correction • Network error correction generalizes classical pointto-point error correction. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Network Codec Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 What Does Network Error Correction Do? • • • A distributed error-correcting scheme over the network. Does not explicitly decode at intermediate nodes as in point-to-point error correction. At a sink node t, if c errors can be corrected, it means that the transmitted message can be decoded correctly as long as the total number of errors, which can happen anywhere in the network, is at most c. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Classical Algebraic Coding y=x+z received vector codeword error vector y, x, and z are all in the same space. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Minimum Distance: Classical Case • • • Hamming distance is the most natural distance measure. For a code C, dmin = min d(v1,v2), where v1,v2 C and v1 v2. If dmin = 2c+1, then C can – – – Correct c errors Detect 2c errors Correct 2c erasures NetCod 2008, Hong Kong January 3-4, 2008 Raymond W. Yeung, CUHK Sphere Packing dmin Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Coding Bounds: Classical Case • Upper bounds – Hamming bound – Singleton bound • Lower bound – Gilbert-Varsharmov bound Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Network Coding yt t s x yu u yv z Raymond W. Yeung, CUHK NetCod 2008, Hong Kong v January 3-4, 2008 Input/Output Relation • • • • The network code is specified by the local encoding kernels at each non-source node. Fix a sink node t. The codeword x, the error vector z, and the received vectors yt are all in different spaces. In this tutorial, we consider only linear network codes. Then yt = x Fs,t + z Ft • where Fs,t and Ft depend on t. In the classical case, Fs,t = Ft are the identity matrix. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Distance Properties of Linear Network Codes (Yang, Y, Zhang 07) • • The network Hamming distance can be defined for linear network codes. Many concepts in algebra coding based on the Hamming distance can be extended to network coding. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 How to Measure the Distance between Two Codewords? • • • • • Fix both the network code and the codebook C, i.e., the set of all possible codewords transmitted into the network. For a sink node t, yt(x,z) = x Fs,t + z Ft For two codewords x1, x2 C , define their distance by Dtmsg(x1,x2) = arg minz wH(z) where the minimum is taken over all error vectors z such that yt(x1,0) = yt(x2,z) , or yt(x1,z) = yt(x2,0) Idea Dtmsg(x1,x2) is the minimum Hamming weight of an error vector z that makes x1 and x2 indistinguishable at node t. Dtmsg defines a metric on the input space of the linear network code. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Minimum Distance for a Sink Node • For a sink node t, dmin,t = minx1x2 Dtmsg(x1,x2) • Each sink node has a different view of the codebook as each is associated with a different distance measure. • dmin,t is the minimum distance as seen by sink node t. • If the codebook C is linear, dmin,t has the following equivalent definition: dmin,t = min { wH(z) : z At } where At = { z : yt(x,z) = 0 for some x C }. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Error Correction/Detection and Erasure Correction for a Linear Network Code • If dmin,t = 2c+1, then sink node t can – Correct c errors – – Detect 2c errors Correct 2c erasures • • Some form of “sphere packing” is at work. Much more complicated when the network code is nonlinear. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Sphere Packing dmin Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Remark on Error Detection • In network coding, some error patterns have no effect on the sink nodes. These are “invisible” error patterns that cannot be (or do not need to be) detected. • Also called “Byzantine modification detection” (Ho et al, ISIT 04) Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Remark on Erasure Correction • In classical algebraic coding, erasure correction has three equivalent interpretation: – A symbol is erased means that it is not available – A symbol is erased means that the erasure symbol is received – The error locations are known. • In our context, erasure correction means that the locations of the errors are known by the sink nodes but not the intermediate nodes. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Coding Bounds for Network Codes • Cai & Y (02, 06) obtained the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for network codes. • These bounds are natural extension of the bounds for algebraic codes. • Let the base field be GF(q), n = mint maxflow(t) and dmin = mint dmin,t Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Upper Bounds • Hamming bound where • Singleton bound . • The Singleton bound is asymptotically tight, i.e., when q is sufficiently large. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Refined Coding Bounds • • Observation Sink nodes with larger maximum flow can have better error correction capability. For a given linear network code, refined Hamming bounds and Singleton bounds specific to the individual sink nodes can be obtained. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Refined Hamming Bound • A network code with rank(Fs,t) = mt, codebook C, and dmin,t > 0, satisfies where , for all sink node t. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Refined Singleton Bound • A network code with rank(Fs,t) = mt, codebook C, and dmin,t > 0, satisfies for all sink node t. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Remark • Note that mt maxflow(t) for all sink nodes t. • Thus the refined Hamming bounds imply the Hamming bound, and the refined Singleton bounds imply the Singleton bound. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Tightness of the Refined Singleton Bounds • These bounds are shown to be asymptotically tight for linear network codes by construction, i.e., it is possible to construct a codebook that achieves tightness of the individual bound at every sink node t for any given linear network code provided that q is sufficiently large. • This implies that for large base fields, only linear transformations need to be performed at the intermediate nodes! No decoding needed. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Construction of Network Codes that Achieve the Refined Singleton bounds • Deterministic algorithms – – – Alg1: Yang, Ngai and Y (ISIT 07) Alg2: Matsumoto (IEICE, 07) obtained an algorithm based on robust network codes. Alg3: Yang and Y (ITW, Bergen 07) • • All these algorithms have almost the same complexity in terms of the field size requirement and time complexity. These algorithms imply that when q is very large, network codes satisfying these bounds can be constructed randomly with high probability. NetCod 2008, Hong Kong January 3-4, 2008 Raymond W. Yeung, CUHK Gilbert Bound • Let ns be the outgoing degree of source node s. • Let t (x, d ) x' F ns : Dtmsg (x' , x) d be the d-ball about x with respect to the metric Dtmsg. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Gilbert Bound • Given a network code, let |C|max be the maximum possible size of the codebook such that dmin,t ≥ dt for each sink node t. Then, where Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Idea of the Gilbert Bound • If dmin,t ≥ dt for each sink node t, then for any x, there exists a codeword v such that Dtmsg(v,x) < dt , otherwise can add one more codeword to the codebook. Thus all the (dt-1)-balls around the codewords cover the whole input space. • v x dt -1 Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Varshamov Bound • Given a set of local encoding kernels, let ωmax be the maximum possible dimension of the linear codebook such that dmin,t ≥ dt for each sink node t. Then, where Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Error Correction Capability of Random Network Codes • Balli, Yan and Zhang 07 – Study the distribution of dmin,t for random network codes based on a refined bound on the probability of decoding error for a random linear network code for multicast. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Algorithms for Network Error Correction Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 For Deterministic and Random Network Codes • Zhang 07 (to appear in IT) – Proposed the minimum rank decoding principle which is equivalent to minimum distance decoding. 1 d – Can decode up to 2 errors for each sink node t. – A fast decoding algorithm for packet networks with random network coding (the same network code is used repeatedly). • Yan, Balli and Zhang 07 – Decoding beyond the error correction capability. • Balli, Yan and Zhang 07 – A hybrid approach that combines link-by-link error detection and network erasure correction. min,t Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 For Random Network Codes • Jaggi, Langberg et al. (INFOCOM 07) – Consider packet networks (the same network code is used repeatedly). – Scenario 1: Alice and Bob has a low-rate secret channel. • A polynomial-time algorithm that achieves the optimal rate asymptotically. – Scenario 2: Alice and Bob has no shared secret. • A polynomial-time algorithm that achieves the Singleton bound asymptotically. • Extendable to the refined Singleton bounds? Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 For Random Network Codes • Koetter and Kschischang (ISIT 07) – Let the input space of the random network code be Fn, where n = mint maxflow(t). – At a sink node t, the transfer matrix is likely to be full rank. – The codebook is the collection of all k-dimensional subspaces of Fn, each called a codeword. – If a codeword A is chosen, then transmit a set of vectors in A that span A. Does not matter which set. – If the transfer matrix at a sink node t is full-rank (with high probability), the received vectors also spans A. – Can be regarded as a more general theoretical framework for random linear network coding. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 • Koetter and Kschischang (cont.) – Thus the codeword can be decoded correctly in the absence of error. – In the presence of error, decoding is done according to a distance measure between subspaces. – Yet to understand the performance of such codes in a given network. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Applications of Network Error Correction Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Errors due to Noise in Channels • Separation of channel coding and network coding is asymptotically optimal provided two conditions are satisfied: 1. All channels are memoryless. 2. The channels are independent. (Borade 02, Song & Y 06) If not, there is no separation theorem. Then applying turbo codes link-by-link does not guarantee optimality. Linear network error-correcting code is an attractive solution for its low encoding complexity. • • • Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Malicious Injection of Errors • Malicious nodes in the network may inject errors deliberately to disturb data transmission. • Classical error correction does not help because redundancy is injected only in time. • Network error correction is a natural solution because redundancy is injected in both time and space. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Further Reading for Network Error Correction • • • • • • • • • • R. W. Yeung and N. Cai, “Network error correction, Part I & II,” Communications in Information and Systems, 2006. First presented at ITW 2002. Ho et al, “Byzantine modification detection in multicast networks using randomized network coding,” ISIT 2004. R. W. Yeung, S.-Y. R. Li, N. Cai and Z. Zhang, Network Coding Theory, now Publishers, 2005 (Foundation and Trends in Communications and Information Theory). S. Yang and R. W. Yeung, “Characterizations of network error correction/detection and erasure correction,” NetCod 2007. Z. Zhang, “Linear network error correction codes in packet networks,” to appear in IEEE IT. S. Yang, C. K. Ngai, and R. W. Yeung, “Construction of linear network codes that achieve a refined Singleton bound,” ISIT 2007. R. Koetter and F. Kschischang, “Coding for errors and erasures in random network coding,” ISIT 2007. S. Yang and R. W. Yeung, “Refined coding bounds for network error correction,” ITW, Bergen 2007. S. Jaggi et al., “Resilient network coding In the presence of Byzantine adversaries”, INFOCOM 2007. Z. Zhang, “Some recent progress in network error correction progress,” NetCod 2008. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 • • • H. Balli, X. Yan, and Z. Zhang, “Error correction capability of random network error correction codes,” submitted to IT. X. Yan, H. Balli, and Z. Zhang, “Decode network error correction codes beyond error correction capability,” submitted to IT. H. Balli, X. Yan, and Z. Zhang, “A hybrid network error correction coding system,” in preparation. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Secure Network Coding Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Problem Formulation • The underlying model is the same as network multicast using network coding except that some sets of channels can be wiretapped. • Let A be a collection of subsets of the edge set E. • A subset in A is called a wiretap set. • Each wiretap set may be fully accessed by a wiretapper. • No wiretapper can access more than one wiretap set. • The network code needs to be designed in a way such that no matter which wiretap set the wiretapper has access to, the multicast message is information-theoretically secure. • The model is a network generalization of secret sharing (Blakley, Shamir, 78) and wiretap channel II (Ozarow and Wyner 84). Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Coding Scheme (Cai-Y 02) • The multicast message is (m,k), where – m is the secure message – k is the key (randomness) • Both m and k are generated at the source node. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Example of a Secure Network Code Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 m-k m+k m-k m+k m-k k k k m+k One of the 3 red channels can be wiretapped m is the secure message k is the key Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Another Example of Secure Network Coding The (1,2)-threshold Secret Sharing Scheme Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 k m+k m-k One of the 3 red channels can be wiretapped m is the secure message k is the key Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Construction of Secure Network Codes • Let n = mint maxflow(t). • A sufficient condition under which a secure linear network code can be constructed has been obtained (Cai and Y, 02 and 07). • Important Special Case If A consists of all the r-subsets of E, where r < n, then we can construct a secure network code with multicast message (m,k) such that |m| = n - r and |k| = r. • For this case, the condition is also necessary. • Interpretation For a sink node t, if r channels in the network are wiretapped, the number of “secure paths” from the source node to T is still at least n - r. So n - r symbols can go through securely. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Idea of Code Construction • Start with a linear network code for multicasting n symbols. • For all wiretap set A A, let fA = { fe : e A }, the set of global encoding kernels of the channels in A. • Let dim(span(fA)) r for all A A. [sufficient condition] • When the base field F is sufficiently large, we can find b1, b2, …, bn-r Fn such that b1, b2, …, bn-r are linearly independent of fA for all A A. • Extend b1, b2, …, bn-r to b1, b2, …, bn-r , bn-r+1 , …, bn to form a basis for Fn, and let let M = [b1 b2 … bn ]. • M is invertible. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 • Let the multicast message be (m,k), with |m| = n-r and |k| = r. • Take a linear transformation of the given linear network code by the matrix M-1 to obtain the desired secure network code. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Optimality of the Cai-Yeung Construction • When the wiretap set A consists of all r-subsets of E, the construction is optimal in terms of – the size of the message (maximum) – the size of the key (minimum). • The proof of the latter involves a set of inequalities due to T. S. Han. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Han’s Inequalities • Let gk H ( X | X ) 1 k n :| | k k • Then g1 g2 … gn. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Algorithms for Secure Network Coding • Jain 2004 – A security protocal that uses both network coding and one-way functions. • Feldman et al, 2004 – A characterization of secure network codes in terms of a generalized distance measure. – A smaller field size can be used by giving up a small amount of overall capacity. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Algorithms for Secure Network Coding • Bhattad and Narayanan 05 – Propose weakly secure network coding for which the wiretaper cannot obtain any “useful” information. – Very simple scheme. – Not information-theoretically secure. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Algorithms for Secure Network Coding • Jaggi, Langberg et al., 07 – An efficient algorithm using random network coding in an unknown network topology that achieves asymptotically the same optimal rate as Cai-Yeung. – Requires repeated use of the same random network code. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Further Reading for Secure Network Coding • • N. Cai and R. W. Yeung, “Secure network coding,” ISIT 02. Full version available upon request. K. Jain, “Security based on network topology against the wiretapping attacking,” IEEE Wireless Comm., Feb 2004. J. Feldman, T. Malkin, C. Stein, R. A. Servedio “On the capacity of secure network coding”, 2004 Allerton Conference. K. Bhattad and K.R. Nayayanan, “Weakly secure network coding,” NetCod 2005. N. Cai and R. W. Yeung, “A security condition for multi-source linear network coding”, ISIT 2007. S. Jaggi et al., “Resilient Network Coding In the Presence of Byzantine Adversaries”, INFOCOM 2007. E. Soljanin and S. El Rouayheb. “On wiretap networks II,” ISIT 2007. • • • • • Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Applications of Random Network Coding in P2P Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 What is Peer-to-Peer (P2P)? • Client-Server is the traditional architecture for content distribution in a computer network. • P2P is the new architecture in which users who download the file also help disseminating it. • Extremely efficiently for large-scale content distribution, i.e., when there are a lot of clients. • P2P traffic occupies at least 70% of Internet bandwidth. • BitTorrent is the most popular P2P system. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 What is Avalanche? • Avalanche is a Microsoft P2P prototype that uses random linear network coding. • It is one of the first applications / implementations of network coding by Gkantsidis and Rodriguez 05. • It has recently been further developed into Microsoft Secure Content Distribution (MSCD). Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 How Avalanche Works? • When the server or a client uploads to a neighbor, it transmits a random linear combination of the blocks it possesses. The linear coefficients are attached with the transmitted block. • Analogy: Color-mixing. • Each transmitted block is some linear combination of the original blocks of the seed file. • Download is complete if enough linearly independent blocks have been received, and decoding can be done accordingly. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 The Butterfly Network: A Review b1 b1 b2 b2 b1 b1+b2 b1+b2 b1+b2 b2 Synchronization here Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 What Exactly is Avalanche Doing? • In Avalanche, there does not seem to be any need of synchronization. • Is Avalanche doing the same kind of network coding we have been talking about? • If not, what is it doing and is it optimal in any sense? Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 A Time-Parametrized Graph t=0 Server 2 1 t=1 t=2 t=3 Client A 4 2 1 1 Client B 2 1 Client C Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Analysis of Avalanche (Y, NetCod 2007) • The time-parametrized graph, not the physical network, is the graph to look at. • By examining the maximum flows in this graph, the following questions can be answered: – When can a client receive the whole file? – If the server and/or some clients leave the system, can the remaining clients recover the whole file? – If some blocks are lost at some clients, how does it affect the recovery of the whole file? Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Some Remarks • Avalanche is not doing the usual kind of random network coding, but it can be analyzed by the tools we are familiar with. • Avalanche minimizes delay with respect to the given transmission schedule if computation is ignored. • Extra computation is the price to pay. • Avalanche provides the maximum possible robustness for the system. • P2P is perhaps the most natural environment for applying random network coding because the subnet is formed on the fly. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Networks with Packet Loss: A Toy Example A B C • One packet is sent on each channel per unit time. • Packet loss rate = 0.1 on each channel. • By using a fountain code, information can be transmitted from A to C at rate (0.9)2 = 0.81. • By using an Avalanche-type system, information can be transmitted from A to C at rate 0.9 = maxflow from A to C. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 An Explanation Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Networks with Packet Loss • By using an Avalanche-type system, the maxflow from the source to the sink (amortized by the packet loss rate) can be achieved automatically, which is the fundamental limit. • Virtually nothing needs to be done. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008 Concluding Remarks • The theory of network coding naturally ramifies in the direction of error correction and information-theoretic cryptography. • The development in these areas of network coding are still in its infancy. • Many potential applications in networking, wireless, information security, etc. • Applications are driven theory. • A lot of very exciting research ahead. Raymond W. Yeung, CUHK NetCod 2008, Hong Kong January 3-4, 2008

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