Adversarial Network Coding

A combinatorial framework for adversarial network coding is presented. Channels are described by specifying the possible actions that one or more (possibly coordinated) adversaries may take. Upper bounds on three notions of capacity—the one-shot capacity, the zero-error capacity, and the compound zero-error capacity—are obtained for point-to-point channels, and generalized to corresponding capacity regions appropriate for multi-source networks. A key result of this paper is a general method by which bounds on these capacities in point-to-point channels may be ported to networks. This technique is illustrated in detail for Hamming-type channels with multiple adversaries operating on specific coordinates, which correspond, in the context of networks, to multiple adversaries acting on specific network edges. Capacity-achieving coding schemes are described for some of the considered adversarial models.

[1]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

[2]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[3]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[4]  Zhen Zhang,et al.  Weight properties of network codes , 2008, Eur. Trans. Telecommun..

[5]  Christina Fragouli,et al.  Code construction for multiple sources network coding , 2009, MobiHoc S3 '09.

[6]  Jörg Kliewer,et al.  Equivalence for networks with adversarial state , 2014, 2014 IEEE International Symposium on Information Theory.

[7]  Ryutaroh Matsumoto.  Construction algorithm for network error-correcting codes attaining the Singleton bound , 2007, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[8]  Sidharth Jaggi,et al.  Multiple-Access Network Information-Flow and Correction Codes , 2010, IEEE Transactions on Information Theory.

[9]  Alon Orlitsky,et al.  Zero-Error Information Theory , 1998, IEEE Trans. Inf. Theory.

[10]  Philippe Delsarte,et al.  Bilinear Forms over a Finite Field, with Applications to Coding Theory , 1978, J. Comb. Theory A.

[11]  R. Yeung,et al.  Characterizations of Network Error Correction / Detection and Erasure Correction , 2007 .

[12]  Frank R. Kschischang,et al.  On Metrics for Error Correction in Network Coding , 2008, IEEE Transactions on Information Theory.

[13]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[14]  Ning Cai,et al.  Network Error Correction, II: Lower Bounds , 2006, Commun. Inf. Syst..

[15]  Tracey Ho,et al.  Byzantine Modification Detection in Multicast Networks With Random Network Coding , 2008, IEEE Transactions on Information Theory.

[16]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2008, IEEE Trans. Inf. Theory.

[17]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[18]  Lang Tong,et al.  Polytope codes against adversaries in networks , 2010, ISIT.

[19]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[20]  Frank R. Kschischang,et al.  An Introduction to Network Coding , 2012 .

[21]  Ning Cai,et al.  Network Error Correction, I: Basic Concepts and Upper Bounds , 2006, Commun. Inf. Syst..

[22]  Ronald L. Rivest,et al.  Introduction to Algorithms, 3rd Edition , 2009 .

[23]  F. Kschischang,et al.  Constricting the Adversary : A Broadcast Transformation for Network Coding , 2007 .

[24]  Suhas N. Diggavi,et al.  Noncoherent multisource network coding , 2008, 2008 IEEE International Symposium on Information Theory.

[25]  Tracey Ho,et al.  On noncoherent correction of network errors and erasures with random locations , 2009, 2009 IEEE International Symposium on Information Theory.

[26]  Zhen Zhang,et al.  Linear Network Error Correction Codes in Packet Networks , 2008, IEEE Transactions on Information Theory.

[27]  Tracey Ho,et al.  Correction of adversarial errors in networks , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[28]  Tracey Ho,et al.  Resilient network coding in the presence of Byzantine adversaries , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.

[29]  R. Koetter,et al.  An algebraic approach to network coding , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[30]  Suhas N. Diggavi,et al.  On the capacity of multisource non-coherent network coding , 2009, 2009 IEEE Information Theory Workshop on Networking and Information Theory.

[31]  Michael Langberg,et al.  Adversarial models and resilient schemes for network coding , 2008, 2008 IEEE International Symposium on Information Theory.

[32]  Shenghao Yang,et al.  Construction of Linear Network Codes that Achieve a Refined Singleton Bound , 2007, 2007 IEEE International Symposium on Information Theory.

[33]  Shenghao Yang,et al.  Refined Coding Bounds for Network Error Correction , 2007, 2007 IEEE Information Theory Workshop on Information Theory for Wireless Networks.