Secure Network Coding via Filtered Secret Sharing

Size: px
Start display at page:

Download "Secure Network Coding via Filtered Secret Sharing"

Transcription

1 Secure Network Coding via Filtered Secret Sharing Jon Feldman, Tal Malkin, Rocco Servedio, Cliff Stein (Columbia University) columbiaedu Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p1/21

2 Network Coding and Security Network coding: new model of transmission how do we make it secure? 1 Cai and Yeung[02] wire-tap adversary: can look at any Suff conditions for 2 Jain[04]: More precise cond (one terminal) 3 Ho, Leong, Koetter, Médard, Effros, Karger [04]: Byzantine modification detection edges secure multicast code Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p2/21

3 Network Coding and Security Network coding: new model of transmission how do we make it secure? 1 Cai and Yeung[02] wire-tap adversary: can look at any Suff conditions for 2 Jain[04]: More precise cond (one terminal) 3 Ho, Leong, Koetter, Médard, Effros, Karger [04]: Byzantine modification detection edges secure multicast code This talk: precise analysis of wire-tap adversary, balance between security, rate, edge bandwidth Related: robustness [Koetter Médard 02] Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p2/21

4 Making our Example Secure Use Less ambitious goal: Send one symbol to both sinks Choose randomly Can define symbols st any single wire-tapper learns nothing about, both sinks can compute Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p3/21

5 Linear Multicast Network Coding (No Security) Given: Network min-cut value = Goal: get message, source, sinks to every sink Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p4/21

6 Linear Multicast Network Coding (No Security) Given: Network min-cut value = Goal: get message, source Network code: Define coding vectors Edge carries symbol, sinks to every sink for each edge Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p4/21

7 Linear Multicast Network Coding (No Security) Given: Network min-cut value = Goal: get message, source Network code: Define coding vectors Edge carries symbol Feasibility of transmission: (i) Every spanned by, sinks to every sink for each edge (or ) Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p4/21

8 Linear Multicast Network Coding (No Security) Given: Network min-cut value = Goal: get message, source Network code: Define coding vectors Edge carries symbol Feasibility of transmission: (i) Every spanned by Recoverability at sinks: (ii) For all, the vectors, sinks to every sink for each edge (or span ) Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p4/21

9 Wire-Tap Model, Randomness at the Source Adversary has access to any set of knows symbol transmitted along edge, knows network code, topology, has unlimited computational power edges, Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p5/21

10 Wire-Tap Model, Randomness at the Source Adversary has access to any set of knows symbol transmitted along edge, knows network code, topology, has unlimited computational power Source allowed to generate random symbols ( ) (Jain [04]: random bits at intermediate nodes) edges, Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p5/21

11 Wire-Tap Model, Randomness at the Source Adversary has access to any set of knows symbol transmitted along edge, knows network code, topology, has unlimited computational power Source allowed to generate random symbols ( ) (Jain [04]: random bits at intermediate nodes) edges, Task: design function at source, coding vectors on edges st: Coding vectors satisfy feasibility, Information recoverable at each sink, Information secure against adversary Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p5/21

12 Wire-Tap Model, Randomness at the Source Adversary has access to any set of knows symbol transmitted along edge, knows network code, topology, has unlimited computational power Source allowed to generate random symbols ( ) (Jain [04]: random bits at intermediate nodes) edges, Task: design function at source, coding vectors on edges st: Coding vectors satisfy feasibility, Information recoverable at each sink, Information secure against adversary Goal: information-theoretic security Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p5/21

13 Security, Rate and Bandwidth We study possible trade-offs between security, rate and bandwidth: Security = Rate = = # edges tapped = # information symbols multicast Edge Bandwidth =, where symbols in Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p6/21

14 Security, Rate and Bandwidth We study possible trade-offs between security, rate and bandwidth: Security = Rate = = # edges tapped = # information symbols multicast Edge Bandwidth =, where symbols in Easy to show: Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p6/21

15 Security, Rate and Bandwidth We study possible trade-offs between security, rate and bandwidth: Security = Rate = = # edges tapped = # information symbols multicast Edge Bandwidth =, where symbols in Easy to show: Cai and Yeung [02]: If symbols securely Construction time, can send Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p6/21

16 Our Results If you give up a little capacity, bandwidth requirement reduced significantly: Thm: For any, if, can send symbols securely Algorithm: poly-time, secure whp If, only need Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p7/21

17 Our Results If you give up a little capacity, bandwidth requirement reduced significantly: Thm: For any, if, can send symbols securely Algorithm: poly-time, secure whp If, only need If you do not give up capacity, then bandwidth might have to be large: Thm: If, then there are examples where all solutions (using this method) must have Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p7/21

18 Relation w/ Cai & Yeung Core Lemma of Cai and Yeung: If one can construct a matrix with certain independence properties relative to the coding vectors, then the network code can be altered to achieve security Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p8/21

19 Relation w/ Cai & Yeung Core Lemma of Cai and Yeung: If one can construct a matrix with certain independence properties relative to the coding vectors, then the network code can be altered to achieve security Our extensions: Independence properties are also necessary Using orthogonal space, re-cast as coding theory problem Give up some capacity to make coding problem solvable Use necessary direction, covering radius, to prove negative result Observation: don t alter code, just input Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p8/21

20 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21

21 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Set, with info, random Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21

22 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Set Send message, with info, random normally using network code Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21

23 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Set Send message, with info, random normally using network code Security condition: If adversary looks at any edges, can learn nothing about (More formally, for all sets with is a random vector in, the random variable is independent of ), If Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21

24 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Set Send message, with info, random normally using network code Security condition: If adversary looks at any edges, can learn nothing about (More formally, for all sets with is a random vector in, the random variable is independent of ) Recoverability, feasibility follow immediately (as long as can be determined from ), If Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21

25 Making a Given Network Code Secure [CY02] Given a linear solutionto a network coding problem, can we use it securely? Set Send message, with info, random normally using network code Security condition: If adversary looks at any edges, can learn nothing about (More formally, for all sets with is a random vector in, the random variable is independent of ) Recoverability, feasibility follow immediately (as long as can be determined from ) Advantage: don t need to alter network code Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p9/21, If

26 Secret Sharing (Shamir) Secret Dealer has secret Distribute shares Given any Given any st shares, can recover secret shares, learn nothing Computational/info-theoretic security Access pattern for recoverability/security Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p10/21

27 Connection to Secret Sharing Simple case: single source/sink, adversary has any set of s edges parallel edges, t Modest goal : send one symbol Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p11/21

28 Connection to Secret Sharing Suppose Encoding: Choose = is the dealer in secret sharing at random Let Set message (Encode (x,r) using a Reed-Solomon code) Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p12/21

29 Connection to Secret Sharing Suppose Encoding: Choose = is the dealer in secret sharing at random Let Set message (Encode (x,r) using a Reed-Solomon code) Decoding: Knowing all Knowing Works for any or fewer reveals (interpolation) s tells you nothing Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p12/21

30 Filtered Secret Sharing Classical secret sharing: adversary has shares Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p13/21

31 Filtered Secret Sharing Classical secret sharing: adversary has Filtered secret sharing: Menu of lin combos of all shares Adversary chooses items from the menu shares Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p13/21

32 Filtered Secret Sharing Classical secret sharing: adversary has Filtered secret sharing: Menu of lin combos of all shares Adversary chooses items from the menu Secret is shares Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p13/21

33 Filtered Secret Sharing Classical secret sharing: adversary has Filtered secret sharing: Menu of lin combos of all shares Adversary chooses items from the menu Secret is Given: -by- full-rank filter matrix Find: Function For all -by- over random we have submatrices, indep of over such that: of, ( ) shares Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p13/21

34 Filtered Secret Sharing Given: -by- full-rank filter matrix over over random we have submatrices, Classical: special case indep of, such that: of,, Find: Function For all -by- For network coding:, is -by- matrix of coding vectors Ignores network topology Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p14/21

35 Design of Linear Error-Correcting Codes = linear subspace generated by the rows of Distance( ) = -by- matrix Goal in coding theory: For given rate, find code with large distance Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p15/21

36 Design of Linear Error-Correcting Codes = linear subspace generated by the rows of Distance( ) = -by- matrix Goal in coding theory: For given rate, find code with large distance - Generalization: Designed code must be far from other given points, and some Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p15/21

37 Generalized Coding Problem For generators,, define: Note: Asymmetric Note: min-dist( ) Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p16/21

38 Generalized Coding Problem For generators,, define: Note: Asymmetric Note: min-dist( ) -by- Span Distance Problem (over field Given an -by- matrix, find a matrix where ): Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p16/21

39 Generalized Coding Problem Goal: Design code For generators,, define: Note: Asymmetric Note: min-dist( ) -by- Span Distance Problem (over field Given an -by- matrix, find a matrix where ): codeword in with distance to every Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p16/21

40 Filtered Secret sharing Span Distance Filtered secret sharing (linear of the span distance problem: ) is a special case (linear) fss solution with (for all ) solution to the span distance problem with,, ( generates null space of Required distance = ) Parameter for network coding application: amount of capacity given up Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p17/21

41 Positive result Given any, choose randomly Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p18/21

42 Positive result Given any, choose randomly Analysis like random lin codes on G-V bound: vector in Each has Vol prob of having dist from Union bound over codewords : Random has w/ prob Vol Vol Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p18/21

43 Positive result Given any, choose randomly Analysis like random lin codes on G-V bound: vector in Each has Vol prob of having dist from Union bound over codewords : Random has w/ prob Vol Vol In general, Pr For if, need only Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p18/21

44 Negative Result: Using the Covering Radius Covering radius( ) = Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p19/21

45 Negative Result: Using the Covering Radius Covering radius( ) = If has covering radius alone subspace (let ) has dist no from Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p19/21

46 Negative Result: Using the Covering Radius ) = Covering radius( no has covering radius alone subspace (let ) has dist If from, using result of [Cohen, Frankl 85]: Setting ) Vol st ( st ), Vol Thm: ( and where Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p19/21

47 Negative Result: Using the Covering Radius ) = Covering radius( no has covering radius alone subspace (let ) has dist If from, using result of [Cohen, Frankl 85]: Setting ) Vol st ( st ), Vol Thm: ( and where, where st exists (Contrast to C/Y upper reasonable settings of if ) bound Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p19/21

48 Conclusions Given a fixed linear network code, the problem of making it secure (using linear filtered secret sharing) is a generalized [classical] code design problem To achieve security: trade-off between rate and required link bandwidth Sacrificing small amount of capacity allows large savings in required bandwidth Secret sharing can be extended from [adversary gets [adversary gets linear combinations (from a given set) of all shares] shares],to Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p20/21

49 Future Work Better upper/lower bounds (, or in general) Consider (network topology, code design, security, robustness) simultaneously Allow more power at nodes [Jain: random bits] Relax notion of security [Jain: computationally bounded adversary] Different adversaries, non-linear network codes, non-multicast? Feldman, Malkin, Servedio, Stein: Secure Network Coding via Filtered Secret Sharing p21/21

Weakly Secure Network Coding

Weakly Secure Network Coding Weakly Secure Network Coding Kapil Bhattad, Student Member, IEEE and Krishna R. Narayanan, Member, IEEE Department of Electrical Engineering, Texas A&M University, College Station, USA Abstract In this

More information

Security for Wiretap Networks via Rank-Metric Codes

Security for Wiretap Networks via Rank-Metric Codes Security for Wiretap Networks via Rank-Metric Codes Danilo Silva and Frank R. Kschischang Department of Electrical and Computer Engineering, University of Toronto Toronto, Ontario M5S 3G4, Canada, {danilo,

More information

Secure Network Coding for Wiretap Networks of Type II

Secure Network Coding for Wiretap Networks of Type II 1 Secure Network Coding for Wiretap Networks of Type II Salim El Rouayheb, Emina Soljanin, Alex Sprintson Abstract We consider the problem of securing a multicast network against a wiretapper that can

More information

On Secure Communication over Wireless Erasure Networks

On Secure Communication over Wireless Erasure Networks On Secure Communication over Wireless Erasure Networks Andrew Mills Department of CS amills@cs.utexas.edu Brian Smith bsmith@ece.utexas.edu T. Charles Clancy University of Maryland College Park, MD 20472

More information

Achievable Strategies for General Secure Network Coding

Achievable Strategies for General Secure Network Coding Achievable Strategies for General Secure Network Coding Tao Cui and Tracey Ho Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125, USA Email: {taocui, tho}@caltech.edu

More information

Network Monitoring in Multicast Networks Using Network Coding

Network Monitoring in Multicast Networks Using Network Coding Network Monitoring in Multicast Networks Using Network Coding Tracey Ho Coordinated Science Laboratory University of Illinois Urbana, IL 6181 Email: trace@mit.edu Ben Leong, Yu-Han Chang, Yonggang Wen

More information

Efficient weakly secure network coding scheme against node conspiracy attack based on network segmentation

Efficient weakly secure network coding scheme against node conspiracy attack based on network segmentation Du et al. EURASIP Journal on Wireless Communications and Networking 2014, 2014:5 RESEARCH Open Access Efficient weakly secure network coding scheme against node conspiracy attack based on network segmentation

More information

Network Coding for Security and Error Correction

Network Coding for Security and Error Correction Network Coding for Security and Error Correction NGAI, Chi Kin A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Information Engineering c The Chinese

More information

Secure Network Coding on a Wiretap Network

Secure Network Coding on a Wiretap Network IEEE TRANSACTIONS ON INFORMATION THEORY 1 Secure Network Coding on a Wiretap Network Ning Cai, Senior Member, IEEE, and Raymond W. Yeung, Fellow, IEEE Abstract In the paradigm of network coding, the nodes

More information

Network coding for security and robustness

Network coding for security and robustness Network coding for security and robustness 1 Outline Network coding for detecting attacks Network management requirements for robustness Centralized versus distributed network management 2 Byzantine security

More information

Secure Network Coding: Bounds and Algorithms for Secret and Reliable Communications

Secure Network Coding: Bounds and Algorithms for Secret and Reliable Communications Secure Network Coding: Bounds and Algorithms for Secret and Reliable Communications S. Jaggi and M. Langberg 1 Introduction Network coding allows the routers to mix the information content in packets before

More information

On Secure Network Coding With Nonuniform or Restricted Wiretap Sets

On Secure Network Coding With Nonuniform or Restricted Wiretap Sets 166 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 1, JANUARY 2013 On Secure Network Coding With Nonuniform or Restricted Wiretap Sets Tao Cui, Student Member, IEEE, TraceyHo, Senior Member, IEEE,

More information

A Network Flow Approach in Cloud Computing

A Network Flow Approach in Cloud Computing 1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges

More information

Topology-based network security

Topology-based network security Topology-based network security Tiit Pikma Supervised by Vitaly Skachek Research Seminar in Cryptography University of Tartu, Spring 2013 1 Introduction In both wired and wireless networks, there is the

More information

Comparison of Network Coding and Non-Network Coding Schemes for Multi-hop Wireless Networks

Comparison of Network Coding and Non-Network Coding Schemes for Multi-hop Wireless Networks Comparison of Network Coding and Non-Network Coding Schemes for Multi-hop Wireless Networks Jia-Qi Jin, Tracey Ho California Institute of Technology Pasadena, CA Email: {jin,tho}@caltech.edu Harish Viswanathan

More information

A Numerical Study on the Wiretap Network with a Simple Network Topology

A Numerical Study on the Wiretap Network with a Simple Network Topology A Numerical Study on the Wiretap Network with a Simple Network Topology Fan Cheng and Vincent Tan Department of Electrical and Computer Engineering National University of Singapore Mathematical Tools of

More information

Linear Codes. Chapter 3. 3.1 Basics

Linear Codes. Chapter 3. 3.1 Basics Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

More information

Secure Network Coding: Dependency of Efficiency on Network Topology,

Secure Network Coding: Dependency of Efficiency on Network Topology, 13 IEEE. Reprinted, with permission, from S. Pfennig and E. Franz, Secure Network Coding: Dependency of Efficiency on Network Topology, in IEEE International Conference on Communications (ICC 13), pp.

More information

Lattice-Based Threshold-Changeability for Standard Shamir Secret-Sharing Schemes

Lattice-Based Threshold-Changeability for Standard Shamir Secret-Sharing Schemes Lattice-Based Threshold-Changeability for Standard Shamir Secret-Sharing Schemes Ron Steinfeld (Macquarie University, Australia) (email: rons@ics.mq.edu.au) Joint work with: Huaxiong Wang (Macquarie University)

More information

On the Multiple Unicast Network Coding Conjecture

On the Multiple Unicast Network Coding Conjecture On the Multiple Unicast Networ Coding Conjecture Michael Langberg Computer Science Division Open University of Israel Raanana 43107, Israel miel@openu.ac.il Muriel Médard Research Laboratory of Electronics

More information

On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective

On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective Weiyu Xu and Babak Hassibi EE Department California Institute of Technology

More information

Energy Benefit of Network Coding for Multiple Unicast in Wireless Networks

Energy Benefit of Network Coding for Multiple Unicast in Wireless Networks Energy Benefit of Network Coding for Multiple Unicast in Wireless Networks Jasper Goseling IRCTR/CWPC, WMC Group Delft University of Technology The Netherlands j.goseling@tudelft.nl Abstract Jos H. Weber

More information

Link Loss Inference in Wireless Sensor Networks with Randomized Network Coding

Link Loss Inference in Wireless Sensor Networks with Randomized Network Coding Link Loss Inference in Wireless Sensor Networks with Randomized Network Coding Vahid Shah-Mansouri and Vincent W.S. Wong Department of Electrical and Computer Engineering The University of British Columbia,

More information

Exact Maximum-Likelihood Decoding of Large linear Codes

Exact Maximum-Likelihood Decoding of Large linear Codes On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes arxiv:cs/0702147v1 [cs.it] 25 Feb 2007 Abstract Since the classical work of Berlekamp, McEliece

More information

Kodo: An Open and Research Oriented Network Coding Library Pedersen, Morten Videbæk; Heide, Janus; Fitzek, Frank Hanns Paul

Kodo: An Open and Research Oriented Network Coding Library Pedersen, Morten Videbæk; Heide, Janus; Fitzek, Frank Hanns Paul Aalborg Universitet Kodo: An Open and Research Oriented Network Coding Library Pedersen, Morten Videbæk; Heide, Janus; Fitzek, Frank Hanns Paul Published in: Lecture Notes in Computer Science DOI (link

More information

Efficient Recovery of Secrets

Efficient Recovery of Secrets Efficient Recovery of Secrets Marcel Fernandez Miguel Soriano, IEEE Senior Member Department of Telematics Engineering. Universitat Politècnica de Catalunya. C/ Jordi Girona 1 i 3. Campus Nord, Mod C3,

More information

ON THE APPLICATION OF RANDOM LINEAR NETWORK CODING FOR NETWORK SECURITY AND DIAGNOSIS

ON THE APPLICATION OF RANDOM LINEAR NETWORK CODING FOR NETWORK SECURITY AND DIAGNOSIS ON THE APPLICATION OF RANDOM LINEAR NETWORK CODING FOR NETWORK SECURITY AND DIAGNOSIS by Elias Kehdi A thesis submitted in conformity with the requirements for the degree of Master of Applied Science,

More information

How To Encrypt Data With A Power Of N On A K Disk

How To Encrypt Data With A Power Of N On A K Disk Towards High Security and Fault Tolerant Dispersed Storage System with Optimized Information Dispersal Algorithm I Hrishikesh Lahkar, II Manjunath C R I,II Jain University, School of Engineering and Technology,

More information

Network Coding for Distributed Storage

Network Coding for Distributed Storage Network Coding for Distributed Storage Alex Dimakis USC Overview Motivation Data centers Mobile distributed storage for D2D Specific storage problems Fundamental tradeoff between repair communication and

More information

Functional-Repair-by-Transfer Regenerating Codes

Functional-Repair-by-Transfer Regenerating Codes Functional-Repair-by-Transfer Regenerating Codes Kenneth W Shum and Yuchong Hu Abstract In a distributed storage system a data file is distributed to several storage nodes such that the original file can

More information

Signatures for Content Distribution with Network Coding

Signatures for Content Distribution with Network Coding Signatures for Content Distribution with Network Coding Fang Zhao, Ton Kalker, Muriel Médard, and Keesook J. Han Lab for Information and Decision Systems MIT, Cambridge, MA 02139, USA E-mail: zhaof/medard@mit.edu

More information

Communication Overhead of Network Coding Schemes Secure against Pollution Attacks

Communication Overhead of Network Coding Schemes Secure against Pollution Attacks Communication Overhead of Network Coding Schemes Secure against Pollution Attacks Elke Franz, Stefan Pfennig, and André Fischer TU Dresden, Faculty of Computer Science 01062 Dresden, Germany {elke.franz

More information

Mathematical Modelling of Computer Networks: Part II. Module 1: Network Coding

Mathematical Modelling of Computer Networks: Part II. Module 1: Network Coding Mathematical Modelling of Computer Networks: Part II Module 1: Network Coding Lecture 3: Network coding and TCP 12th November 2013 Laila Daniel and Krishnan Narayanan Dept. of Computer Science, University

More information

In-Network Coding for Resilient Sensor Data Storage and Efficient Data Mule Collection

In-Network Coding for Resilient Sensor Data Storage and Efficient Data Mule Collection In-Network Coding for Resilient Sensor Data Storage and Efficient Data Mule Collection Michele Albano Jie Gao Instituto de telecomunicacoes, Aveiro, Portugal Stony Brook University, Stony Brook, USA Data

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

SECRET sharing schemes were introduced by Blakley [5]

SECRET sharing schemes were introduced by Blakley [5] 206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has

More information

On network coding for security

On network coding for security On network coding for security Keesook Han Tracey Ho Ralf Koetter Muriel Medard AFRL Computer Science ICE EECS Caltech TUM MIT Fang Zhao EECS MIT Abstract The use of network coding in military networks

More information

CIS 700: algorithms for Big Data

CIS 700: algorithms for Big Data CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/big-data-class.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

Linear Network Coding on Multi-Mesh of Trees (MMT) using All to All Broadcast (AAB)

Linear Network Coding on Multi-Mesh of Trees (MMT) using All to All Broadcast (AAB) 462 Linear Network Coding on Multi-Mesh of Trees (MMT) using All to All Broadcast (AAB) Nitin Rakesh 1 and VipinTyagi 2 1, 2 Department of CSE & IT, Jaypee University of Information Technology, Solan,

More information

Design of LDPC codes

Design of LDPC codes Design of LDPC codes Codes from finite geometries Random codes: Determine the connections of the bipartite Tanner graph by using a (pseudo)random algorithm observing the degree distribution of the code

More information

To Provide Security & Integrity for Storage Services in Cloud Computing

To Provide Security & Integrity for Storage Services in Cloud Computing To Provide Security & Integrity for Storage Services in Cloud Computing 1 vinothlakshmi.s Assistant Professor, Dept of IT, Bharath Unversity, Chennai, TamilNadu, India ABSTRACT: we propose in this paper

More information

Notes 11: List Decoding Folded Reed-Solomon Codes

Notes 11: List Decoding Folded Reed-Solomon Codes Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded Reed-Solomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,

More information

Adaptive Linear Programming Decoding

Adaptive Linear Programming Decoding Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

Secret Key Generation from Reciprocal Spatially Correlated MIMO Channels,

Secret Key Generation from Reciprocal Spatially Correlated MIMO Channels, 203 IEEE. Reprinted, with permission, from Johannes Richter, Elke Franz, Sabrina Gerbracht, Stefan Pfennig, and Eduard A. Jorswieck, Secret Key Generation from Reciprocal Spatially Correlated MIMO Channels,

More information

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE/ACM TRANSACTIONS ON NETWORKING 1 A Greedy Link Scheduler for Wireless Networks With Gaussian Multiple-Access and Broadcast Channels Arun Sridharan, Student Member, IEEE, C Emre Koksal, Member, IEEE,

More information

File Sharing between Peer-to-Peer using Network Coding Algorithm

File Sharing between Peer-to-Peer using Network Coding Algorithm File Sharing between Peer-to-Peer using Network Coding Algorithm Rathod Vijay U. PG Student, MBES College Of Engineering, Ambajogai V.R. Chirchi PG Dept, MBES College Of Engineering, Ambajogai ABSTRACT

More information

A Robust and Secure Overlay Storage Scheme Based on Erasure Coding

A Robust and Secure Overlay Storage Scheme Based on Erasure Coding A Robust and Secure Overlay Storage Scheme Based on Erasure Coding Chuanyou Li Yun Wang School of Computer Science and Engineering, Southeast University Key Lab of Computer Network and Information Integration,

More information

Growth Codes: Maximizing Sensor Network Data Persistence

Growth Codes: Maximizing Sensor Network Data Persistence Growth Codes: Maximizing Sensor Network Data Persistence ABSTRACT Abhinav Kamra Columbia University New York, USA kamra@cs.columbia.edu Jon Feldman Google Inc. New York, USA jonfeld@google.com Sensor networks

More information

N O T E S. A Reed Solomon Code Magic Trick. The magic trick T O D D D. M A T E E R

N O T E S. A Reed Solomon Code Magic Trick. The magic trick T O D D D. M A T E E R N O T E S A Reed Solomon Code Magic Trick T O D D D. M A T E E R Howard Community College Columbia, Maryland 21044 tmateer@howardcc.edu Richard Ehrenborg [1] has provided a nice magic trick that can be

More information

A Practical Scheme for Wireless Network Operation

A Practical Scheme for Wireless Network Operation A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving

More information

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

Privacy and Security in the Internet of Things: Theory and Practice. Bob Baxley; bob@bastille.io HitB; 28 May 2015

Privacy and Security in the Internet of Things: Theory and Practice. Bob Baxley; bob@bastille.io HitB; 28 May 2015 Privacy and Security in the Internet of Things: Theory and Practice Bob Baxley; bob@bastille.io HitB; 28 May 2015 Internet of Things (IoT) THE PROBLEM By 2020 50 BILLION DEVICES NO SECURITY! OSI Stack

More information

Physical Layer Security in Wireless Communications

Physical Layer Security in Wireless Communications Physical Layer Security in Wireless Communications Dr. Zheng Chang Department of Mathematical Information Technology zheng.chang@jyu.fi Outline Fundamentals of Physical Layer Security (PLS) Coding for

More information

A Short Summary on What Happens When One Dies

A Short Summary on What Happens When One Dies Beyond the MDS Bound in Distributed Cloud Storage Jian Li Tongtong Li Jian Ren Department of ECE, Michigan State University, East Lansing, MI 48824-1226 Email: {lijian6, tongli, renjian}@msuedu Abstract

More information

Optimizing Congestion in Peer-to-Peer File Sharing Based on Network Coding

Optimizing Congestion in Peer-to-Peer File Sharing Based on Network Coding International Journal of Emerging Trends in Engineering Research (IJETER), Vol. 3 No.6, Pages : 151-156 (2015) ABSTRACT Optimizing Congestion in Peer-to-Peer File Sharing Based on Network Coding E.ShyamSundhar

More information

Diversity Coloring for Distributed Data Storage in Networks 1

Diversity Coloring for Distributed Data Storage in Networks 1 Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Low Delay Network Streaming Under Burst Losses

Low Delay Network Streaming Under Burst Losses Low Delay Network Streaming Under Burst Losses Rafid Mahmood, Ahmed Badr, and Ashish Khisti School of Electrical and Computer Engineering University of Toronto Toronto, ON, M5S 3G4, Canada {rmahmood, abadr,

More information

3/25/2014. 3/25/2014 Sensor Network Security (Simon S. Lam) 1

3/25/2014. 3/25/2014 Sensor Network Security (Simon S. Lam) 1 Sensor Network Security 3/25/2014 Sensor Network Security (Simon S. Lam) 1 1 References R. Blom, An optimal class of symmetric key generation systems, Advances in Cryptology: Proceedings of EUROCRYPT 84,

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net

More information

954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005

954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 Using Linear Programming to Decode Binary Linear Codes Jon Feldman, Martin J. Wainwright, Member, IEEE, and David R. Karger, Associate

More information

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm. Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

More information

Enhancing Wireless Security with Physical Layer Network Cooperation

Enhancing Wireless Security with Physical Layer Network Cooperation Enhancing Wireless Security with Physical Layer Network Cooperation Amitav Mukherjee, Ali Fakoorian, A. Lee Swindlehurst University of California Irvine The Physical Layer Outline Background Game Theory

More information

CODEC: CONTENT DISTRIBUTION WITH (N,K) ERASURE CODE IN MANET

CODEC: CONTENT DISTRIBUTION WITH (N,K) ERASURE CODE IN MANET CODEC: CONTENT DISTRIBUTION WITH (N,K) ERASURE CODE IN MANET Bin Dai, Wenwen Zhao, Jun Yang and Lu Lv Department of Electronic and Information Engineering, Huazhong University of Science and Technology

More information

Ensuring Data Storage Security in Cloud Computing

Ensuring Data Storage Security in Cloud Computing Ensuring Data Storage Security in Cloud Computing Cong Wang 1, Qian Wang 1, Kui Ren 1, and Wenjing Lou 2 1 ECE Department, Illinois Institute of Technology 2 ECE Department, Worcester Polytechnic Institute

More information

Lecture 14: Data transfer in multihop wireless networks. Mythili Vutukuru CS 653 Spring 2014 March 6, Thursday

Lecture 14: Data transfer in multihop wireless networks. Mythili Vutukuru CS 653 Spring 2014 March 6, Thursday Lecture 14: Data transfer in multihop wireless networks Mythili Vutukuru CS 653 Spring 2014 March 6, Thursday Data transfer over multiple wireless hops Many applications: TCP flow from a wireless node

More information

Erasure Coding for Cloud Storage Systems: A Survey

Erasure Coding for Cloud Storage Systems: A Survey TSINGHUA SCIENCE AND TECHNOLOGY ISSNll1007-0214ll06/11llpp259-272 Volume 18, Number 3, June 2013 Erasure Coding for Cloud Storage Systems: A Survey Jun Li and Baochun Li Abstract: In the current era of

More information

Coding and decoding with convolutional codes. The Viterbi Algor

Coding and decoding with convolutional codes. The Viterbi Algor Coding and decoding with convolutional codes. The Viterbi Algorithm. 8 Block codes: main ideas Principles st point of view: infinite length block code nd point of view: convolutions Some examples Repetition

More information

The Complexity of Online Memory Checking

The Complexity of Online Memory Checking The Complexity of Online Memory Checking Moni Naor Guy N. Rothblum Abstract We consider the problem of storing a large file on a remote and unreliable server. To verify that the file has not been corrupted,

More information

On the Locality of Codeword Symbols

On the Locality of Codeword Symbols On the Locality of Codeword Symbols Parikshit Gopalan Microsoft Research parik@microsoft.com Cheng Huang Microsoft Research chengh@microsoft.com Sergey Yekhanin Microsoft Research yekhanin@microsoft.com

More information

We shall turn our attention to solving linear systems of equations. Ax = b

We shall turn our attention to solving linear systems of equations. Ax = b 59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

More information

MOST error-correcting codes are designed for the equal

MOST error-correcting codes are designed for the equal IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 3, MARCH 2007 387 Transactions Letters Unequal Error Protection Using Partially Regular LDPC Codes Nazanin Rahnavard, Member, IEEE, Hossein Pishro-Nik,

More information

A Survey of the Theory of Error-Correcting Codes

A Survey of the Theory of Error-Correcting Codes Posted by permission A Survey of the Theory of Error-Correcting Codes by Francis Yein Chei Fung The theory of error-correcting codes arises from the following problem: What is a good way to send a message

More information

Network Design and Protection Using Network Coding

Network Design and Protection Using Network Coding Network Design and Protection Using Network Coding Salah A. Aly Electrical & Computer Eng. Dept. New Jersey Institute of Technology salah@njit.edu Ahmed E. Kamal Electrical & Computer Eng. Dept. Iowa State

More information

Image Compression through DCT and Huffman Coding Technique

Image Compression through DCT and Huffman Coding Technique International Journal of Current Engineering and Technology E-ISSN 2277 4106, P-ISSN 2347 5161 2015 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet Research Article Rahul

More information

An Application of Visual Cryptography To Financial Documents

An Application of Visual Cryptography To Financial Documents An Application of Visual Cryptography To Financial Documents L. W. Hawkes, A. Yasinsac, C. Cline Security and Assurance in Information Technology Laboratory Computer Science Department Florida State University

More information

Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

More information

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear

More information

SYSTEMATIC NETWORK CODING FOR LOSSY LINE NETWORKS. (Paresh Saxena) Supervisor: Dr. M. A. Vázquez-Castro

SYSTEMATIC NETWORK CODING FOR LOSSY LINE NETWORKS. (Paresh Saxena) Supervisor: Dr. M. A. Vázquez-Castro SYSTEMATIC NETWORK CODING FOR LOSSY LINE NETWORKS Paresh Saxena Supervisor: Dr. M. A. Vázquez-Castro PhD Programme in Telecommunications and Systems Engineering Department of Telecommunications and Systems

More information

On the Security and Efficiency of Content Distribution Via Network Coding

On the Security and Efficiency of Content Distribution Via Network Coding 1 On the Security and Efficiency of Content Distribution Via Network Coding Qiming Li John C.S. Lui + Dah-Ming Chiu Crypto. & Security Department, Institute for Infocomm Research, Singapore + Computer

More information

Network Coding Applications

Network Coding Applications Foundations and Trends R in Networking Vol. 2, No. 2 (2007) 135 269 c 2007 C. Fragouli and E. Soljanin DOI: 10.1561/1300000013 Network Coding Applications Christina Fragouli 1 and Emina Soljanin 2 1 École

More information

Performance Evaluation of Network Coding: Effects of Topology and Network Traffic for Linear and XOR Coding

Performance Evaluation of Network Coding: Effects of Topology and Network Traffic for Linear and XOR Coding JOURNAL OF COMMUNICATIONS, VOL., NO. 11, DECEMBER 9 88 Performance Evaluation of Network Coding: Effects of Topology and Network Traffic for Linear and XOR Coding Borislava Gajic, Janne Riihijärvi and

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

More information

Quantum Network Coding

Quantum Network Coding Salah A. Aly Department of Computer Science Texas A& M University Quantum Computing Seminar April 26, 2006 Network coding example In this butterfly network, there is a source S 1 and two receivers R 1

More information

956 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 4, DECEMBER 2009

956 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 4, DECEMBER 2009 956 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 4, DECEMBER 2009 Biometric Systems: Privacy and Secrecy Aspects Tanya Ignatenko, Member, IEEE, and Frans M. J. Willems, Fellow,

More information

CS263: Wireless Communications and Sensor Networks

CS263: Wireless Communications and Sensor Networks CS263: Wireless Communications and Sensor Networks Matt Welsh Lecture 4: Medium Access Control October 5, 2004 2004 Matt Welsh Harvard University 1 Today's Lecture Medium Access Control Schemes: FDMA TDMA

More information

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION Sérgio Pequito, Stephen Kruzick, Soummya Kar, José M. F. Moura, A. Pedro Aguiar Department of Electrical and Computer Engineering

More information

CODING THEORY a first course. Henk C.A. van Tilborg

CODING THEORY a first course. Henk C.A. van Tilborg CODING THEORY a first course Henk C.A. van Tilborg Contents Contents Preface i iv 1 A communication system 1 1.1 Introduction 1 1.2 The channel 1 1.3 Shannon theory and codes 3 1.4 Problems 7 2 Linear

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

Efficient Data Retrival - A Case Study on Network Coding and Recovery

Efficient Data Retrival - A Case Study on Network Coding and Recovery Sensors 2012, 12, 17128-17154; doi:10.3390/s121217128 Article OPEN ACCESS sensors ISSN 1424-8220 www.mdpi.com/journal/sensors Prioritized Degree Distribution in Wireless Sensor Networks with a Network

More information

Maximum Entropy Functions: Approximate Gács-Körner for Distributed Compression

Maximum Entropy Functions: Approximate Gács-Körner for Distributed Compression Maximum Entropy Functions: Approximate Gács-Körner for Distributed Compression Salman Salamatian MIT Asaf Cohen Ben-Gurion University of the Negev Muriel Médard MIT ariv:60.03877v [cs.it] Apr 06 Abstract

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

On Quality of Monitoring for Multi-channel Wireless Infrastructure Networks

On Quality of Monitoring for Multi-channel Wireless Infrastructure Networks On Quality of Monitoring for Multi-channel Wireless Infrastructure Networks Arun Chhetri, Huy Nguyen, Gabriel Scalosub*, and Rong Zheng Department of Computer Science University of Houston, TX, USA *Department

More information

Secure Way of Storing Data in Cloud Using Third Party Auditor

Secure Way of Storing Data in Cloud Using Third Party Auditor IOSR Journal of Computer Engineering (IOSR-JCE) e-issn: 2278-0661, p- ISSN: 2278-8727Volume 12, Issue 4 (Jul. - Aug. 2013), PP 69-74 Secure Way of Storing Data in Cloud Using Third Party Auditor 1 Miss.

More information