A Numerical Study on the Wiretap Network with a Simple Network Topology
|
|
|
- Elinor Allen
- 8 years ago
- Views:
Transcription
1 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 Information-Theoretic Security Workshop September, 23 25, 2015
2 Problem Statement Source Sink
3 Problem Statement 1: 2: Wiretappers Source Sink
4 Problem Statement Wiretap Network Strategy Source M: message K: random key, I(M; K) = 0 Intermediate: mix the ciphertext Sink: recover M A: no information about M
5 Problem Statement Wiretap Network Strategy Source M: message K: random key, I(M; K) = 0 Intermediate: mix the ciphertext Sink: recover M A: no information about M Goal Maximize H(M) Minimize H(K)
6 Problem Statement Wiretap Network Strategy Source M: message K: random key, I(M; K) = 0 Intermediate: mix the ciphertext Sink: recover M A: no information about M Goal Maximize H(M) Minimize H(K) Very hard!
7 Problem Statement Our Model X 1 X 4 S T R X 2 X 5 X 3 X 6
8 Problem Statement Our Model X 1 X 4 S T R X 2 X 5 X 3 X 6 Objective: For a given wiretap pattern A: τ A = min H(K) H(M)
9 Problem Statement Our Model X 1 X 4 S T R X 2 X 5 X 3 X 6 Objective: For a given wiretap pattern A: τ A = min H(K) H(M) Question: Is routing optimal?
10 Outline Literature review Problem formulation and methods Results and open problems
11 Wiretap Network N. Cai and R. W. Yeung, Secure network coding, 2002 IEEE International Symposium on Information Theory. N. Cai and R. W. Yeung, Secure Network Coding on a Wiretap Network, IEEE Transactions on Information Theory, vol. 57, no. 1, pp , Jan Sources Sinks Wiretap network (G = (V, E), S, U, A)
12 Wiretapper, Wiretap Set, Wiretap Pattern X 1 X 4 S T R X 2 X 5 X 3 X 6 Wiretapper Wiretap Set Wiretapper Pattern A, wiretap pattern: A = {A 1, A 2,..., }, where A i E
13 Wiretapper, Wiretap Set, Wiretap Pattern X 1 X 4 S T R X 2 X 5 X 3 X 6 Wiretapper Wiretap Set Wiretapper Pattern A, wiretap pattern: A = {A 1, A 2,..., }, where A i E Example Three Wiretappers with wiretap sets A 1 = {X 1, X 2 }, A 2 = {X 2, X 3, X 5 }, A 3 = {X 1, X 2, X 4, X 6 }. A = {A 1, A 2, A 3 }. (H(M), H(K))? when A is given.
14 Wiretapper, Wiretap Set, Wiretap Pattern X 1 X 4 S T R X 2 X 5 X 3 X 6 Wiretapper Wiretap Set Wiretapper Pattern A, wiretap pattern: A = {A 1, A 2,..., }, where A i E Example Three Wiretappers with wiretap sets A 1 = {X 1, X 2 }, A 2 = {X 2, X 3, X 5 }, A 3 = {X 1, X 2, X 4, X 6 }. A = {A 1, A 2, A 3 }. (H(M), H(K))? when A is given. 63 subsets of E 2 63 wiretap patterns.
15 Existing Results on Wiretap Networks 1. Single source multiple sinks, A = {W : W = r, W E}: H(M) (n r) log q; H(K) r/(n r)h(m). (Cai and Yeung, 2010) 2. General multi-source multi-sink wiretap network General entropic region problem. (Chan and Grant, 2014)
16 Existing Results on Wiretap Networks 1. Single source multiple sinks, A = {W : W = r, W E}: H(M) (n r) log q; H(K) r/(n r)h(m). (Cai and Yeung, 2010) 2. General multi-source multi-sink wiretap network General entropic region problem. (Chan and Grant, 2014) Open problem Single source and single sink network with arbitrary A.
17 CutSet Bound F. Cheng and R. W. Yeung, Performance Bounds on a Wiretap Network with Arbitrary Wiretap Sets, IEEE Trans. Inform. Theory, vol. 60, no. 6, pp , Jun Graph Cut x 1 x 2 T Cutset bound H(K) H(M) max s.t. n x i 1 i=1 0 x i 1, 1 i n x j 1, A A e j A x n 1 x n Assume H(K) = 1 S
18 Problem Formulation: a Simple Network Network model X 1 X 4 S T R X 2 X 5 X 3 X 6 (3, 3) wiretap network G = (V, E) S = {S}, U = {R} A 2 E
19 Entropy Equations Network model X 1 X 4 S T R X 2 X 5 X 3 X 6 (3, 3) wiretap network Source: I(M; K) = 0 H(X 1, X 2, X 3 M, K) = 0 Node T: H(X 4, X 5, X 6 X 1, X 2, X 3 ) = 0 Sink: H(M X 4, X 5, X 6 ) = 0 (Level-I) or H(M, K X 4, X 5, X 6 ) = 0 (Level-II) Security: I(X A ; M) = 0, A A Question τ A = min H(K) H(M) For a fixed A, is routing optimal? Numerical Study!
20 Lower and Upper Bounds on τ A Lower bounds: Cutset bound, Shannon Bound Upper bounds: Routing bound, Linear network coding bound Cutset Shannon τ A Linear network coding Routing
21 Shannon Bound All information measures H( ), I( ; ) are non-negative; H(X 1 X 2 ), H(X 1, X 2, X 3 ), I(X; Y) 0 Linearity exists among information measures; H(X 1, X 2 ) = H(X 1 X 2 ) + H(X 2 ) All information inequalities are linear; H(K) H(M) Idea: choose some as decision variables to represent the others.
22 Shannon Bound All information measures H( ), I( ; ) are non-negative; H(X 1 X 2 ), H(X 1, X 2, X 3 ), I(X; Y) 0 Linearity exists among information measures; H(X 1, X 2 ) = H(X 1 X 2 ) + H(X 2 ) All information inequalities are linear; H(K) H(M) Idea: choose some as decision variables to represent the others. Shannon Cipher System Message: X 1, Key: X 2, Ciphertext: X 3 : X 2 X 1 S X 3 R X 1 I(X 1 ; X 2 ) = 0; H(X 3 X 1, X 2 ) = 0; H(X 1 X 2, X 3 ) = 0; I(X 1 ; X 3 ) = 0 Objective: H(X 2 ) H(X 1 ) (Perfect Secrecy Theorem)
23 Shannon Bound (cont d) Decision variables: H(X 1 X 2, X 3 ), H(X 2 X 1, X 3 ), H(X 3 X 1, X 2 ), I(X 1 ; X 2 ), I(X 1 ; X 2 X 3 ),I(X 1 ; X 3 ), I(X 1 ; X 3 X 2 ),I(X 2 ; X 3 ), I(X 2 ; X 3 X 1 ) Denote z := H(X 1 X 2, X 3 ) H(X 2 X 1, X 3 )... I(X 2 ; X 3 X 1 ) H(X 2 ) H(X 1 ) c T z 0 I(X 1 ; X 2 ) = 0; H(X 3 X 1, X 2 ) = 0; H(X 1 X 2, X 3 ) = 0; I(X 1 ; X 3 ) = 0 a z 0 Az 0 min c T z, s.t. Az 0
24 Shannon Bound (cont d) Minimal representation Let [n] = {1, 2,..., n}. X 1, X 2,..., X n. The minimal decision variables: (i) H(X i X [n] {i} ), i [n]; (ii) I(X i ; X j X K ), where i j and K [n] {i, j}. Shannon bound: min c T z s.t. Az 0
25 Shannon Bound (cont d) Minimal representation Let [n] = {1, 2,..., n}. X 1, X 2,..., X n. The minimal decision variables: (i) H(X i X [n] {i} ), i [n]; (ii) I(X i ; X j X K ), where i j and K [n] {i, j}. Shannon bound: min c T z s.t. Az 0 Remark Always exists Hard to compute, hard to understand, optimality is unknown Automatical tools
26 Shannon Bound: ITIP/Xitip Information Theoretic Inequalities Prover For n random variables, a linear program in n + ( n 2) 2 n 2 variables
27 Routing Bound and Linear Network Coding Bound X 1 X 4 S T R X 2 X 5 X 3 X 6 Routing bound Special Cutset bound Linear network coding bound (LNCB): V 1, V 2,..., V m F n q rank(v 1 ), rank(v 1, V 2 ) entropy functions. rank(v 1, V 2 ) rank(v 1 ) + rank(v 2 ) LNCB All the inequalities over V 1, V 2,..., V m m = 4, Ingleton m = 5, Dougherty, Freiling, Zeger, 28 m = 6, unknown, at least 1 million
28 Reduction of Wiretap Patterns Network model X 1 X 4 S T R X 2 X 5 X 3 X 6 A 2 E, A 1 = {1, 2, 4}, A 2 = {2, 4} delete A 2 from A A: A i A j (i j) (antichain) E #
29 Numerical Study: Algorithm Cut-set Shannon τ A Linear Network Coding Routing Algorithm for assessing the tightness of the routing bound For each wiretap pattern, 1. Compute the cut-set bounds on (S, T) and (T, R): l Compute the routing bound on S T R: l If l 1 == l 2, then τ A = l 1. Proceed to Step Compare l 2 with the Shannon bound in ITIP/Xitip: If equal, then τ A = l 2. Otherwise, a gap is detected. 5. Proceed to the next wiretap pattern.
30 Numerical Study: Result Gaps exist only if 4 A 12 For almost 80% of the wiretap patterns, cut-set bounds = routing bounds In the Level-I (3, 3) network, there are around 159, 258 wiretap patterns (2% of all the wiretap patterns) routing bounds Shannon bounds In the Level-II (3, 3) network, there are around 32, 472 wiretap patterns (0.4% of all the wiretap patterns) routing bounds Shannon bounds
31 Numerical Study: Result Gaps exist only if 4 A 12 For almost 80% of the wiretap patterns, cut-set bounds = routing bounds In the Level-I (3, 3) network, there are around 159, 258 wiretap patterns (2% of all the wiretap patterns) routing bounds Shannon bounds In the Level-II (3, 3) network, there are around 32, 472 wiretap patterns (0.4% of all the wiretap patterns) routing bounds Shannon bounds Problem How to fill this gap? Cut-set Shannon τ A Linear Network Coding Routing
32 Case Study: Routing Vs. Coding X 1 X 4 S T R X 2 X 5 X 3 X 6 A 1 = {2, 3, 5}, A 2 = {1, 4, 5}, A 3 = {1, 3, 6}, and A 4 = {2, 4, 6}. Routing bound: 3. Both of the Shannon bounds (Level I/II): 2.
33 Case Study: Routing Vs. Coding X 1 X 4 S T R X 2 X 5 X 3 X 6 A 1 = {2, 3, 5}, A 2 = {1, 4, 5}, A 3 = {1, 3, 6}, and A 4 = {2, 4, 6}. Routing bound: 3. Both of the Shannon bounds (Level I/II): 2. X 1 = K 1 X 4 = M + 2K 1 + 2K 2 X 2 = K 2 X 5 = M + K 1 + 2K 2 X 3 = M + K 1 + K 2 X 6 = M + 2K 1 + K 2
34 Case Study: Level I Vs. Level II X 1 X 4 S T R X 2 X 5 X 3 X 6 A 1 = {1, 4}, A 2 = {2, 3, 4}, A 3 = {1, 2, 5, 6}, and A 4 = {3, 5, 6}. Routing bound = 3. Shannon bounds: 2 (Level-I) and 3 (Level-II).
35 Case Study: Level I Vs. Level II X 1 X 4 S T R X 2 X 5 X 3 X 6 A 1 = {1, 4}, A 2 = {2, 3, 4}, A 3 = {1, 2, 5, 6}, and A 4 = {3, 5, 6}. Routing bound = 3. Shannon bounds: 2 (Level-I) and 3 (Level-II). X 1 = M + K 1 X 4 = K 1 + K 2 X 2 = K 2 X 5 = M + K 1 + K 2 X 3 = K 1
36 Heuristic Observations Objective: H(K) 3H(M) Constraints from network: I(M; K) = 0 H(X 1, X 2, X 3 M, K) = 0 H(X 4, X 5, X 6 X 1, X 2, X 3 ) = 0 H(M, K X 4, X 5, X 6 ) = 0 I(M; X 2, X 4, X 5 ) = 0 I(M; X 2, X 3, X 6 ) = 0 I(M; X 1, X 5, X 6 ) = 0 I(M; X 1, X 3, X 4 ) = 0 I(M; X 1, X 2, X 4, X 6 ) = 0 More useful constraints: H(X 1, X 2, X 3 ) = H(X 1 ) + H(X 2 ) + H(X 3 ) H(X 4, X 5, X 6 ) = H(X 4 ) + H(X 5 ) + H(X 6 ) H(X 4 X 1, X 3 ) = 0 H(X 5 X 1, X 2, X 3 ) = 0 H(X 6 X 2, X 3 ) = 0 H(X 3 ) = 2H(X 1 ) H(X 1 ) = H(X 2 ) H(X 5 ) = 2H(X 4 ) H(X 4 ) = H(X 6 ) Helpful in designing a LNC!
37 Heuristic Observations (cont d) Initial constraints: H(K) = 3H(M) I(M; K) = 0 H(X 1, X 2, X 3 M, K) = 0 H(X 4, X 5, X 6 X 1, X 2, X 3 ) = 0 H(M, K X 4, X 5, X 6 ) = 0 I(M; X 2, X 4, X 5 ) = 0 I(M; X 2, X 3, X 6 ) = 0 I(M; X 1, X 5, X 6 ) = 0 I(M; X 1, X 3, X 4 ) = 0 I(M; X 1, X 2, X 4, X 6 ) = 0 Implications: H(X 1, X 2, X 3 ) = H(X 1 ) + H(X 2 ) + H(X 3 ) H(X 4, X 5, X 6 ) = H(X 4 ) + H(X 5 ) + H(X 6 ) H(X 4 X 1, X 3 ) = 0 H(X 5 X 1, X 2, X 3 ) = 0 H(X 6 X 2, X 3 ) = 0 H(X 3 ) = 2H(X 1 ) H(X 1 ) = H(X 2 ) H(X 5 ) = 2H(X 4 ) H(X 4 ) = H(X 6 ) Necessary conditions!
38 A Hard Example: A = 12 X 1 X 4 S T R X 2 X 5 X 3 X 6 A 1 = {3, 5, 6}, A 2 = {3, 4, 6}, A 3 = {3, 4, 5}, A 4 = {2, 5, 6}, A 5 = {2, 4, 6}, A 6 = {2, 3, 6}, A 7 = {2, 3, 5}, A 8 = {2, 3, 4}, A 9 = {1, 5, 6}, A 10 = {1, 3, 5}, A 11 = {1, 3, 4}, A 12 = {1, 2, 4, 5} Routing bound: 4. Shannon bound: 19/5. LNC on F 24 q. H(X 1, X 2, X 3 ) = H(X 1 ) + H(X 2 ) + H(X 3 ) H(X 4, X 5, X 6 ) = H(X 4 ) + H(X 5 ) + H(X 6 ) 7H(X 1 ) = 9H(X 2 ) 8H(X 1 ) = 9H(X 3 ) 9 : 7 : 8 3H(X 4 ) = 4H(X 5 ) 5H(X 4 ) = 4H(X 6 ) 8 : 6 : 10
39 Conclusion F. Cheng and V. Y. F. Tan, A Numerical Study on the Wiretap Network with a Simple Network Topology, arxiv: A simple network model Numerical study by ITIP/Xitip Initial progress: A = 4 Some very interesting wiretap patterns Open A = 5,..., 12
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
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
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,
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
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
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
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
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
Linear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
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,
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
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
Secure Network Coding via Filtered Secret Sharing
Secure Network Coding via Filtered Secret Sharing Jon Feldman, Tal Malkin, Rocco Servedio, Cliff Stein (Columbia University) jonfeld@ieor, tal@cs, rocco@cs, cliff@ieor columbiaedu Feldman, Malkin, Servedio,
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
Scheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
On the effect of forwarding table size on SDN network utilization
IBM Haifa Research Lab On the effect of forwarding table size on SDN network utilization Rami Cohen IBM Haifa Research Lab Liane Lewin Eytan Yahoo Research, Haifa Seffi Naor CS Technion, Israel Danny Raz
Adaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
Single machine models: Maximum Lateness -12- Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...
Lecture 4 Scheduling 1 Single machine models: Maximum Lateness -12- Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule 0 Q 1100 11 00 11 000 111 0 0 1 1 00 11 00 11 00
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
Further Analysis Of A Framework To Analyze Network Performance Based On Information Quality
Further Analysis Of A Framework To Analyze Network Performance Based On Information Quality A Kazmierczak Computer Information Systems Northwest Arkansas Community College One College Dr. Bentonville,
Competitive Analysis of On line Randomized Call Control in Cellular Networks
Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising
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
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
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
Proximal mapping via network optimization
L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
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
Towards a Tight Finite Key Analysis for BB84
The Uncertainty Relation for Smooth Entropies joint work with Charles Ci Wen Lim, Nicolas Gisin and Renato Renner Institute for Theoretical Physics, ETH Zurich Group of Applied Physics, University of Geneva
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
4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
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
MILP Model And Models For Cloud Network Backup
Minimizing Disaster Backup Window for Geo-Distributed Multi-Datacenter Cloud Systems Jingjing Yao, Ping Lu, Zuqing Zhu School of Information Science and Technology University of Science and Technology
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
I. INTRODUCTION. of the biometric measurements is stored in the database
122 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL 6, NO 1, MARCH 2011 Privacy Security Trade-Offs in Biometric Security Systems Part I: Single Use Case Lifeng Lai, Member, IEEE, Siu-Wai
Load Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
Multi-layer Structure of Data Center Based on Steiner Triple System
Journal of Computational Information Systems 9: 11 (2013) 4371 4378 Available at http://www.jofcis.com Multi-layer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1,
Basic Components of an LP:
1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic
Message Authentication Code
Message Authentication Code Ali El Kaafarani Mathematical Institute Oxford University 1 of 44 Outline 1 CBC-MAC 2 Authenticated Encryption 3 Padding Oracle Attacks 4 Information Theoretic MACs 2 of 44
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
Capacity Limits of MIMO Channels
Tutorial and 4G Systems Capacity Limits of MIMO Channels Markku Juntti Contents 1. Introduction. Review of information theory 3. Fixed MIMO channels 4. Fading MIMO channels 5. Summary and Conclusions References
Routing in Line Planning for Public Transport
Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany MARC E. PFETSCH RALF BORNDÖRFER Routing in Line Planning for Public Transport Supported by the DFG Research
Branch and Cut for TSP
Branch and Cut for TSP jla,jc@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark 1 Branch-and-Cut for TSP Branch-and-Cut is a general technique applicable e.g. to solve symmetric
Online Scheduling for Cloud Computing and Different Service Levels
2012 IEEE 201226th IEEE International 26th International Parallel Parallel and Distributed and Distributed Processing Processing Symposium Symposium Workshops Workshops & PhD Forum Online Scheduling for
Compressing Forwarding Tables for Datacenter Scalability
TECHNICAL REPORT TR12-03, TECHNION, ISRAEL 1 Compressing Forwarding Tables for Datacenter Scalability Ori Rottenstreich, Marat Radan, Yuval Cassuto, Isaac Keslassy, Carmi Arad, Tal Mizrahi, Yoram Revah
ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015
ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving
Zeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property
A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property Venkat Chandar March 1, 2008 Abstract In this note, we prove that matrices whose entries are all 0 or 1 cannot achieve
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
Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs
Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS
USING EXCEL SOLVER IN OPTIMIZATION PROBLEMS Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 445 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu
HYBRID GENETIC ALGORITHMS FOR SCHEDULING ADVERTISEMENTS ON A WEB PAGE
HYBRID GENETIC ALGORITHMS FOR SCHEDULING ADVERTISEMENTS ON A WEB PAGE Subodha Kumar University of Washington subodha@u.washington.edu Varghese S. Jacob University of Texas at Dallas vjacob@utdallas.edu
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
SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis
SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 17, 2015 Outline
A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
Decentralized Utility-based Sensor Network Design
Decentralized Utility-based Sensor Network Design Narayanan Sadagopan and Bhaskar Krishnamachari University of Southern California, Los Angeles, CA 90089-0781, USA narayans@cs.usc.edu, bkrishna@usc.edu
Profit Maximization and Power Management of Green Data Centers Supporting Multiple SLAs
Profit Maximization and Power Management of Green Data Centers Supporting Multiple SLAs Mahdi Ghamkhari and Hamed Mohsenian-Rad Department of Electrical Engineering University of California at Riverside,
On Adaboost and Optimal Betting Strategies
On Adaboost and Optimal Betting Strategies Pasquale Malacaria School of Electronic Engineering and Computer Science Queen Mary, University of London Email: pm@dcs.qmul.ac.uk Fabrizio Smeraldi School of
Optimal Multicast in Dense Multi-Channel Multi-Radio Wireless Networks
Optimal Multicast in Dense Multi-Channel Multi-Radio Wireless Networks Rahul Urgaonkar IBM TJ Watson Research Center Yorktown Heights, NY 10598 Email: rurgaon@us.ibm.com Prithwish Basu and Saikat Guha
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
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 of
Connected Identifying Codes for Sensor Network Monitoring
Connected Identifying Codes for Sensor Network Monitoring Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email:
A comprehensive survey on various ETC techniques for secure Data transmission
A comprehensive survey on various ETC techniques for secure Data transmission Shaikh Nasreen 1, Prof. Suchita Wankhade 2 1, 2 Department of Computer Engineering 1, 2 Trinity College of Engineering and
Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
Ex. 2.1 (Davide Basilio Bartolini)
ECE 54: Elements of Information Theory, Fall 00 Homework Solutions Ex.. (Davide Basilio Bartolini) Text Coin Flips. A fair coin is flipped until the first head occurs. Let X denote the number of flips
Multi-layer MPLS Network Design: the Impact of Statistical Multiplexing
Multi-layer MPLS Network Design: the Impact of Statistical Multiplexing Pietro Belotti, Antonio Capone, Giuliana Carello, Federico Malucelli Tepper School of Business, Carnegie Mellon University, Pittsburgh
Improved Algorithms for Data Migration
Improved Algorithms for Data Migration Samir Khuller 1, Yoo-Ah Kim, and Azarakhsh Malekian 1 Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported by NSF Award
Establishing a Mobile Conference Call Under Delay and Bandwidth Constraints
Establishing a Mobile Conference Call Under Delay and Bandwidth Constraints Amotz Bar-Noy Computer and Information Science Department Brooklyn College, CUNY, New York Email: amotz@sci.brooklyn.cuny.edu
Large-Scale IP Traceback in High-Speed Internet
2004 IEEE Symposium on Security and Privacy Large-Scale IP Traceback in High-Speed Internet Jun (Jim) Xu Networking & Telecommunications Group College of Computing Georgia Institute of Technology (Joint
! 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
Physical-Layer Security: Combining Error Control Coding and Cryptography
1 Physical-Layer Security: Combining Error Control Coding and Cryptography Willie K Harrison and Steven W McLaughlin arxiv:09010275v2 [csit] 16 Apr 2009 Abstract In this paper we consider tandem error
The Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
Information Theory and Stock Market
Information Theory and Stock Market Pongsit Twichpongtorn University of Illinois at Chicago E-mail: ptwich2@uic.edu 1 Abstract This is a short survey paper that talks about the development of important
Secure Physical-layer Key Generation Protocol and Key Encoding in Wireless Communications
IEEE Globecom Workshop on Heterogeneous, Multi-hop Wireless and Mobile Networks Secure Physical-layer ey Generation Protocol and ey Encoding in Wireless Communications Apirath Limmanee and Werner Henkel
EFFICIENT KNOWLEDGE BASE MANAGEMENT IN DCSP
EFFICIENT KNOWLEDGE BASE MANAGEMENT IN DCSP Hong Jiang Mathematics & Computer Science Department, Benedict College, USA jiangh@benedict.edu ABSTRACT DCSP (Distributed Constraint Satisfaction Problem) has
3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
Batch Scheduling of Deteriorating Products
Decision Making in Manufacturing and Services Vol. 1 2007 No. 1 2 pp. 25 34 Batch Scheduling of Deteriorating Products Maksim S. Barketau, T.C. Edwin Cheng, Mikhail Y. Kovalyov, C.T. Daniel Ng Abstract.
A linear algebraic method for pricing temporary life annuities
A linear algebraic method for pricing temporary life annuities P. Date (joint work with R. Mamon, L. Jalen and I.C. Wang) Department of Mathematical Sciences, Brunel University, London Outline Introduction
Planning and Scheduling in the Digital Factory
Institute for Computer Science and Control Hungarian Academy of Sciences Berlin, May 7, 2014 1 Why "digital"? 2 Some Planning and Scheduling problems 3 Planning for "one-of-a-kind" products 4 Scheduling
Classification - Examples
Lecture 2 Scheduling 1 Classification - Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
Random graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks
Load-Balanced Virtual Backbone Construction for Wireless Sensor Networks Jing (Selena) He, Shouling Ji, Yi Pan, Zhipeng Cai Department of Computer Science, Georgia State University, Atlanta, GA, USA, {jhe9,
Capacity of the Multiple Access Channel in Energy Harvesting Wireless Networks
Capacity of the Multiple Access Channel in Energy Harvesting Wireless Networks R.A. Raghuvir, Dinesh Rajan and M.D. Srinath Department of Electrical Engineering Southern Methodist University Dallas, TX
Cyber-Security Analysis of State Estimators in Power Systems
Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute
OPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
Max Flow, Min Cut, and Matchings (Solution)
Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes
On Orchestrating Virtual Network Functions
On Orchestrating Virtual Network Functions Presented @ CNSM 2015 Md. Faizul Bari, Shihabur Rahman Chowdhury, and Reaz Ahmed, and Raouf Boutaba David R. Cheriton School of Computer science University of
Chapter 10: Network Flow Programming
Chapter 10: Network Flow Programming Linear programming, that amazingly useful technique, is about to resurface: many network problems are actually just special forms of linear programs! This includes,
A Model For Revelation Of Data Leakage In Data Distribution
A Model For Revelation Of Data Leakage In Data Distribution Saranya.R Assistant Professor, Department Of Computer Science and Engineering Lord Jegannath college of Engineering and Technology Nagercoil,
Pacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
2004 Networks UK Publishers. Reprinted with permission.
Riikka Susitaival and Samuli Aalto. Adaptive load balancing with OSPF. In Proceedings of the Second International Working Conference on Performance Modelling and Evaluation of Heterogeneous Networks (HET
A Game Theoretic Model to Handle Network Intrusions over Multiple Packets
A Game Theoretic Model to Handle Network Intrusions over Multiple Packets Mona Mehrandish, Chadi M. Assi, and Mourad Debbabi Concordia Institute for Information Systems Engineering Concordia University,
Multiobjective Multicast Routing Algorithm
Multiobjective Multicast Routing Algorithm Jorge Crichigno, Benjamín Barán P. O. Box 9 - National University of Asunción Asunción Paraguay. Tel/Fax: (+9-) 89 {jcrichigno, bbaran}@cnc.una.py http://www.una.py
6.231 Dynamic Programming and Stochastic Control Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.231 Dynamic Programming and Stochastic Control Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.231
SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
How To Find The Network Lifetime Of A Base Station With An Approximation Algorithm
Optimal Base Station Placement in Wireless Sensor Networks YI SHI and Y. THOMAS HOU Virginia Polytechnic Institute and State University Base station location has a significant impact on network lifetime
Algorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
Security-Aware Beacon Based Network Monitoring
Security-Aware Beacon Based Network Monitoring Masahiro Sasaki, Liang Zhao, Hiroshi Nagamochi Graduate School of Informatics, Kyoto University, Kyoto, Japan Email: {sasaki, liang, nag}@amp.i.kyoto-u.ac.jp
Mathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
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
SECURITY EVALUATION OF EMAIL ENCRYPTION USING RANDOM NOISE GENERATED BY LCG
SECURITY EVALUATION OF EMAIL ENCRYPTION USING RANDOM NOISE GENERATED BY LCG Chung-Chih Li, Hema Sagar R. Kandati, Bo Sun Dept. of Computer Science, Lamar University, Beaumont, Texas, USA 409-880-8748,
REAL-TIME PRICE FORECAST WITH BIG DATA
REAL-TIME PRICE FORECAST WITH BIG DATA A STATE SPACE APPROACH Lang Tong (PI), Robert J. Thomas, Yuting Ji, and Jinsub Kim School of Electrical and Computer Engineering, Cornell University Jie Mei, Georgia
SECRET sharing schemes were introduced by Blakley [5]
![SECRET sharing schemes were introduced by Blakley [5]](/thumbs/24/4404980.jpg "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