Download - Secure Communication for Distributed Systems
Overview• Application• A framework for secrecy of distributed systems
• Theoretical result• Information theory in a competitive context (zero-sum game)
• Two methods of coordination
Main Idea• Secrecy for distributed systems
• Design encryption specifically for a system objective
Node A
Node BMessageInformation
Action
Adversary
Distributed System
Attack
Communication in Distributed Systems
“Smart Grid”
Image from http://www.solarshop.com.au
Example: Rate-Limited Control
Adversary
00101110010010111
Signal (sensor)Communication
Signal (control)
Attack Signal
Example: Feedback Stabilization
• “Data Rate Theorem” [Wong-Brockett 99, Baillieul 99]
Controller Dynamic System
EncoderDecoder 10010011011010101101010100101101011
SensorAdversary
Feedback
Shannon Analysis• 1948• Channel Capacity• Lossless Source Coding• Lossy Compression
• 1949 - Perfect Secrecy• Adversary learns nothing about the information• Only possible if the key is larger than the information
C. Shannon, "Communication Theory of Secrecy Systems," Bell Systems Technical Journal, vol. 28, pp. 656-715, Oct. 1949.
Shannon Model• Schematic
• Assumption• Enemy knows everything about the system except the key
• Requirement• The decipherer accurately reconstructs the information
C. Shannon, "Communication Theory of Secrecy Systems," Bell Systems Technical Journal, vol. 28, pp. 656-715, Oct. 1949.
Encipherer DeciphererCiphertext
Key Key
Plaintext Plaintext
Adversary
For simple substitution:
Shannon Analysis• Equivocation vs Redundancy• Equivocation is conditional entropy:• Redundancy is lack of entropy of the source:• Equivocation reduces with redundancy:
C. Shannon, "Communication Theory of Secrecy Systems," Bell Systems Technical Journal, vol. 28, pp. 656-715, Oct. 1949.
Computational Secrecy• Assume limited computation resources• Public Key Encryption• Trapdoor Functions
• Difficulty not proven• Can become a “cat and mouse” game
• Vulnerable to quantum computer attack
W. Diffie and M. Hellman, “New Directions in Cryptography,” IEEE Trans. on Info. Theory, 22(6), pp. 644-654, 1976.
1125897758 834 689524287
2147483647X
Information Theoretic Secrecy• Achieve secrecy from randomness (key or channel), not from
computational limit of adversary.
• Physical layer secrecy• Wyner’s Wiretap Channel [Wyner 1975]
• Partial Secrecy• Typically measured by “equivocation:”• Other approaches:• Error exponent for guessing eavesdropper [Merhav 2003]• Cost inflicted by adversary [this talk]
Equivocation• Not an operationally defined quantity
• Bounds:• List decoding• Additional information needed for decryption
• Not concerned with structure
Our Framework• Assume secrecy resources are available (secret key, private
channel, etc.)
• How do we encode information optimally?
• Game Theoretic• Eavesdropper is the adversary• System performance (for example, stability) is the payoff• Bayesian games• Information structure
Competitive Distributed System
Node A Node BMessage
Key
Information Action
AdversaryAttack
Encoder:
System payoff: .
Decoder:
Adversary:
Zero-Sum Game• Value obtained by system:• Objective• Maximize payoff
Node A Node BMessage
Key
Information Action
AdversaryAttack
Secrecy-Distortion Literature
• [Yamamoto 97]:• Cause an eavesdropper to have high reconstruction distortion
• Replace payoff (π) with distortion• No causal information to the eavesdropper
• Warning: Problem statement can be too optimistic!
How to Force High Distortion• Randomly assign bins• Size of each bin is • Adversary only knows bin
• Reconstruction of only depends on the marginal posterior distribution of
Example (Bern(1/3)):
Two Categories of Results
Lossless Transmission
• Simplex interpretation• Linear program
• Hamming Distortion
General Reward Function
• Common Information• Secret Key
Competitive Distributed System
Node A Node BMessage
Key
Information Action
AdversaryAttack
Encoder:
System payoff: .
Decoder:
Adversary:
Zero-Sum Game• Value obtained by system:• Objective• Maximize payoff
Node A Node BMessage
Key
Information Action
AdversaryAttack
Theorem: [Cuff 10]
Lossless Case• Require Y=X• Assume a payoff function
• Related to Yamamoto’s work [97]• Difference: Adversary is more capable with more information
Also required:
Linear Program on the Simplex
Constraint:
Minimize:
Maximize:
U will only have mass at a small subset of points (extreme points)
Linear Program on the Simplex
Catego
ry 1
Catego
ry 2
Catego
ry 3
Catego
ry 4
00.4
0.8
Series 1
Series 2
Series 3
Series 4
Series 5 Series 1Series 2Series 3Series 4Series 5
Binary-Hamming Case• Binary Source:• Hamming Distortion
• Optimal approach• Reveal excess 0’s or 1’s to condition the hidden bits
0 1 0 0 1 0 0 0 0 1
* * 0 0 * * 0 * 0 *Source
Public message
Binary Source (Example)• Information source is Bern(p)• Usually zero (p < 0.5)• Hamming payoff
• Secret key rate R0 required to guarantee eavesdropper error
R0
p
Eavesdropper Error
General Payoff FunctionNo requirement for lossless transmission.
• Any payoff function π(x,y,z)• Any source distribution (i.i.d.)
Adversary:
Unlimited Public Communication• Maximum achievable average payoff
• Conditional common information:
Theorem (R=∞):
Coordination Capacity• References:• [C., Permuter, Cover – IT Trans. 09]• [C. - ISIT 08]• [Bennett, Shor, Smolin, Thapliyal – IT Trans. 02]• [C., Zhao – ITW 11]
• Ability to coordinate sequences (“actions”) with communication limitations.• Empirical Coordination• Strong Coordination
X1 X2 X3 X4 X5 X6 … Xn
Empirical Coordination
Y1 Y2 Y3 Y4 Y5 Y6 … Yn
Z1 Z2 Z3 Z4 Z5 Z6 … Zn
Empirical Distribution
Empirical Distribution
1 0 1 1 0 0 0 1
0 1 1 0 1 0 1 1
1 1 0 1 0 0 1 0
000 001 010 011 100 101 110 111
Average Distortion
• Average values are a function of the empirical distribution
• Example: Squared error distortion
• Rate distortion theory fits in the empirical coordination context.
No Rate – No Channel• No explicit communication channel
• Signal “A” serves an analog and information role.• Analog: symbol-by-symbol relationship• (Digital): uses complex structure to carry information.
Processor 1 Processor 2
Source
Actuator 1 Actuator 2
Coordination Region
• The coordination region
gives us all results concerning average distortion.
Processor 1 Processor 2
Source
Result – No constraints
Processor 1 Processor 2
Source
Achievability: Make a codebook of (An , Bn ) pairs
General Results• Variety of causality constraints (delay)
Finite Look-ahead
Processor 1 Processor 2
Source
Alice and Bob Game• Alice and Bob want to cooperatively score points by both
correctly guessing a sequence of random binary numbers (one point if they both guess correctly).
• Alice gets entire sequence ahead of time• Bob only sees that past binary numbers and guesses of Alice.• What is the optimal score in the game?
Alice and Bob Game (answer)• Online Matching Pennies• [Gossner, Hernandez, Neyman, 2003]• “Online Communication”
• Solution
General (causal) solution
• Score in Alice and Bob Game is a first-order statistic
• Achievable empirical distributions• (Processor 2 is strictly causal)
• Surprise: Bob doesn’t need to see the past of the sequence.
X1 X2 X3 X4 X5 X6 … Xn
Strong Coordination
Y1 Y2 Y3 Y4 Y5 Y6 … Yn
Z1 Z2 Z3 Z4 Z5 Z6 … Zn
Joint distribution of sequences is i.i.d.with respect to the desired joint distribution.
(Allow epsilon total variation distance.)
Point-to-point Coordination
• Theorem [C. 08]:• Strong Coordination involves picking a V such that X-V-Y• Message: R > I(X;V)• Common Randomness: R0 + R > I(X,Y;V)• Uses randomized decoder (channel from V to Y)
Node A Node BMessage
Common Randomness
Source Output
Synthetic Channel p(y|x)
Zero-Sum Game• Value obtained by system:• Objective• Maximize payoff
Node A Node BMessage
Key
Information Action
AdversaryAttack
Encoding Scheme
• Coordination Strategies• Empirical coordination for U• Strong coordination for Y
K