Paul Cuff
THE SOURCE CODING SIDE OF SECRECY
Game Theoretic Secrecy
• Motivating Problem
• Mixed Strategy• Non-deterministic
• Requires randomdecoder• Dual to wiretap channel
Encoder
Communication
leakage
Eavesdropping
Zero-sumRepeated
Game
Player 1
Player 2
State
Main Topics of this Talk
• Achievability Proof Techniques:1. Pose problems in terms of existence of joint distributions
2. Relax Requirements to “close in total variation”
3. Main Tool --- Reverse Channel Encoder
4. Easy Analysis of Optimal Adversary
X n > B
Restate Problem---Example 1 (RD Theory)
• Can we design:
such that
• Does there exists a distribution:
Standard Existence of Distributions
f g
Restate Problem---Example 2 (Secrecy)
• Can we design:
such that
• Does there exists a distribution:
Standard Existence of Distributions
f g
Eve
Score
[Cuff 10]
Tricks with Total Variation
• Technique• Find a distribution p1 that is easy to analyze and satisfies the
relaxed constraints.
• Construct p2 to satisfy the hard constraints while maintaining small total variation distance to p1.
How?
Property 1:
Tricks with Total Variation
• Technique• Find a distribution p1 that is easy to analyze and satisfies the
relaxed constraints.
• Construct p2 to satisfy the hard constraints while maintaining small total variation distance to p1.
Why?
Property 2 (bounded functions):
Summary• Achievability Proof Techniques:
1. Pose problems in terms of existence of joint distributions
2. Relax Requirements to “close in total variation”
3. Main Tool --- Reverse Channel Encoder
4. Easy Analysis of Optimal Adversary
• Secrecy Example: For arbitrary ², does there exist a distribution satisfying:
Cloud Overlap Lemma• Previous Encounters• Wyner, 75 --- used divergence• Han-Verdú, 93 --- general channels, used total variation• Cuff 08, 09, 10, 11 --- provide simple proof and utilize for
secrecy encoding
PX|U(x|u)
Memoryless Channel
Reverse Channel Encoder
• For simplicity, ignore the key K, and consider Ja to be the part of the message that the adversary obtains. (i.e. J = (Ja, Js), and ignore Js for now)
• Construct a joint distribution between the source Xn and the information Ja (revealed to the Adversary) using a memoryless channel.
PX|U(x|u)
Memoryless Channel
Simple Analysis• This encoder yields a very simple analysis and convenient properties
1. If |Ja| is large enough, then Xn will be nearly i.i.d. in total variation
2. Performance:
PX|U(x|u)
Memoryless Channel
Summary
• Achievability Proof Techniques:1. Pose problems in terms of existence of joint distributions
2. Relax Requirements to “close in total variation”
3. Main Tool --- Reverse Channel Encoder
4. Easy Analysis of Optimal Adversary
X n > B
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