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