ECE 6605 - Information Theory

Prof. Matthieu Bloch

Thursday, November 30, 2023

Announcements

Project
- Send me an email to confirm your choices (the sooner the better)
- I'll hold project office hours to help you
- This is meant to be fun
Exam options
- Oral: list of problems published, scheduling options coming
- Written: Thursday, December 14 11:20 am - 2:10 pm
Last time
- Wiretap channel
Today
- Secret key generation
- Covert communications

Secret key generation

Public authenticated channel of unlimited capacity available for reocnciliation

A $(n, K_{n})$ code $C$ for secret-key generation consists of an encoding function $f_{n} : X^{n} \to {1, \dots, K_{n}} \times F$ and a decoding function $g_{n} : Y^{n} \times F \to {1, \dots, K_{n}}$

The performance of an $(n, M_{n})$ code is measured in terms of
1. The rate of the code $R ≜ \frac{\log_{2} K_{n}}{n}$ (bits/source symbol)
2. Reliability and secrecy $P_{e}^{(n)} ≜ P_{} (\hat{K} \neq K)$ and $S^{(n)} ≜ I (K; (F, Z^{n}))$

Secret key generation

Define $C_{sk} ≜ sup {R : \exists {(f_{n}, g_{n})}_{n \geq 1} with lim_{n \to \infty} \frac{\log K_{n}}{n} \geq R and lim_{n \to \infty} P_{e}^{(n)} = lim_{n \to \infty} S^{(n)} = 0}$

For a physically degraded discrete memoryless channel characterized by $P_{Y | X}$ and $P_{Z | X}$ , $C_{sk} = H (X | Z) - H (X | Y) = I (X; Y | Z)$

Not different from secrecy capacity, but this is an artefact of physically degraded channels
- In general, secret key generation rates are higher than wiretap coding rates.
Challenges
- How do we operationalize secrecy?
- Can we leverage the coding mechanisms studied earlier?

Covert communications

Pioneering work paper: B.A Bash et al., Limits of Reliable Communication with Low Probability of Detection on AWGN Channels, IEEE Journal on Selected Areas in Communications, (2013)

Covertness metric: $lim_{n \to \infty} D ({\hat{Q}}^{n} ‖ Q_{0}^{\otimes n}) = 0$ (asymptotic independence)

A $(n, M_{n})$ code $C$ for a wiretap channel consists of an encoding function $f_{n} : {1, \dots, M_{n}} \to X^{n}$ and a decoding function $g_{n} : Y^{n} \to {1, \dots, M_{n}}$

The performance of an $(n, M_{n})$ code is measured in terms of
1. The rate of the code $R ≜ \frac{\log_{2} M_{n}}{n}$ (bits/source symbol)
2. Reliability and covertness $P_{e}^{(n)} ≜ P_{} (\hat{M} \neq M)$ and $D^{(n)} ≜ D ({\hat{Q}}^{n} ‖ Q_{0}^{\otimes n})$