Physical Layer Security in Wireless Networks
An Information Theoretic Perspective
Vince Poor([email protected])
Thanks to:
Arsenia Chorti (ENSEA), Mahdi Shakiba Herfeh (ENSEA) and Miroslav G. Mitev (Essex)
M. S. Herfeh, A. Chorti & H. V. Poor, “Physical Layer Security: Authentication, Integrity andConfidentiality.” In Physical Layer Security, Khoa N. Le, Ed. (Springer Nature, 2021), to appear.
Today’s Security Architecture
Application
Transport
Network
Link
Physical
Secured
Why do we need PLS?
Standard Crypto Systems
• No known feasible attacks
• Widely employed and tested
• Trustworthy authentication
DC• Built on unproven assumptions
• PHY connection assumed established
• Computationally intensive
• Require significant infrastructure
Physical Layer Security
• Quantum secure
• On-the-fly secret key generation
• Fast and lightweight implementation
DC• Assumptions on Eve’s channel
• Bounds not categorized on finite blocks
• Testing – still in infancy
A Prime Example: IoT
Figure source: https://www.computer.org/
• IoT vulnerabilities to cyber attacks à Mostly concern personal privacy and security
IoT Security – A Major Concern
An Example of What Can Go Wrong[Soltan, Mittal, Poor, USENIX’18]
• Manipulation of demand via IoT: Botnets controlling high-wattage IoTdevices (air conditioners, refrigerators, etc.) can disrupt the power grid.
• A Mirai-sized (600,000 bots) botnet of water heaters can change thedemand instantly by 3GW – similar to having access to the largestcurrently deployed nuclear plant!
Compromised High Wattage IoT Devices
Target of the Attack
Power Grid
Compromised High Wattage IoT Devices
Adversary
Synchronous On/Off
PLS Can Offer
PLS
Node authentication
Messageintegrity
Message confidentiality
Secure session
initiation
Completeness of security protocols
Secure messageexchange
• Physical unclonable functions• Biometrics• Wireless identification
• Wiretap channel model• Secret key generation
• Physical unclonable functions• Wireless identification
The theme:
- A role for information theory in this area
Focus mainly on message confidentiality and privacy:
- Message confidentiality in data transmission systems
- Privacy in sensing systems
Other issues – some newer some older (briefly):
- Authentication, security in MANETs, attacks on sensor networks
Overview of Today’s Talk
Message Confidentiality
in
Transmission Systems
• Traditionally security is treated as a higher-layer issue (e.g.,
at the APP)
• Encryption can be complex (and add delay), and difficult
without infrastructure
• Information theoretic security examines the fundamental
ability of the medium (i.e., the PHY) to provide security
(primarily data confidentiality)
Information Theoretic SecurityRethinking Security Design
• Physical layer security provides security by exploiting imperfections in physical channels: noise, fading, …
• Joint encoding for reliability and security.
• Conventionally a higher layer issue: encryption, key distributions, …
• Difficult with massive number of devices (esp. with no infrastructure), low cost, low latency.
!X
Shannon (1949): For cipher, perfect secrecy requires a one-time pad.
[I.e., the entropy of the key must be at least the entropy of the source: H(K) ≥ H(M)]
Information Theoretic Security: Shannon’s Model
Alice Bob
Eve
Secret Key K
Message M M
?
X X
X
Information Theoretic Security: Wyner’s Model
Alice Bob
Eve
Message M M
?
X Y
Z
Noisy Channel
Noisy Channel
“The Wiretap Channel”
• Tradeoff: reliable rate R to Bob vs. the equivocation H(M|Z) at Eve
• Secrecy capacity = maximum R such that R = H(M|Z)
• Wyner (1975): Secrecy capacity > 0 iff. Z is degraded relative to Y
Physical Layer Security
• In general, the legitimate receiver needs an advantage overthe eavesdropper – either a secret shared with thetransmitter, or a better channel.
• There has been a resurgence of interest in these ideas.
Physical Layer Security
• In general, the legitimate receiver needs an advantage overthe eavesdropper – either a secret shared with thetransmitter, or a better channel.
• There has been a resurgence of interest in these ideas.
• The physical properties of radio propagation (diffusion &superposition) provide opportunities for this, via
- fading: provides natural degradedness over time
- interference: allows active countermeasures to eavesdropping
- spatial diversity (MIMO, relays): creates “secrecy degrees of freedom”
- random channels: sources of common randomness for key generation
Physical Layer Security
• In general, the legitimate receiver needs an advantage overthe eavesdropper – either a secret shared with thetransmitter, or a better channel.
• There has been a resurgence of interest in these ideas.
• The physical properties of radio propagation (diffusion &superposition) provide opportunities for this, via
- fading: provides natural degradedness over time
- interference: allows active countermeasures to eavesdropping
- spatial diversity (MIMO, relays): creates “secrecy degrees of freedom”
- random channels: sources of common randomness for key generation
• The first three of these phenomena lead to rich secrecy capacityregions for the fundamental channel models used to understandwireless networks.
• Interference Channels:
• Multiple-Access Channels:
• Relay Channels, MIMO Channels, etc.
Secrecy in Fundamental Channel Models
Alice1/Eve2
Bob
Message M1
M1,M2
X1
Y
Alice2/Eve1Message M2 X2
Z1
Z2
Alice1 Bob1/Eve2Message M1 M1,?X1
Alice2Message M2X2
Y1
Y2Bob2/Eve1 M2,?
Y1
Y2
Alice
Bob1
Bob2/Eve
M1,M2
M1,?
Messages M1, M2• Broadcast Channels:
Poor, Schaefer (2017) Wireless Physical Layer Security PNAS
Y1
Y2
Alice
Bob1
Bob2/Eve
M1,M2
M1,?
Messages M1, M2
E.g., the Gaussian Broadcast Channel
Y1
Y2
Alice
Bob1
Bob2/Eve
M1,M2
M1,?
Messages M1, M2
E.g., the Gaussian Broadcast Channel
Y1
Y2
Alice
Bob1
Bob2/Eve
M1,M2
M1,?
Messages M1, M2
E.g., the Fading Gaussian BC
Wiretap Channel and Secrecy Capacity
• Secrecy capacity: largest rate in the asymptotic regime of
• Blocklength
• Probability of error
• Information leakage
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P⇣W 6= W
⌘! 0
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Cs = maxPX
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• Limitation: not suitable for low-latency applications as in IoT.
Channel
BobXn
ŴYn
W
Zn Leakage ≤ dEve
Alice
Encoder Channel DecoderSource
1, 2, … , MW Xn Yn Ŵ
• (n,M,ε) code: P(W≠Ŵ) ≤ ε
• Fundamental limit: M*(n,ε) = max{M: ∃ an (n,M,ε) code}
Finite Blocklength Information Theory
log M*(n,ε) = n C - √nV Q-1(ε) + O(log n)
C = E[i(X*,Y*)] (Shannon’s capacity); V = Var[i(X*,Y*)] (“dispersion”)
[Polyanskiy, et al. IT’10, etc.]
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5
Blocklength, n
Rat
e R
, bi
ts/c
h.us
eCapacity
Converse
Best achievability
Normal approximation
Example: AWGN (SNR = 0 dB; ε = 10-3)
[Polyanskiy, et al. (2010)]
PHY Layer Security: Finite Blocklength
• secrecy code:
• Message
• Encoder ; decoder
• Average error probability:
• Secrecy constraint: information leakage
!X
• maximum secret rate at a given blocklength.
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W 2 {1, . . . ,M}<latexit sha1_base64="X0p1Ew3qIQr+meQx9STwv+hyi1k=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclJKIoMuiGzdCBfuAJpTJdNoOnUzCzI1SYj/FjQtF3Pol7vwbp20W2npg4HDOPdw7J0wE1+i639bK6tr6xmZhq7i9s7u3b5cOmjpOFWUNGotYtUOimeCSNZCjYO1EMRKFgrXC0fXUbz0wpXks73GcsCAiA8n7nBI0UtcutXwu/cyr+L0YdeXWn3Ttslt1Z3CWiZeTMuSod+0vk6VpxCRSQbTueG6CQUYUcirYpOinmiWEjsiAdQyVJGI6yGanT5wTo/ScfqzMk+jM1N+JjERaj6PQTEYEh3rRm4r/eZ0U+5dBxmWSIpN0vqifCgdjZ9qD0+OKURRjQwhV3Nzq0CFRhKJpq2hK8Ba/vEyaZ1XP8Lvzcu0qr6MAR3AMp+DBBdTgBurQAAqP8Ayv8GY9WS/Wu/UxH12x8swh/IH1+QN6wZN7</latexit><latexit sha1_base64="X0p1Ew3qIQr+meQx9STwv+hyi1k=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclJKIoMuiGzdCBfuAJpTJdNoOnUzCzI1SYj/FjQtF3Pol7vwbp20W2npg4HDOPdw7J0wE1+i639bK6tr6xmZhq7i9s7u3b5cOmjpOFWUNGotYtUOimeCSNZCjYO1EMRKFgrXC0fXUbz0wpXks73GcsCAiA8n7nBI0UtcutXwu/cyr+L0YdeXWn3Ttslt1Z3CWiZeTMuSod+0vk6VpxCRSQbTueG6CQUYUcirYpOinmiWEjsiAdQyVJGI6yGanT5wTo/ScfqzMk+jM1N+JjERaj6PQTEYEh3rRm4r/eZ0U+5dBxmWSIpN0vqifCgdjZ9qD0+OKURRjQwhV3Nzq0CFRhKJpq2hK8Ba/vEyaZ1XP8Lvzcu0qr6MAR3AMp+DBBdTgBurQAAqP8Ayv8GY9WS/Wu/UxH12x8swh/IH1+QN6wZN7</latexit><latexit sha1_base64="X0p1Ew3qIQr+meQx9STwv+hyi1k=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclJKIoMuiGzdCBfuAJpTJdNoOnUzCzI1SYj/FjQtF3Pol7vwbp20W2npg4HDOPdw7J0wE1+i639bK6tr6xmZhq7i9s7u3b5cOmjpOFWUNGotYtUOimeCSNZCjYO1EMRKFgrXC0fXUbz0wpXks73GcsCAiA8n7nBI0UtcutXwu/cyr+L0YdeXWn3Ttslt1Z3CWiZeTMuSod+0vk6VpxCRSQbTueG6CQUYUcirYpOinmiWEjsiAdQyVJGI6yGanT5wTo/ScfqzMk+jM1N+JjERaj6PQTEYEh3rRm4r/eZ0U+5dBxmWSIpN0vqifCgdjZ9qD0+OKURRjQwhV3Nzq0CFRhKJpq2hK8Ba/vEyaZ1XP8Lvzcu0qr6MAR3AMp+DBBdTgBurQAAqP8Ayv8GY9WS/Wu/UxH12x8swh/IH1+QN6wZN7</latexit><latexit sha1_base64="X0p1Ew3qIQr+meQx9STwv+hyi1k=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclJKIoMuiGzdCBfuAJpTJdNoOnUzCzI1SYj/FjQtF3Pol7vwbp20W2npg4HDOPdw7J0wE1+i639bK6tr6xmZhq7i9s7u3b5cOmjpOFWUNGotYtUOimeCSNZCjYO1EMRKFgrXC0fXUbz0wpXks73GcsCAiA8n7nBI0UtcutXwu/cyr+L0YdeXWn3Ttslt1Z3CWiZeTMuSod+0vk6VpxCRSQbTueG6CQUYUcirYpOinmiWEjsiAdQyVJGI6yGanT5wTo/ScfqzMk+jM1N+JjERaj6PQTEYEh3rRm4r/eZ0U+5dBxmWSIpN0vqifCgdjZ9qD0+OKURRjQwhV3Nzq0CFRhKJpq2hK8Ba/vEyaZ1XP8Lvzcu0qr6MAR3AMp+DBBdTgBurQAAqP8Ayv8GY9WS/Wu/UxH12x8swh/IH1+QN6wZN7</latexit>
PX|W : {1, . . . ,M} ! A<latexit sha1_base64="oLPhmXseA6rY/U43AFvI0bvwtdE=">AAACFHicbVDLSsNAFJ34rPUVdekmWATBUhIRFFdVN26ECvYBTQiT6aQdOnkwc6OUmI9w46+4caGIWxfu/BsnbRbaemDgcM69d+49XsyZBNP81ubmFxaXlksr5dW19Y1NfWu7JaNEENokEY9Ex8OSchbSJjDgtBMLigOP07Y3vMz99h0VkkXhLYxi6gS4HzKfEQxKcvXDhpt2HtrZmZ1aVbsXgaxe25ktWH8AWIjo3g4wDAjm6Xnm6hWzZo5hzBKrIBVUoOHqX2oiSQIaAuFYyq5lxuCkWAAjnGZlO5E0xmSI+7SraIgDKp10fFRm7CulZ/iRUC8EY6z+7khxIOUo8FRlvqKc9nLxP6+bgH/qpCyME6AhmXzkJ9yAyMgTMnpMUAJ8pAgmgqldDTLAAhNQOZZVCNb0ybOkdVSzFL85rtQvijhKaBftoQNkoRNUR1eogZqIoEf0jF7Rm/akvWjv2sekdE4renbQH2ifP3PKnxU=</latexit><latexit sha1_base64="oLPhmXseA6rY/U43AFvI0bvwtdE=">AAACFHicbVDLSsNAFJ34rPUVdekmWATBUhIRFFdVN26ECvYBTQiT6aQdOnkwc6OUmI9w46+4caGIWxfu/BsnbRbaemDgcM69d+49XsyZBNP81ubmFxaXlksr5dW19Y1NfWu7JaNEENokEY9Ex8OSchbSJjDgtBMLigOP07Y3vMz99h0VkkXhLYxi6gS4HzKfEQxKcvXDhpt2HtrZmZ1aVbsXgaxe25ktWH8AWIjo3g4wDAjm6Xnm6hWzZo5hzBKrIBVUoOHqX2oiSQIaAuFYyq5lxuCkWAAjnGZlO5E0xmSI+7SraIgDKp10fFRm7CulZ/iRUC8EY6z+7khxIOUo8FRlvqKc9nLxP6+bgH/qpCyME6AhmXzkJ9yAyMgTMnpMUAJ8pAgmgqldDTLAAhNQOZZVCNb0ybOkdVSzFL85rtQvijhKaBftoQNkoRNUR1eogZqIoEf0jF7Rm/akvWjv2sekdE4renbQH2ifP3PKnxU=</latexit><latexit sha1_base64="oLPhmXseA6rY/U43AFvI0bvwtdE=">AAACFHicbVDLSsNAFJ34rPUVdekmWATBUhIRFFdVN26ECvYBTQiT6aQdOnkwc6OUmI9w46+4caGIWxfu/BsnbRbaemDgcM69d+49XsyZBNP81ubmFxaXlksr5dW19Y1NfWu7JaNEENokEY9Ex8OSchbSJjDgtBMLigOP07Y3vMz99h0VkkXhLYxi6gS4HzKfEQxKcvXDhpt2HtrZmZ1aVbsXgaxe25ktWH8AWIjo3g4wDAjm6Xnm6hWzZo5hzBKrIBVUoOHqX2oiSQIaAuFYyq5lxuCkWAAjnGZlO5E0xmSI+7SraIgDKp10fFRm7CulZ/iRUC8EY6z+7khxIOUo8FRlvqKc9nLxP6+bgH/qpCyME6AhmXzkJ9yAyMgTMnpMUAJ8pAgmgqldDTLAAhNQOZZVCNb0ybOkdVSzFL85rtQvijhKaBftoQNkoRNUR1eogZqIoEf0jF7Rm/akvWjv2sekdE4renbQH2ifP3PKnxU=</latexit><latexit sha1_base64="oLPhmXseA6rY/U43AFvI0bvwtdE=">AAACFHicbVDLSsNAFJ34rPUVdekmWATBUhIRFFdVN26ECvYBTQiT6aQdOnkwc6OUmI9w46+4caGIWxfu/BsnbRbaemDgcM69d+49XsyZBNP81ubmFxaXlksr5dW19Y1NfWu7JaNEENokEY9Ex8OSchbSJjDgtBMLigOP07Y3vMz99h0VkkXhLYxi6gS4HzKfEQxKcvXDhpt2HtrZmZ1aVbsXgaxe25ktWH8AWIjo3g4wDAjm6Xnm6hWzZo5hzBKrIBVUoOHqX2oiSQIaAuFYyq5lxuCkWAAjnGZlO5E0xmSI+7SraIgDKp10fFRm7CulZ/iRUC8EY6z+7khxIOUo8FRlvqKc9nLxP6+bgH/qpCyME6AhmXzkJ9yAyMgTMnpMUAJ8pAgmgqldDTLAAhNQOZZVCNb0ybOkdVSzFL85rtQvijhKaBftoQNkoRNUR1eogZqIoEf0jF7Rm/akvWjv2sekdE4renbQH2ifP3PKnxU=</latexit>
g : B ! {1, . . . ,M}<latexit sha1_base64="kIkUZumVtJN/ZqOausLxVev1qS0=">AAACDnicbVC7SgNBFL0bXzG+opY2S0JAMIRdG8UqxMZGiGAekF3C7GSSDJl9MHNXCct+gY2f4C/YWChia23n3zh5FJp4YOBwzj3MvceLBFdoWd9GZmV1bX0ju5nb2t7Z3cvvHzRVGEvKGjQUoWx7RDHBA9ZAjoK1I8mI7wnW8kaXE791x6TiYXCL44i5PhkEvM8pQS1186XBheMTHFIiklrqSD4YIpEyvHcSu+z0QlTlayft5otWxZrCXCb2nBSrBefkCQDq3fyXztLYZwFSQZTq2FaEbkIkcipYmnNixSJCR2TAOpoGxGfKTabnpGZJKz2zH0r9AjSn6u9EQnylxr6nJyerq0VvIv7ndWLsn7sJD6IYWUBnH/VjYWJoTroxe1wyimKsCaGS611NOiSSUNQN5nQJ9uLJy6R5WrE1v7GL1RrMkIUjKMAx2HAGVbiCOjSAwgM8wyu8GY/Gi/FufMxGM8Y8cwh/YHz+AKvfnfU=</latexit><latexit sha1_base64="HC2Oz3F2nue3ZWVlbYjQxdKJwfE=">AAACDnicbVDLSsNAFJ34rPUVdekmtBQES0ncKK5K3bgRKtgHNKVMppN06GQSZm6UEPoFblz4I25cKOLWtbv+jdPHQlsPDBzOuYe593gxZwpse2ysrK6tb2zmtvLbO7t7++bBYVNFiSS0QSIeybaHFeVM0AYw4LQdS4pDj9OWN7ya+K17KhWLxB2kMe2GOBDMZwSDlnpmKbh0QwwDgnlWG7mSBQPAUkYPbuaU3X4Eqnzjjnpm0a7YU1jLxJmTYrXgnj6Pq2m9Z37rLElCKoBwrFTHsWPoZlgCI5yO8m6iaIzJEAe0o6nAIVXdbHrOyCpppW/5kdRPgDVVfycyHCqVhp6enKyuFr2J+J/XScC/6GZMxAlQQWYf+Qm3ILIm3Vh9JikBnmqCiWR6V4sMsMQEdIN5XYKzePIyaZ5VHM1vnWK1hmbIoWNUQCfIQeeoiq5RHTUQQY/oBb2hd+PJeDU+jM/Z6IoxzxyhPzC+fgC1SZ97</latexit><latexit sha1_base64="HC2Oz3F2nue3ZWVlbYjQxdKJwfE=">AAACDnicbVDLSsNAFJ34rPUVdekmtBQES0ncKK5K3bgRKtgHNKVMppN06GQSZm6UEPoFblz4I25cKOLWtbv+jdPHQlsPDBzOuYe593gxZwpse2ysrK6tb2zmtvLbO7t7++bBYVNFiSS0QSIeybaHFeVM0AYw4LQdS4pDj9OWN7ya+K17KhWLxB2kMe2GOBDMZwSDlnpmKbh0QwwDgnlWG7mSBQPAUkYPbuaU3X4Eqnzjjnpm0a7YU1jLxJmTYrXgnj6Pq2m9Z37rLElCKoBwrFTHsWPoZlgCI5yO8m6iaIzJEAe0o6nAIVXdbHrOyCpppW/5kdRPgDVVfycyHCqVhp6enKyuFr2J+J/XScC/6GZMxAlQQWYf+Qm3ILIm3Vh9JikBnmqCiWR6V4sMsMQEdIN5XYKzePIyaZ5VHM1vnWK1hmbIoWNUQCfIQeeoiq5RHTUQQY/oBb2hd+PJeDU+jM/Z6IoxzxyhPzC+fgC1SZ97</latexit><latexit sha1_base64="VrUuQzVFyl0znKS5PmV0oNzbfnE=">AAACDnicbVC7SgNBFJ31GeNr1dJmMAQsQti1UaxCbGyECOYB2SXMTmY3Q2YfzNxVwrJfYOOv2FgoYmtt5984m6TQxAMDh3PuYe49XiK4Asv6NlZW19Y3Nktb5e2d3b198+Cwo+JUUtamsYhlzyOKCR6xNnAQrJdIRkJPsK43vir87j2TisfRHUwS5oYkiLjPKQEtDcxqcOmEBEaUiKyZO5IHIyBSxg9OZtecYQyqduPkA7Ni1a0p8DKx56SC5mgNzC+dpWnIIqCCKNW3rQTcjEjgVLC87KSKJYSOScD6mkYkZMrNpufkuKqVIfZjqV8EeKr+TmQkVGoSenqyWF0teoX4n9dPwb9wMx4lKbCIzj7yU4EhxkU3eMgloyAmmhAqud4V0xGRhIJusKxLsBdPXiads7qt+a1daTTndZTQMTpBp8hG56iBrlELtRFFj+gZvaI348l4Md6Nj9noijHPHKE/MD5/AJlgnGw=</latexit>
P⇣W 6= W
⌘ ✏
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�<latexit sha1_base64="2UpktDjIXDw6QkA0AS30lOtYgkk=">AAAB8XicbZC7TsNAEEXH4RXCKzw6GosIiSqyaaC0oKGgCBJJELEVrdfjZJX12uyukSIrf0FDAUK0/AWfQMffsHkUkHCllY7undHOTJhxprTjfFulpeWV1bXyemVjc2t7p7q711JpLik2acpTeRcShZwJbGqmOd5lEkkScmyHg8tx3n5EqVgqbvUwwyAhPcFiRok21r3P8cGPkGvSrdacujORvQjuDGrewfWnBwCNbvXLj1KaJyg05USpjutkOiiI1IxyHFX8XGFG6ID0sGNQkARVUEwmHtnHxonsOJXmCW1P3N8dBUmUGiahqUyI7qv5bGz+l3VyHZ8HBRNZrlHQ6Udxzm2d2uP17YhJpJoPDRAqmZnVpn0iCdXmSBVzBHd+5UVondZdwzduzbuAqcpwCEdwAi6cgQdX0IAmUBDwBC/wainr2Xqz3qelJWvWsw9/ZH38ANhwkoE=</latexit><latexit sha1_base64="hahxMb32b62Aj9TJQ3lu1YNwsTI=">AAAB8XicbZC7TsNAEEXH4RXCKzw6GosIiSqyaaCMoKGgCBJJELYVrdfjZJX12uyukSIrf0FDAUK0/AWfQMffsHkUkHCllY7undHOTJhxprTjfFulpeWV1bXyemVjc2t7p7q711ZpLim2aMpTeRcShZwJbGmmOd5lEkkScuyEg8tx3nlEqVgqbvUwwyAhPcFiRok21r3P8cGPkGvSrdacujORvQjuDGqNg+vPpix7zW71y49SmicoNOVEKc91Mh0URGpGOY4qfq4wI3RAeugZFCRBFRSTiUf2sXEiO06leULbE/d3R0ESpYZJaCoTovtqPhub/2VeruPzoGAiyzUKOv0ozrmtU3u8vh0xiVTzoQFCJTOz2rRPJKHaHKlijuDOr7wI7dO6a/jGrTUuYKoyHMIRnIALZ9CAK2hCCygIeIIXeLWU9Wy9We/T0pI169mHP7I+fgAKs5Nm</latexit><latexit sha1_base64="hahxMb32b62Aj9TJQ3lu1YNwsTI=">AAAB8XicbZC7TsNAEEXH4RXCKzw6GosIiSqyaaCMoKGgCBJJELYVrdfjZJX12uyukSIrf0FDAUK0/AWfQMffsHkUkHCllY7undHOTJhxprTjfFulpeWV1bXyemVjc2t7p7q711ZpLim2aMpTeRcShZwJbGmmOd5lEkkScuyEg8tx3nlEqVgqbvUwwyAhPcFiRok21r3P8cGPkGvSrdacujORvQjuDGqNg+vPpix7zW71y49SmicoNOVEKc91Mh0URGpGOY4qfq4wI3RAeugZFCRBFRSTiUf2sXEiO06leULbE/d3R0ESpYZJaCoTovtqPhub/2VeruPzoGAiyzUKOv0ozrmtU3u8vh0xiVTzoQFCJTOz2rRPJKHaHKlijuDOr7wI7dO6a/jGrTUuYKoyHMIRnIALZ9CAK2hCCygIeIIXeLWU9Wy9We/T0pI169mHP7I+fgAKs5Nm</latexit><latexit sha1_base64="8WZk1MgupJfKQNNxF6qpvzGBZWs=">AAAB8XicbZA9T8MwEIYv5auUrwIji0WFxFQlLDBWsDAWiX6IJqoc59JadZxgO0hV1X/BwgBCrPwbNv4NbpsBWl7J0qP37uS7N8wE18Z1v53S2vrG5lZ5u7Kzu7d/UD08aus0VwxbLBWp6oZUo+ASW4Ybgd1MIU1CgZ1wdDOrd55QaZ7KezPOMEjoQPKYM2qs9eALfPQjFIb2qzW37s5FVsEroAaFmv3qlx+lLE9QGiao1j3PzUwwocpwJnBa8XONGWUjOsCeRUkT1MFkvvGUnFknInGq7JOGzN3fExOaaD1OQtuZUDPUy7WZ+V+tl5v4KphwmeUGJVt8FOeCmJTMzicRV8iMGFugTHG7K2FDqigzNqSKDcFbPnkV2hd1z/KdV2tcF3GU4QRO4Rw8uIQG3EITWsBAwjO8wpujnRfn3flYtJacYuYY/sj5/AGkDJDf</latexit>
R⇤(n, ✏, �) :<latexit sha1_base64="dyu12PUwZGuowFqbHHoaArjGN7s=">AAACAHicdZBNSwMxEIZn/bZ+VT148BJaBL8ou7VY9VT04lHFqtDWkk2nbWg2uyRZoZRe/CWCFw+KePVnePPfmG0VVPSFwMM7M0zm9SPBtXHdd2dkdGx8YnJqOjUzOze/kF5cutBhrBiWWShCdeVTjYJLLBtuBF5FCmngC7z0O0dJ/fIGleahPDfdCGsBbUne5Iwaa9XTK2fXm+tyu4qR5iK00EBh6MZBPZ11c/t7u/nCLnFzrlv08l4C+WJhp0A86yTKljLVrTsAOKmn36qNkMUBSsME1briuZGp9agynAnsp6qxxoiyDm1hxaKkAepab3BAn6xZp0GaobJPGjJwv0/0aKB1N/BtZ0BNW/+uJeZftUpsmnu1HpdRbFCy4aJmLIgJSZIGaXCFzIiuBcoUt38lrE0VZcZmlrIhfF1K/oeLfM6zfOplS4cw1BSsQgbWwYMilOAYTqAMDPpwD4/w5Nw6D86z8zJsHXE+Z5bhh5zXD5hUl0A=</latexit><latexit sha1_base64="dz7RVEy9cdzpYFf7gvFeIsuRsKs=">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</latexit><latexit sha1_base64="dz7RVEy9cdzpYFf7gvFeIsuRsKs=">AAACAHicdZDLSgMxFIYzXmu9VV24cBMsgjfKTC1tdTXoxmUVq0JbSyY9bUMzmSHJCGXoxmfwDdy4UMStj+Gub2OmVVDRHwIf/zmHk/N7IWdK2/bQmpicmp6ZTc2l5xcWl5YzK6uXKogkhSoNeCCvPaKAMwFVzTSH61AC8T0OV17vJKlf3YJULBAXuh9CwycdwdqMEm2sZmb9/GZ3W+zXIVSMBwZawDXZOWpmsnbusFzMF4rYztl2yck7CeRLhYMCdoyTKOtu1vfuh26/0sy811sBjXwQmnKiVM2xQ92IidSMchik65GCkNAe6UDNoCA+qEY8OmCAt4zTwu1Amic0HrnfJ2LiK9X3PdPpE91Vv2uJ+VetFul2uREzEUYaBB0vakcc6wAnaeAWk0A17xsgVDLzV0y7RBKqTWZpE8LXpfh/uMznHMNnTtY9RmOl0AbaRNvIQSXkolNUQVVE0QA9oCf0bN1Zj9aL9TpunbA+Z9bQD1lvH6G+mMY=</latexit><latexit sha1_base64="6ME4DRxcE1+uAoSHTxHFzC6uXAY=">AAACAHicdZDLSgMxFIYzXmu9VV24cBMsQhUpM7W01VXRjcsq9gJtLZn0TBuayQxJRiilG1/FjQtF3PoY7nwbM20FFf0h8PGfczg5vxtyprRtf1hz8wuLS8uJleTq2vrGZmpru6aCSFKo0oAHsuESBZwJqGqmOTRCCcR3OdTdwUVcr9+BVCwQN3oYQtsnPcE8Rok2Vie1e317lBHHLQgV44GBLnBNDs86qbSdPS0VcvkCtrO2XXRyTgy5Yv4kjx3jxEqjmSqd1HurG9DIB6EpJ0o1HTvU7RGRmlEO42QrUhASOiA9aBoUxAfVHk0OGOMD43SxF0jzhMYT9/vEiPhKDX3XdPpE99XvWmz+VWtG2iu1R0yEkQZBp4u8iGMd4DgN3GUSqOZDA4RKZv6KaZ9IQrXJLGlC+LoU/w+1XNYxfOWky+ezOBJoD+2jDHJQEZXRJaqgKqJojB7QE3q27q1H68V6nbbOWbOZHfRD1tsnhdWVtw==</latexit>
Channel
BobXn
ŴYn
W
Zn Leakage ≤ dEve
Alice
[Yang, Schaefer, Poor, IT’19]
• Legitimate channel is deterministic, eavesdropper channel is BSC:
Semi-deterministic Wiretap Channel (BSC):
!X
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5CS = 0.5
Second-order Approximation
Converse
Converse [Y.-Schaefer-Poor’16]
Achievability
Blocklength, n
Secrecy
rate,bit/(ch.use)
1
� = ✏ = 10�3<latexit sha1_base64="tpkhU/E/IBNWo1ZiGXNq+neYhCo=">AAACAHicbZDLSsNAFIYn9VbrLerChZtgEdxYEhV0Uyi6cVnBXqCJZTI5aYdOJmFmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/H7CqFS2/W2UlpZXVtfK65WNza3tHXN3ry3jVBBokZjFoutjCYxyaCmqGHQTATjyGXT80c2k3nkEIWnM79U4AS/CA05DSrDSVt88cANgCtddSCRlMa879kN2ep73zapds6eyFsEpoIoKNfvmlxvEJI2AK8KwlD3HTpSXYaEoYZBX3FRCgskID6CnkeMIpJdND8itY+0EVhgL/biypu7viQxHUo4jX3dGWA3lfG1i/lfrpSq88jLKk1QBJ7NFYcosFVuTNKyACiCKjTVgIqj+q0WGWGCidGYVHYIzf/IitM9qjua7i2rjuoijjA7RETpBDrpEDXSLmqiFCMrRM3pFb8aT8WK8Gx+z1pJRzOyjPzI+fwDLxpXe</latexit><latexit sha1_base64="tpkhU/E/IBNWo1ZiGXNq+neYhCo=">AAACAHicbZDLSsNAFIYn9VbrLerChZtgEdxYEhV0Uyi6cVnBXqCJZTI5aYdOJmFmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/H7CqFS2/W2UlpZXVtfK65WNza3tHXN3ry3jVBBokZjFoutjCYxyaCmqGHQTATjyGXT80c2k3nkEIWnM79U4AS/CA05DSrDSVt88cANgCtddSCRlMa879kN2ep73zapds6eyFsEpoIoKNfvmlxvEJI2AK8KwlD3HTpSXYaEoYZBX3FRCgskID6CnkeMIpJdND8itY+0EVhgL/biypu7viQxHUo4jX3dGWA3lfG1i/lfrpSq88jLKk1QBJ7NFYcosFVuTNKyACiCKjTVgIqj+q0WGWGCidGYVHYIzf/IitM9qjua7i2rjuoijjA7RETpBDrpEDXSLmqiFCMrRM3pFb8aT8WK8Gx+z1pJRzOyjPzI+fwDLxpXe</latexit><latexit sha1_base64="tpkhU/E/IBNWo1ZiGXNq+neYhCo=">AAACAHicbZDLSsNAFIYn9VbrLerChZtgEdxYEhV0Uyi6cVnBXqCJZTI5aYdOJmFmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/H7CqFS2/W2UlpZXVtfK65WNza3tHXN3ry3jVBBokZjFoutjCYxyaCmqGHQTATjyGXT80c2k3nkEIWnM79U4AS/CA05DSrDSVt88cANgCtddSCRlMa879kN2ep73zapds6eyFsEpoIoKNfvmlxvEJI2AK8KwlD3HTpSXYaEoYZBX3FRCgskID6CnkeMIpJdND8itY+0EVhgL/biypu7viQxHUo4jX3dGWA3lfG1i/lfrpSq88jLKk1QBJ7NFYcosFVuTNKyACiCKjTVgIqj+q0WGWGCidGYVHYIzf/IitM9qjua7i2rjuoijjA7RETpBDrpEDXSLmqiFCMrRM3pFb8aT8WK8Gx+z1pJRzOyjPzI+fwDLxpXe</latexit><latexit sha1_base64="tpkhU/E/IBNWo1ZiGXNq+neYhCo=">AAACAHicbZDLSsNAFIYn9VbrLerChZtgEdxYEhV0Uyi6cVnBXqCJZTI5aYdOJmFmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/H7CqFS2/W2UlpZXVtfK65WNza3tHXN3ry3jVBBokZjFoutjCYxyaCmqGHQTATjyGXT80c2k3nkEIWnM79U4AS/CA05DSrDSVt88cANgCtddSCRlMa879kN2ep73zapds6eyFsEpoIoKNfvmlxvEJI2AK8KwlD3HTpSXYaEoYZBX3FRCgskID6CnkeMIpJdND8itY+0EVhgL/biypu7viQxHUo4jX3dGWA3lfG1i/lfrpSq88jLKk1QBJ7NFYcosFVuTNKyACiCKjTVgIqj+q0WGWGCidGYVHYIzf/IitM9qjua7i2rjuoijjA7RETpBDrpEDXSLmqiFCMrRM3pFb8aT8WK8Gx+z1pJRzOyjPzI+fwDLxpXe</latexit>
R⇤(n, ✏, �) = Cs �
rV
nQ�1
✓�
1� ✏
◆+O
✓log n
n
◆
<latexit sha1_base64="2XYj1zXI74+cqD57yOAJUqzVM0k=">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</latexit><latexit sha1_base64="2XYj1zXI74+cqD57yOAJUqzVM0k=">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</latexit><latexit sha1_base64="2XYj1zXI74+cqD57yOAJUqzVM0k=">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</latexit><latexit sha1_base64="2XYj1zXI74+cqD57yOAJUqzVM0k=">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</latexit>
[Yang, et al. (2017)]
4078 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 65, NO. 7, JULY 2019
Fig. 2. Secrecy rate for the Gaussian wiretap channel with P/N1 = 3 dB, P/N2 = −3 dB, ε = δ = 10−3.
where CS is given in (114) and
Vi = log2 e2
P2 + 2P Ni
(P + Ni )2 , i ∈ {1, 2} (117)
Vc = V1 + V2 − P N1
P + N1
(1
N2+ 1
P + N2
)log2 e. (118)
Proof: See Appendix VI.
C. Numerical Results and Discussions
In this section, we compare the bounds proposed in thispaper with existing bounds in [25] and [35], and with theapproximations provided in Theorem 14 for a Gaussian wire-tap channel (with the O(·) terms omitted). The results areshown in Fig. 2.
Let us first explain how each bound was computed. Notethat, every achievability bound for the wiretap channel consistsof two parts: a reliability bound on the decoding performanceof the legitimate receiver and a secrecy bound on the informa-tion leakage to the eavesdropper (see, e.g., Theorem 7). Forthe reliability part, we have used Shannon’s channel codingachievability bound [49], which is the tightest achievabilitybound for Gaussian channels [7, Sec. III.J-4]. In particular,let MShannon(ε, n) be the number of codewords for a givenprobability of error ε and blocklength n implied by Shannon’sachievability bound. The quantity MShannon(ε, n) can be com-puted via the numerical routine in [50]. The maximal secrecyrate R∗
avg(n, ε, δ) can be lower-bounded by
R∗avg(n, ε, δ) ≥ 1
nlog
MShannon(ε, n)
L(119)
where L may be interpreted as the number of “dummy”messages that must be conveyed through the wiretap channelto achieve the desired secrecy. To compute an achievabilitybound on R∗
avg(n, ε, δ), it suffices to determine an upper bound
on L. Note that, each of the achievability bound plottedin Fig. 2 is computed using (119) and corresponds to arespective method for upper-bounding L.
1) Achievability (Theorem 7): The result is an applicationof the secrecy bound (45). To compute (45), we have chosenQZ to be a multivariate Gaussian distribution with i.i.d.N (0, P + N2) elements, and have chosen A to be the powersphere Sn = {xn : ‖xn‖2 = n P}. By the rotational symmetryof the set A and of the distributions PZ |X=x and QZ , the twosupremums on the right-hand side of (45) are both achievedby an arbitrary x ∈ A. In particular, we have set
x = x n ! [√
P, . . . ,√
P] (120)
in the computation. With these choices, (45) becomes
δ ≤ 1 − E[exp
(−|Bn − log γ |+
)]
+ 12
√γ
LE
[exp(−|Bn − log γ |)
](121)
where
Bn ! n2
log(
1 + PN2
)+ log e
2
n∑
i=1
(
1 − (√
P Zi − √N2)2
P + N2
)
(122)
with {Zi} i.i.d. CN (0, 1) distributed. The inequality (121)implies the following upper bound on L:
√L ≤ min
γ
√γ E
[exp(−|Bn − log γ |)
]
2(δ + E[exp
(−|Bn − log γ |+
)]− 1)
(123)
where the minimization is over all γ > 0 such that thedenominator in (123) is positive. Substituting (123) into (119)we obtain the desired result.
Gaussian Wiretap Channel(SNRBob = 3db; SNREve = - 3db)
[Yang, Schaefer, Poor, IT’19]
Key Generation from Common Randomness
• Passive Eavesdropper:
– Public discussion
– Channel reciprocity: joint source-channel model
– Relay assisted: trusted or oblivious
• Active Eavesdropper:
– Channel reciprocity: joint source-channel model
Lai, Liang, Du, Poor (2015)Key Generation from Random Channelsin Physical Layer Security inWireless Communications (CRC)
Privacy-Utility Tradeoffs
in
Sensing Systems
• Privacy is not secrecy:
Privacy vs. Secrecy
• Privacy is not secrecy:
• Denial of access (secrecy) makes a data source useless.
= Eve ?
Privacy vs. Secrecy
• Sensing systems generate considerable electronic data:
• Data’s utility depends on its accessibility.
• Accessibility endangers privacy.
• This fundamental tradeoff can becharacterized via information theory.
Privacy pEquivocation E
Utility UDistortion D
Utility-PrivacyTradeoff Region
Privacy-Utility Tradeoff
• Smart meter data is useful for price-aware usage, load balancing
• But, it leaks information about in-home activity
Example: Smart Meter Privacy
Giaconi, Gunduz, Poor (2021) Smart Meter Privacy, in Advanced Data Analytics forPower Systems (Cambridge)
P-U tradeoff leads to a spectral ‘reverse water-filling’ solution
The following theorem captures our main result.
Theorem 3: The utility-privacy tradeoff for smart metermeasurements modeled as a Gaussian source with memory isgiven by the leakage function λ(D) which results from choos-ing the distribution p (xn|xn) as the rate-distortion (withoutprivacy) optimal distribution.
Proof: The proof follows directly from noting that, for agiven jointly Gaussian distribution of the source and correlatedhidden sequence, pXnY n , the infimum in (8) and (9) is strictlyover the space of conditional distributions of the revealedsequence given the original source sequence as a result ofthe Markov chain relationship Y n − Xn − Xn. Expandingthe leakage as I(Y n; Xn) = h(Y n) − h(Y n|Xn), and usingthe fact for correlated Gaussian processes, Yk = αkXk + Zk,for all k, where {Zk} is a sequence independent of {Xk}and αk is a constant for each k, one can show that the jointlyGaussian distribution of Xn and Xn which minimizes (8) alsominimizes (9).
Remark 2: Theorem 3 simplifies the development of theRDL region for Gaussian sources with memory for which therate-distortion function is known. For Gaussian sources withmemory the rate-distortion function is known and lends itselfto a straightforward practical implementation that we discussin the following section.
F. Rate-Distortion for Gaussian Sources with Memory
In general, the rate distortion functions for sources withmemory are not straightforward to compute. However, forGaussian sources, the rate-distortion function R(D) (withoutthe additional privacy constraint) is known and can be obtainedvia a transformation of the correlated source sequence Xn
to its eigen-space in which the resulting sequence Xn isuncorrelated (and hence, independent for jointly Gaussiansources); let SX(ω), SY (ω), and SXY (ω) denote the two-sided power spectral densities (PSDs) of the {Xk} , {Yk},and {XkYk} processes, respectively [16]. Let φ denote theLagrangian parameter for the distortion constraint (4) in therate minimization problem. Explicitly denoting the dependenceon the water-level φ, the rate-distortion function Rφ (D) andthe average distortion function D (φ) are given by
Rφ (D) =
∫ π
−π
max
(
0,1
2log
SX(ω)
φ
)
dω
2π(10)
D (φ) =
∫ π
−π
min (SX(ω),φ)dω
2π. (11)
Note that the water-level φ is determined by the desiredaverage distortion D (φ) = D. Thus, R(D) for a Gaussiansource with memory can be expressed as an infinite sum of therate-distortion functions for independent Gaussian variables,one for each angular frequency ω ∈ [−π,π]. The “water-level” φ captures the average time-domain distortion constraintacross the spectrum such that the distortion for any ω is theminimum of the water-level and the PSD. The privacy leakageλ(D (φ)) is then the infinite sum of the information leakage
0 0 0
0.5
1
1.5
2
2.5
3
ωπ−π
S (ω)
φ
Fig. 1. The PSD of {Xk}. The area below the curve and the horizontal lineis equal to D.
about {Yk} for each ω, and is given by
λ (D (φ)) =
∫ π
−π
1
2log
(
SY (ω)
SXY (ω)g (ω) + SY (ω)
)
dω
2π(12)
where g (ω) ≡ (min (SX(ω),φ) − 1) .Remark 3: The transform domain “water-filling” solution
suggests that in practice the time-series data can be filteredfor a desired level of fidelity (distortion) and privacy (leak-age) using Fourier transforms. The privacy-preserving rate-distortion optimal scheme thus reveals only those frequencycomponents with power above the water-level φ. Furthermore,at every frequency only the portion of the signal energy whichis above the water level φ is preserved by the minimum-ratesequence from which the source can be generated with anaverage distortion D.
IV. ILLUSTRATION
The following example illustrates our results. We assumethat the private information to be hidden is the measurementsequence itself, i.e., Yk = Xk, for all k. For the metermeasurements modeled as a stationary Gaussian time series{Xk} , we choose Xk ∼ N (0, 1) for all k ∈ I, and anautocorrelation function
cm = E[XkXk+m] =
1 m = 0,0.3 m = ±1,0.4 m = ±2,0 otherwise.
The power spectral density PSD (frequency domain represen-tation of the autocorrelation function) of this process is givenby
S(ω) =∞∑
m=−∞
cm exp(imω) = 1+0.6 cos(ω)+0.8 cos(2ω),
− π ≤ ω ≤ π. (13)
In order to obtain the rate-distortion function Rφ(D) for thissource, for a given D we have to find the water-level φsatisfying (11).
Source Coding Solution:Hidden Gauss-Markov Model (protection of the hidden
intermittency state)
wasted energyversus
information leakageTradeoff:
A Control Approach:Energy Harvesting and Storage
º
ENERGY MANAGEMENT UNIT (EMU)
APPLIANCES
ENERGY HARVESTING (EH)
DEVICE
Char
ging
Disc
harg
ing
Harv
este
dEn
ergy
Output LoadInput Load
UTILITY PROVIDER
(UP)
RECHARGEABLE BATTERY (RB)
iX iY
iZ
SMAR
T M
ETER
(S
M)
Info
rmat
ion
Leak
age
Rat
e, I
(pX = pZ = ½)
Privacy-Utility Tradeoff:Binary Variables
Competitive Privacy: Privacy-Utility Tradeoffs for Interacting Agents
• Multiple interacting, but competing,agents (or groups of agents) withcoupled measurements.
• Each wants to estimate its ownparameters, or state.
• They can help each other by sharingdata, but wish to preserve privacy.
• Each has a privacy-utility tradeoff, butthey are competitive ones.
• How should they interact?
Motivating Examples
Electricity Grids: grid management
Radar: untrustworthy allies
Sensor Networks: resource localization
Yk = Hk ,mXm + Zk
m=1
M
∑ , k = 1,2,…,M
mth system state
• Noisy measurements at agent k:
• Utility for agent k: mean-square error for its own state Xk
• Privacy for agent k: leakage of information about Xk to other agents
Linear Measurement Model
How Should Agents Exchange Data?
• This is a classical problem in information theory – theWyner-Ziv problem (optimal distributed sourcecoding) – which tells how to exchange information.
• But, doesn’t say how much information to exchange.
• Because of the competitive nature, gametheory or prospect theory can illuminate this.
• Leads to a number of interesting solutions:– a basic problem is a prisoners’ dilemma
– with pricing, cooperation or multi-play games,more meaningful solutions arise
Poor (2018)Privacy in Networks of Interacting Agentsin Emerging Applications of Control and SystemTheory (Springer)
Other Issues
Other Issues
• Authentication
– Information theoretic bounds on the probabilities of successful impersonation and
substitution attacks [Lai, et al. IT’09]
• Attacks on MANETs
– Information theoretic guidance on how many malicious nodes can be tolerated
[Liang, et al. IT’11]
• Man-in-the-Middle and Spoofing Attacks on Sensor Nets
– Effects on CRLB in parameter estimation [Zhang, et al. SPM’18]
Authentication with Correlated Sequences
Impersonation attack: O transmits a message before S
Substitution attack: O replaces S’s message with its own
S R
O
s1L s2L
M
Theorem [Lai, et al. IT’09]: If the S-O channel is not less noisy than the S-R channel, then
Eav�
Eav�
Eav�
MANETs with Malicious Nodes
Secrecy Capacity Scaling[Liang, et al. IT’11]
Attacks on Sensor Nets[Zhang, et al. SPM’18]
3
studied in [3], [4], [7]–[9] and references therein. Investigations on cyberattacks on estimation systems
have been studied in [6], [7], [10]–[15], [17], [23] and references therein. The early work in [3], [4],
[6], [7], [7]–[9], [13]–[15], [23] set the tone for many later investigations and influenced most of the
discussions in this paper. In particular, the impact and mitigation of cyberattacks on systems solving
hypothesis testing problems was studied in [3], [4], [7], [9] and references therein. Distributed detection
in tree topologies in the presence of cyberattacks was considered in [8]. Investigations on cyberattacks
on estimation systems have been studied in [6], [7], [10]–[15], [17], [23] and references therein. The
problem of distributed spectrum sensing in a cognitive radio network under cyberattacks was studied in
[4], [5], [25]. Several cyberattack detection techniques were proposed for IoT localization systems in [6],
[12], [18], [19]. More recently, the data-injection attack in smart grids was considered in [13]–[15], [26]
and the references therein.
FusionCenter
QuantizerSensor 1
Physical phenomenon
QuantizerSensor 2
QuantizerSensor 3
QuantizerSensor N
Spoofing Attack
���
Man-in-the-
middle Attack
Fig. 1: Cyberattacks in IoT systems.
According to the place where they occur, cyberattacks in IoT systems can be categorized into two
classes, as illustrated in Fig. 1. We call any attack modifying a signal in the IoT system prior to
quantization a spoofing attack. It has been shown [12] that the same changes in the signals in the
IoT system produced by any spoofing attack can also be produced by changing the data going into
the sensors to be different from that coming from the physical phenomenon being monitored. We call
any attack modifying a signal in the IoT system after quantization a man-in-the-middle attack (MiMA).
The same changes in the signals in the IoT system produced by any MiMA [12] can also be produced
May 22, 2018 DRAFT
Man-in-the-middle attack:- TQA uses attacked data- SEA ignores attacked data
• Information theory can help understand some fundamental limits of
security and privacy in wireless networks
• These are theoretical constructs; although they sometimes point to
potential practical solutions, there are many needs to connect this kind ofanalysis to real networks, e.g.
- more finite-blocklength analysis
- scaling laws for large networks
- practical coding schemes to achieve fundamental limits
- other security primitives (signatures, certificates, etc.)
• Not all security has to be iron-clad: Quality of Security can be aparameter, just like Quality of Service
Closing Thoughts
Some References
Schaefer, Boche, Khisti, Poor (2017) Information Theoretic Security and Privacy ofInformation Systems (Cambridge)
Poor, Schaefer (2017) Wireless Physical Layer Security, PNAS.
Poor (2018) Privacy in Networks of Interacting Agents, in Emerging Applications ofControl and System Theory (Springer)
Yang, Schaefer, Poor (2019) Wiretap Channels: Nonasymptotic FundamentalLimits, IEEETransactions on Information Theory
Yagli, Dytso, Poor (2020) Information-Theoretic Bounds on the GeneralizationError and Privacy Leakage in Federated Learning, Proceedings SPAWC
Giaconi, Gunduz, Poor (2021) Smart Meter Privacy, in Advanced Data Analytics forPower Systems (Cambridge)
Herfeh, Chorti, Poor (2021) Physical Layer Security: Authentication, Integrity andConfidentiality, in Physical Layer Security (Springer)
Thank You!