Information Theoretic Security: Fundamentals and Applications

Information Theoretic Security: Fundamentals and Applications Ashish Khisti University of Toronto

IPSI Seminar Nov 25th 2013

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Layered Architectures Layered architecture for communication systems. Where is Security? Application Layer

Encryption, Authentication

(Semantics of Information)

Transport Layer

Secure Socket Layer

(End to End Connectivity)

Network Layer

Virtual Private Networks Anonymous Routing

(Routing and Path Discovery)

Data Link Layer

Device level Authentication

(Error Correction Codes)

Physical Layer

Anti-Jamming

(Signals, RF hardware)

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Layered Architectures Layered architecture for communication systems. Where is Security? Application Layer

Encryption, Authentication

(Semantics of Information)

Transport Layer

Secure Socket Layer

Wireless Systems

(End to End Connectivity)

Network Layer

Virtual Private Networks Anonymous Routing

(Routing and Path Discovery)

Data Link Layer (Error Correction Codes)

Physical Layer (Signals, RF hardware)

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Traditional Approach A typical graduate level course in computer security introduces Shannon’s notion of security. Shannon’s Notion

Message W

Alice key K

X

X X

Bob

decoded message W

key K

Eve

Perfect Secrecy: p(w|x)=p(w)

Note that Key Size = Message length, hence impractical Focus: computational cryptography Is this all about information theoretic security? Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Outline

Motivating Applications Secure Biometrics Smart-Meter Privacy Wireless Systems

Information Theoretic Models Wiretap Channel Model Secret-key agreement

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Biometric Technologies

Laptop

ATM

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Passport

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Biometric Technologies

Laptop

ATM

Passport

Enrollment Biometric Stored in clear

Feature Extraction

=? Authentication Feature Extraction

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Biometric Technologies

Laptop

ATM

Passport

Enrollment Biometric Stored in clear

Feature Extraction

=? Authentication Feature Extraction

Issue: Biometrics are stored in the clear Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Biometrics: Toy Example Enrolment Biometric 0

1

1

0

1

0

0

Biometric Channel Authentication Biometric No Error

Bit 1 Flipped

Bit 6 Flipped

Bit 7 Flipped

8 Possible Events : All Equally Likely Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Biometrics: Toy Example

X, Y : length seven binary sequence Channel Model: one bit flip (8 possibilities) 3 bits required.

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Biometrics: Toy Example

X, Y : length seven binary sequence Channel Model: one bit flip (8 possibilities) 3 bits required. Syndrome Decoder

Syndrome Encoder

1 1 0

0

1 0

0

0

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Authentication Biometric

1

Syndrome bits

Enrollment Biometric

0

Syndr. 0

0

1

1

0

1

0

1 0 0 1 0 0

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Biometrics: Toy Example

X, Y : length seven binary sequence Channel Model: one bit flip (8 possibilities) 3 bits required. Syndrome Decoder

Syndrome Encoder

1 1 0

0

1 0

0

0

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Authentication Biometric

1

Syndrome bits

Enrollment Biometric

0

Syndr. 1

1

1

1

0

1

0

1 0 0 1 0 0

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Biometrics: Toy Example

X, Y : length seven binary sequence Channel Model: one bit flip (8 possibilities) 3 bits required. Syndrome Decoder

Syndrome Encoder

1 1 0

0

1 0

0

0

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Authentication Biometric

1

Syndrome bits

Enrollment Biometric

0

Syndr. 0

1

1

0

0

1

0

0 0 0 1 0 0

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Biometrics: Toy Example

X, Y : length seven binary sequence Channel Model: one bit flip (8 possibilities) 3 bits required. Syndrome Decoder

Syndrome Encoder

1 1 0

0

1 0

0

0

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Authentication Biometric

1

Syndrome bits

Enrollment Biometric

0

Syndr. 0

1

1

0

0

0

0

1 1 0 1 0 0

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Privacy Preserving Biometrics S. Draper, A. Khisti, et. al “Using distributed source coding to secure fingerprint biometrics” ICASSP, 2007

Encode enrollment biometric

X

Syndrome Encoding

Store syndrome S

S

Original enrollment biometric

Store syndromes Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Privacy Preserving Biometrics S. Draper, A. Khisti, et. al “Using distributed source coding to secure fingerprint biometrics” ICASSP, 2007

Encode enrollment biometric

X

Syndrome Encoding

Original enrollment biometric

Store syndrome S

S

Fingerprint Channel

Decode w/ probe biometric Syndrome Decoding

Y

Original enrollment biometric

Authentication biometric

Store syndromes Reproduce enrollment biometric Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Privacy Preserving Biometrics S. Draper, A. Khisti, et. al “Using distributed source coding to secure fingerprint biometrics” ICASSP, 2007 Biometric Authentication

Encode enrollment biometric

X

Original enrollment biometric

Syndrome Encoding

One way hash Store syndrome S

S

Fingerprint Channel

Decode w/ probe biometric

One way hash

Syndrome Decoding

Y

Original enrollment biometric

Authentication biometric

Store syndromes Reproduce enrollment biometric Authenticate Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Smart-Meter Privacy D. Varodayan and A Khisti, ICASSP 2011

C. Efthymiou and G. Kalogridis, Smart grid privacy via anonymization of smart metering data, Smart Grid Commun. Conf., Gaithersburg, 2010.

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Smart-Meter Privacy D. Varodayan and A Khisti, ICASSP 2011

X User Appliances

Rechargeable Battery

Y

Utility Company

Privacy Leakage: I (X N ; Y N ) Battery: Limited Storage Model Battery as a Finite State Communication Channel “Design the Channel” Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secret-Key Generation in Wireless Fading Channels

Forward Link y B ! hAB x A

mA

n AB

B

A y A ! hBA xB

nBA

mB

Reverse Link KB

KA

Channel Gain Forward Channel Reverse Channel

Fading Reciprocity Spatial Decorrelation Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

time

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Secret-Key Generation in Wireless Fading Channels

Forward Link y B ! hAB x A

mA

n AB

B

A y A ! hBA xB

nBA

Reverse Link KB

KA

z AE ! g AE x A n AE Channel Gain

mB

z BE ! g BE xB

nBE

E

Eavesdropper Link Forward Channel Reverse Channel

Fading Reciprocity Spatial Decorrelation Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

time

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Secret-Key Generation in Wireless Fading Channels A. Khisti 2013

Forward Link y B ! hAB x A

mA

n AB

B

A y A ! hBA xB KA

nBA

Reverse Link KB

z AE ! g AE x A n AE

Start

mB

z BE ! g BE xB

nBE

E Two Phase Approach:

Channel Probing N hˆAB

N hˆBA

Key Extraction

Phase I: Channel Probing and N , h ˆN ) Estimation: (hˆAB BA Phase 2: Source Reconciliation and Key Extraction Secret-Key Generation:

Limits Shared Key Information Theoretic Security: Fundamentals and Applications :

Ashish Khisti (University of Toronto)

Capacity 11 / 35

Secure MIMO Communication

Dec. 101011

Rx.

Enc.

Tx.

101011

?????? Eaves.

Signal of interest: direction of legitimate receiver. Synthetic noise: null-space of legitimate receiver.

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secure MIMO Multicast A. Khisti, 2011

Artificial Noise Alignment Noise Symbols IA Precoder

Information Symbols

Rx1 Ev 1 Rx2

Signal Masking

Transmitter

Align Noise Symbols at Legitimate Receivers Mask Information Symbols at Eavesdroppers Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Outline

Motivating Applications Secure Biometrics Smart-Meter Privacy Wireless Systems

Information Theoretic Models Wiretap Channel Model Secret-key agreement

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Wiretap Channel Wyner’75

AWGN Wiretap Channel Model Zrn Yrn M

Encoder

Xn

Receiver

ˆ M

Eaves.

??

Zen Yen

n ˆ −→ Reliability Constraint : Pr(M 6= M) 0

Secrecy Constraint :

1 n n H(M|Ye )

= n1 H(M) − on (1)

Secrecy Capacity Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secrecy Criterion

1 1 H(M|Yen ) = H(M) −on (1) } |n {z |n {z } Equivocation rate

Information rate

Perfect Secrecy: on (1) ≡ 0, (Shannon ’49) n

Weak Secrecy: on (1) −→ 0, (Wyner ’75)
Strong Secrecy: on (1) ∈ O n1 , (Maurer and Wolf ’00) Guessing approach : (Arikan & Merhav ’02) Focus: Wyner’s notion

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Joint Encryption and Encoding Separation based approach vs. Wiretap codes Traditional Approach : Separation ... Zrn Yrn

Key

M

Encryption

Encoder

Xn

Decoder

Decryption

ˆ M

Zen Yen

Traditional Approach

Key

Eaves.

??

Wiretap Codes

Separation based

Joint encryption/encoding

Requires keys

Channel based secrecy

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Joint Encryption and Encoding Separation based approach vs. Wiretap codes Wiretap Codes: Joint Encryption and Encoding Zrn Yrn M

Secure Encoder

Xn

Decoder

Decryption

ˆ M

Zen Yen

Traditional Approach

Key

Eaves.

??

Wiretap Codes

Separation based

Joint encryption/encoding

Requires keys

Channel based secrecy

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Joint Encryption and Encoding Separation based approach vs. Wiretap codes Wiretap Codes: Joint Encryption and Encoding Zrn Yrn M

Secure Encoder

Xn

ˆ M

Zen Yen

Traditional Approach

Secure Decoder

Eaves.

??

Wiretap Codes

Separation based

Joint encryption/encoding

Requires keys

Channel based secrecy

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Wiretap Codes

Uniform Noise Wiretap Channel Model Zrn Yrn M

Secure Encoder

Xn

ˆ M

Zen Yen

QAM Modulation

Secure Decoder

Recv. Noise

??

Eaves.

Eaves. Noise

Ɣ

Uniform noise model

σe2 = 4σr2

Ɣ

σe2 = 4σr2 Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Wiretap Codes QAM Modulation

Recv. Noise

Eaves. Noise

Ɣ

Uniform noise model

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

σe2 = 4σr2

Ɣ

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Wiretap Codes Recv. Noise

QAM Modulation

σe2 = 4σr2

Eaves. Noise

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Uniform noise model Receiver’s Constellation Ɣ

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Cr = log2 64 = 6 b/s Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Eavesdropper’s Constellation Ɣ

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Ce = log2 16 = 4 b/s 19 / 35

Wiretap Codes Recv. Noise

QAM Modulation

σe2 = 4σr2

Eaves. Noise

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Uniform noise model Receiver’s Constellation Ɣ

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Eavesdropper’s Constellation

Cr = log2 64 = 6 b/s

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Ce = log2 16 = 4 b/s

Cs = Cr − Ce = 2 b/s Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Wiretap Codes Secure QAM Constellation

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Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Ɣ Msg 1 Ɣ Msg 2 Ɣ Msg 3 Ɣ Msg 4

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Wiretap Codes Encoding: Randomly select one candidate

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Ɣ Ɣ Msg 1 Ɣ Msg 2 Ɣ Msg 3 Ɣ Msg 4

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Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Wiretap Codes Decoding at legitimate receiver

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Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Ɣ Msg 1 Ɣ Msg 2 Ɣ Msg 3 Ɣ Msg 4

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Wiretap Codes Confusion at the eavesdropper

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Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Ɣ Msg 1 Ɣ Msg 2 Ɣ Msg 3 Ɣ Msg 4

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Gaussian Wiretap Channel Leung-Yan-Cheong and Hellman’78

Zrn Yrn M

Encoder

Xn

Receiver

ˆ M

Eaves.

??

Zen Yen

Secrecy Capacity Cs = {log(1 + SNRr ) − log(1 + SNRe )}+ = {C (SNRr ) − C (SNRe )}+ SNRr : Legitimate receiver’s signal to noise ratio SNRe : Eavesdropper’s signal to noise ratio Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Other Classical Results

The secrecy capacity was also characterized for: Degraded Memoryless Wiretap Channel(Wyner’75) X → Yr → Ye C = max I (X ; Yr ) − I (X ; Ye ) pX

Discrete Memoryless Wiretap Channel (Csiszar-Korner ’78) C = max I (U; Yr ) − I (U; Ye ), pU,X

U → X → (Yr , Ye ) Cardinality bounds on the alphabet of U

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Gaussian Wiretap Channel

101011

101011

Receiver Transmitter ?????? Eavesdropper

Strong Requirement: Eavesdropper must not be closer to the transmitter Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Gaussian Wiretap Channel

101011

101011

Receiver Transmitter

101011 Eavesdropper

Strong Requirement: Eavesdropper must not be closer to the transmitter Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Solution ... Multiple Antennas Khisti-Wornell 2010

Multi-antenna wiretap channel 1 0 1 1

Receiver 1 0 1 1

? ? ? ?

Transmitter

Eavesdropper

Spatial Diversity: Multiple Antennas Temporal Diversity: Fading Channels Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Solution ... Multiple Antennas Khisti-Wornell 2010

Multi-antenna wiretap channel 1 0 1 1

Receiver 1 0 1 1

Transmitter

? ? ? ?

Eavesdropper

Channel Model Yr = Hr X + Zr Ye = He X + Ze Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Channel matrices: Hr ∈ CNr ×Nt , He ∈ CNe ×Nt Nt : # Tx antennas AWGN noise: Zr , Ze 24 / 35

MIMOME: Secrecy Capacity Khisti-Wornell 2010

Theorem Secrecy capacity of the Multi-antenna wiretap channel is given by, Cs =

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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MIMOME: Secrecy Capacity Khisti-Wornell 2010

Theorem Secrecy capacity of the Multi-antenna wiretap channel is given by, Cs =

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Scalar Gaussian Case (Leung-Yan-Cheong & Hellman ’78), Cs = log(1 + SNRr ) − log(1 + SNRe ) New information theoretic upper-bound Convex Optimization Matrix Analysis Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secrecy Capacity: Remarks

Cs =

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secrecy Capacity: Remarks

Cs = 1

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Convex Reformulation Cs = min max R+ (Φ, Q) Φ∈P Q∈Q

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secrecy Capacity: Remarks

Cs = 1

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Convex Reformulation Cs = min max R+ (Φ, Q) Φ∈P Q∈Q

2

MISOME Case: rank-one covariance is optimal Cs = log+ λmax (I + Phr hr† , I + PHe† He )

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Secrecy Capacity: Remarks

Cs = 1

max

Q0:Tr (Q)≤P

log det(Ir + Hr QHr† ) − log det(Ie + He QHe† )

Convex Reformulation Cs = min max R+ (Φ, Q) Φ∈P Q∈Q

2

MISOME Case: rank-one covariance is optimal Cs = log+ λmax (I + Phr hr† , I + PHe† He )

3

High SNR case: GSVD transform Simultaneous diagonalization: (Hr , He )

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Masked Beamforming Scheme

MISOME Case: Yr = hr† X + Zr , Ye = He X + Ze Dec. 101011

Rx.

Enc.

Tx.

101011

?????? Eaves.

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

27 / 35

Masked Beamforming Scheme

MISOME Case: Yr = hr† X + Zr , Ye = He X + Ze Dec. 101011

Rx.

Enc.

Tx.

101011

?????? Eaves.

Signal of interest: direction of legitimate receiver. Synthetic noise: null-space of legitimate receiver.

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

27 / 35

Masked Beamforming vs. Capacity Achieving Scheme MISOME Case: Yr = hr† X + Zr , Ye = He X + Ze Masked beamforming scheme hr , H e Scalar Wiretap Code

Capacity achieving scheme hr , H e

hr Masked Beam-forming

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

Scalar Wiretap Code

Optimal Beam-forming

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Masked Beamforming vs. Capacity Achieving Scheme MISOME Case: Yr = hr† X + Zr , Ye = He X + Ze Masked beamforming scheme hr , H e Scalar Wiretap Code

Capacity achieving scheme hr , H e

hr Masked Beam-forming

Scalar Wiretap Code

Optimal Beam-forming

    P − RMB (hr , He , P) = 0 lim C hr , He , P→∞ Nt Transmit Power: P Transmit antennas: Nt Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Outline

Motivating Applications Secure Biometrics Smart-Meter Privacy Wireless Systems

Information Theoretic Models Wiretap Channel Model Secret-key agreement

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

29 / 35

Secret Key Generation Maurer ’93, Ahlswede-Csiszar ’93

uN

vN F = f (u N )

A

B k = KA (u N )

kˆ = KB (v N , f )

ˆ ≤ εN Error Probability: Pr(k 6= k) 1 Equivocation: N H(k|f ) ≥ N1 H(k) − εn Rate R = N1 H(k) Ckey = I (u; v ) Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Achievability Random Binning Technique (Slepian-Wolf ’73) uN

Bin-Index Slepain-Wolf Encoder

4

2

2

2 3 1

Bin 1

4

3 1

3 1

Bin 2

No. of Bins: ≈ 2nH(v |u) No. of Sequences/Bin: ≈ 2nI (u;v ) Information Theoretic Security: Fundamentals and Applications :

Ashish Khisti (University of Toronto)

4

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Joint Source and Channel Coding Khisti-Diggavi-Wornell ’08

vN uN yn dec

xn Enc.

p(y , z|x)

zn w.t.

Two types of uncertainty Sources Channel How to combine both these equivocation for secret-key-distillation? Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Achievability Wiretap Codebook Bin

uN

yn

xn

Index

Bin

vN

Wyner-Ziv Codebook W-Z

Wiretap Decoder

Codeword

W-Z

Secret-Key Codebook

k

k

Encoder

Index

Wyner-Ziv Decoder

Codeword

Secret Key Codebook

Decoder

Rkey = max βI (t; v ) + I (x; y ) − I (x; z) t,x | {z } | {z } src. equiv.

t → u → v,

channel equiv.

β{I (t; u) − I (t; v )} ≤ I (x; y )

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Capacity Results

Rkey = max βI (t; v ) + I (x; y |z) t,x

t → u → v,

β{I (t; u) − I (t; v )} ≤ I (x; y )

Upper and lower bounds coincide, when channels are degraded or parallel reversely degraded broadcast. Capacity for Parallel Gaussian broadcast channels and Gaussian sources Extension to side information at the eavesdropper, when sources and channels are degraded. Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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Conclusions

Motivating Applications Secure Biometrics Smart-Meter Privacy Wireless Systems

Information Theoretic Models Wiretap Channel Model Secret-key agreement

Information Theoretic Security: Fundamentals and Applications : Ashish Khisti (University of Toronto)

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