1. Information-Theoretic Security
EE 25N, Science of Information
11/08/2018
Ziv Goldfeld

2. Past Lectures - Communication Despite Noise
Alice the sender and Bob the receiver
Communicate a message as a strings of 0’s and 1’s (bits)
Use longer bit strings (codewords) to protect message carrying bits
Agree on a strategy beforehand:
1) Set of codewords used
2) Encoding for Alice
3) Decoding for Bob
⇒ Error correcting codes for reliability
010001
011000

3. Communication under Eavesdropping
0100
1011…
Alice and bob wish to communicate
Channel is noiseless
But Eve taps their line
They don’t want Eve to decipher their chat
Assumptions on Eve:
She sees their transmitted bit string
She knows their communication strategy (aka code)
She has an extremely powerful computer
Q: Can Alice send Bob a secret message without Eve finding out?
A: Not without an additional recourse!

4. Resource 1: Pre-Eve Secret

5. ? Simple Case Study E B A 𝐾 Alice sends Bob a bit 𝑀∈ 0,1
Hey, do you remember if it was raining on the first day of our last vacation?
Sure I remember E 𝑀 B A 𝐾 𝐶
Alice sends Bob a bit 𝑀∈ 0,1
Bit probability: 𝑃 𝑀 0 = 𝑃 𝑀 1 = 1 2
They share a secret Eve has no access to
⇒ Resource: 1 secret bit 𝐾∈ 0,1
Formally:
Alice: 𝑀,𝐾 →𝐶
Bob: (𝐶,𝐾)→ 𝑀
Eve: Intercepts 𝐶 and tries to figure out 𝑀
Good!
0100
1011… ?

6. Simple Case Study – Modeling Eve𝑀 B A 𝐾 𝐶
Q1: How to model Eve’s perception of 𝐾?
Knows 𝐾 is being used
Doesn’t know its value
⇒ Eve has a guessing probability over 𝐾’s values 0,1 :
Doesn’t have a clue: 𝑃 𝐾 0 = 𝑃 𝐾 1 = 1 2
Knows something: 𝑃 𝐾 0 =𝑝 , 𝑃 𝐾 1 =1−𝑝 , 𝑝≠ 1 2
Q2: Which kind of secret should Alice and Bob favor?

7. Simple Case Study – Modeling Security𝑀 B A 𝐾 𝐶
Q3: What does it mean to secure 𝑀?
Pre-transmission: 𝑃 𝑀 0 = 𝑃 𝑀 1 = 1 2
Eve tries to recover 𝑀 from 𝐶
⇒ 𝑀 is secure if after seeing 𝐶 Eve’s odds don’t improve
Goal: Design functions for Alice and Bob such that:
Bob can decode 𝑀 from 𝐶,𝐾
Eve’s best guess of 𝑀 after seeing 𝐶 is still 50/50

8. Simple Case Study – Binary Operations𝑀 B A 𝐾 𝐶
Assume 𝑀 and 𝐾 are both symmetric (50/50)
Alice gets 𝐶 via binary operation on 𝑀,𝐾
Possible binary operations:
Q4: Which binary operation is better for secrecy? OR
AND
XOR 𝑀 𝐾
𝑀+𝐾 1 𝑀 𝐾
𝑀∙𝐾 1 𝑀 𝐾
𝑀⊕𝐾 1

9. Simple Case Study – Reliability & Optimality𝑀 B A 𝐾 𝐶
⇒ Best function for symmetric 𝑀,𝐾 is XOR:
Eve’s best guess after seeing 𝐶 is 50/50
Same odds like before seeing 𝐶
⇒ Information-theoretic security
Q4: Can Bob decode an XOR–based transmission?
Q5: Can OR or AND operations be used for communication only?
Symmetry is Crucial: Asymmetric keys can’t achieve security with XOR

10. Simple Case Study – General Claim𝑀 B A 𝐾 𝐶
One-Time Pad:
𝑚 messages bits and 𝑘 key bits
All bits are equiprobable
𝑘 ∗ = least 𝑘 s.t. secure communication is possible
Shannon (1949):
Achieving reliability & information-theoretic security over the OTP is:
possible using exactly 𝑚 key bits ⇒ 𝑘 ∗ ≤𝑚
impossible using less than 𝑚 key bits ⇒ 𝑘 ∗ ≥𝑚
𝑘 ∗ =𝑚

11. Resource 2: Noise

12. Error Correcting Codes
Repetition Code: Length 3 for two messages: 0→000 and 1→111
‘1-flip’ ball around 111
111
011
101
000
001
100
110
010
110
111
111
100
101
‘1-flip’ ball around 000
011
010
000
000
001

13. Noisy Channel
In most real-world systems we don’t exactly know the number of bit flips
Common mode of operation is to model noise probabilistically
Contains all sequences with ≈20% flips of the center
Contains all sequences with ≈10% flips of the center 1
0.1
0.9
Alice
Bob

14. Noisy Channel
In most real-world systems we don’t exactly know the number of bit flips
Common mode of operation is to model noise probabilistically 1
0.1
0.9
Alice
Bob

15. Wiretap Channel Noisy communication channel with an eavesdropper
⇒ We can exploit Eve’s extra noise for securing the transmitted message! 1
0.1
0.9
Alice
Bob 1
0.2
0.8
Eve
‘weak’ noise
accumulated ‘strong’ noise

16. Wiretap Codes Eve experiences strong noise ⇒ Large noise balls
Bob experiences weak noise
⇒ Alice can use many codewords
Wiretap Coding:
Hide messages inside Eve’s noise balls
Each message has several codewords one in each noise ball of Eve
message
color 1 2 3 4 5 6 7 8

17. Wiretap Codes Encoding: Alice want to transmit message #5
Chooses one codeword at random
Decoding:
Bob can decode due to fine resolution
Security:
Eve observes an output somewhere in her large noise ball
But all message are there & equally likely!
⇒ Eve’s best guess is equiprobable
⇒ Security is achieved!

18. Information-Theoretic Security Research
Many interesting research questions:
Key agreement over noisy channels
Active Adversaries
Eve not only overhear the transmission but can influence the channel
Has a set of possible actions Alice and Bob know
They don’t know which action is chosen ⇒ Ensure security versus all actions!
Covert Communication:
Communicate without Eve noticing
Many many many many more…