Network Coding and Wireless Physical-layer ... - Jacobs University

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Chapter 7: Physical-layer Key Encoding for Wireless Physical-layer Secret-key
Generation (WPSG) with Unequal Security Protection (USP)
priority class can be weakly secure with the eavesdropper’s guessing success probability
of 0.25. If the number of vulnerable key bits is 3, specify the generator matrix of the
physical-layer key encoding generating four encoded key bits.
Solution. In this problem, we start from the generator matrix in (7.10) for perfect secrecy
with parameters I V K = 3, and n = 4. The overlapping part of the groups of 1-elements
in two adjacent columns consists of two elements such that
⎡
G p =
⎢
⎣
1 1 1 1 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1 0 0
0 0 0 0 0 0 1 1 1 1
⎤
T
. (7.24)
⎥
⎦
For scalable security satisfying the conditions c 2 = 3 and p t2 = 0.25, we reduce the
number of 1-elements in each column from 4 to 3. We also shift each row of the transpose
of the generator matrix to the left such that it becomes
⎡
G p =
⎢
⎣
1 1 1 0 0 0
0 1 1 1 0 0
0 0 1 1 1 0
0 0 0 1 1 1
⎤
T
. (7.25)
⎥
⎦
Now, we will verify that this generator matrix results in weak security corresponding
to Definition 7.4. According to eqs. (7.1), (7.2), (7.7), (7.8), and (7.24), the ciphertext Y i
can be written as
Y i = K i ⊕ K i+1 ⊕ K i+2 ⊕ X i , i = 1, 2, 3, 4. (7.26)
Next, we will check the first condition in Definition 7.4 where c 2 = 3. The best way
for the eavesdropper to guess three symbols is to add together three adjacent ciphertext
symbols, for example,
3∑
Y i = K 1 ⊕ K 3 ⊕ K 5 ⊕
i=1
3∑
X i . (7.27)
i=1