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

96
Chapter 7: Physical-layer Key Encoding for Wireless Physical-layer Secret-key
Generation (WPSG) with Unequal Security Protection (USP)
data and present the second theorem.
Theorem 7.2 If K ∈ GF (2 I K
), in order to generate Z ∈ GF (2 n ), which is a codeword
of n encoded key bits providing perfect secrecy in our system, the following conditions on
I K are necessary and sufficient. [4]
7.2.1. If I V K + 1 is even,
7.2.2 If I V K + 1 is odd,
I K ≥
(n + 1)
(I V K + 1) (7.3)
2
I K ≥
(n + 1)
(n − 1)
(I V K + 1) +
2
2
(7.4)
Proof We prove this theorem by mathematical induction. We first demonstrate that the
theorem is valid for n = 1 and n = 2 before showing that if the theorem is valid for any
n, it will hold true for n + 1. The case of n = 1 can be easily validated by substituting it
into (7.3) and (7.4) and observing that the resulting I K corresponds to that in Theorem
7.1.
Given Z 1 in Eq. (7.2), when L = 2 and I V K + 1 is an even number. Let
Z 2 = K 1
2 (I V K+3) ⊕ K 1 2 (I V K+5) ⊕ ... ⊕ K 3 2 (I V K+1) . (7.5)
Now, the generator matrix generating Z 1 and Z 2 can be written as follows.
⎡
G p = ⎢
⎣
I V K +1
{ }} {
1 1 ... 1 1 1 ... 1
0} 0 {{... 0}
1} 1 ... 1{{ 1 1 ... 1}
1
2 (I V K+1) I V K +1
I K −I V K −1
{ }} {
0 0 ... 0 0 0 ... 0
0 0 ... 0
⎤
⎥
⎦
T
= [g 1 g 2 ] (7.6)
such that
Z = [Z 1 , Z 2 ] = K · G p , (7.7)
where
K = [K 1 , K 2 , ..., K IK ]. (7.8)