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Chapter 6: Wireless Physical-layer Secret-key Generation (WPSG) in Relay Networks:
Information Theoretic Limits, Key Extension, and Security Protocol 81
when there is only one or two paths, the correlation is significantly high.
In [34], two rigorous metrics have been developed for quantifying the information
theoretic limits of key generation in the scenario depicted in Fig. 6.3. Given the users have
noisy channel estimates, which are denoted by ĥa, ĥa ′, ĥb, and ĥc, we refer to the maximum
number of independent key bits that can be generated per channel observation as I K =
I(h a ; h a ′), where I(x; y) denotes the mutual information between x and y. Likewise, the
maximum number of independent key bits that can be generated and are secure from an
eavesdropper is given by I SK = I(ĥa, ĥa ′|ĥb, ĥc) [76]. In [34], closed-form expressions for
I K and I SK are derived for correlated complex Gaussian vector channels. The generated
key bits that are not secure are called vulnerable key bits. The number of vulnerable key
bits is therefore I V K = I K − I SK .
To illustrate the key generation in Fig. 6.3, consider a simple scenario with scalar
channels. Alice sends a pilot signal x to Bob, who derives the channel estimate ĥa of the
channel h a from the received signal y b = h a x + n b , where n b denotes complex Gaussian
noise on Bob’s side. After that, Bob sends x to Alice, who derives the estimate ĥa ′ of h a ′
in a similar manner. The number of available key bits I K per channel observation that can
be generated by Alice and Bob is given by I K = I(ĥa; ĥa ′). If the time between Alice’s
transmission and Bob’s is short and the same frequency band is used, we can assume
reciprocity or h a = h a ′. Assuming further that h a is Rayleigh-distributed with a standard
deviation of 0.5, we obtain a simulation result in Fig. 6.4 showing the relationship between
I K and the signal-to-noise ratio, which is the ratio of the power of x to the Gaussian noise
power at Bob and Alice. We also compute the result when the key is derived from the
envelopes of channel parameters only, such that I K = I(|ĥa|; |ĥa ′|) [76].
6.3 Possible Key Extensions
In a general wireless network, the length of the key can be extended by having it generated
from several transmission routes instead of only one. Although it is obvious that the total
key length is the summation of the lengths from all paths, the key length from each path