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

Chapter 7: Physical-layer Key Encoding for Wireless Physical-layer Secret-key
Generation (WPSG) with Unequal Security Protection (USP) 95
perfect secrecy of Y 1 is achieved.
Now, in order to construct Z 2 , Z 3 , and so on, we perform a linear combination of I V K
key symbols similar to (7.2). Therefore, our physical-layer key encoding can be defined by
a linear block code C p transforming an I K -tuple K over GF (q I K
) of key symbols obtained
from the quantizer into an M-tuple codeword Z over GF (q M ). We propose the following
theorem providing two necessary and sufficient conditions for perfect secrecy.
Theorem 7.1 If I V K out of I K physical-layer key symbols generated by the quantizer can
be correctly estimated by the eavesdropper, the cryptosystem in Fig. 7.1 still maintains
perfect secrecy if and only if each member in the I K -dimensional vector K is statistically
independent of one another and the physical-layer code C p has the following properties. [4]
7.1.1. Every code symbol is a linear combination of at least I V K + 1 input symbols.
7.1.2. Every linear combination of any subset of code symbols results in a linear combination
of at least I V K + 1 input symbols.
Proof From earlier discussion it should be clear that the first condition is necessary.
The necessity of the second condition can be proved by contradiction as follows. If the
combination of ν code symbols ∑ ν
i=1 Z i yields a combination of I V K +1−δ input symbols,
where δ is a positive integer, the eavesdropper who calculates ∑ ν
i=1 Y i may know the exact
value of ∑ ν
i=1 Z i, since the number of input symbols in the combination does not exceed
the number of vulnerable symbols. If ∑ ν
i=1 Z i is known, perfect secrecy is not achieved
because these ν encoded key symbols are not independent.
As for the sufficiency, we can also prove it by contradiction. Now, we have to prove
that if perfect secrecy is not achieved, either the condition 7.1.1 or 7.1.2 is not satisfied.
By definition, perfect secrecy in a one-time-pad system is not achieved only if either 1)
at least one Z i , i = 1, 2, ..., M is known or 2) any linear combination of some Z i is known
implying linear dependence among key symbols. Since 1) and 2) will not happen if the
condition 7.1.1 and 7.1.2 are, respectively, satisfied, the two conditions are sufficient.
Now, we focus our attention on the special case when K and Z are vectors of binary