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) 99
7.3 The Equivalent Number of Vulnerable Bits for
the Design of Physical-Layer Key Encoding
The analysis given in the last section is based on the assumption that the encoder knows
I V K .
However, empirical results in [33, 34] show that the encoder normally can only
estimate the ratio I V K /I K instead of I V K alone. In this section, we derive an equivalent
number of vulnerable bits, denoted by I V ′ K , as a design parameter that can substitute
I V K in the theorems discussed in the last section. To do so, we first propose a model of an
enemy cryptanalyst based on the ratio I V K /I K estimated by the legitimate transmitter.
According to Fig. 7.2 (a), the enemy is modeled to have almost the same structure
as the legitimate transmitter and receiver. However, since it can only estimate channel
coefficients from an imaginary channel which differs from the key-generating channel used
by the legitimate terminals, we propose an equivalent model in Fig. 7.2 (b).
In the
equivalent model, the enemy does estimate the key-generating channel, but the quantized
key K is distorted by a binary symmetric channel (BSC), erring each estimated key bit
with a probability p.
The legitimate transmitter has to guess this error probability p
based on its estimate of 1 − (I V K /I K ). For security, the estimated value p should not be
more than the estimated 1 − (I V K /I K ).
Based on the equivalent model, we can specify the relationship between the estimate
of p and the design parameter I ′ V K
needed for perfect secrecy in Theorem 7.4.
Theorem 7.4 If the enemy cryptanalyst behaves according to the model in Fig. 7.2 (b)
having p as the error probability of the binary symmetric channel, and the generator matrix
prototype in the form of equations (7.10) or (7.11) is used, when I V ′ K + 1, being even or
odd, respectively, substitutes I V K + 1, the following conditions on I ′ V K
are sufficient for
perfect secrecy. [4]