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) 107
Although the source symbols in both Fig. 7.4 (a) and (b) cannot be decrypted by
wiretapping a single edge, the degrees of security in the two figures differ. If we check
the definition of Shannon security and weak security, we find that secure network coding
in Fig. 7.4 (a) is Shannon secure since I(X = [x 1 ]; Y) = 0, where Y is the information
symbol at any single edge.
However, that in Fig. 7.4 (b) is only weakly secure since,
although I(x i ; Y) = 0, i = 2, 3, I(X = [x 2 , x 3 ]; Y) = log 2 |F|, where |F| is the field size
of the symbols x 2 and x 3 . (In this case, x2 and x3 can take their values from the Galois
field with the size of 5 or greater.)
Secure network coding in Fig. 7.4 can be seen as a special case of the scalable security
framework proposed in the previous section. We can see that, when the field size of x 1 ,
x 2 , x 3 , and w is |F|, the guessing success probability p g of x 1 is 1
|F| , and that of [x 2, x 3 ] is
1
as well. The degree of security in Fig. 7.4 (b) is weaker since, with the same p
|F| g , two
symbols are decrypted as compared with one symbol in Fig. 7.4 (a).
The practical implication is that the high-priority symbols should be encrypted according
to Fig. 7.4 (a), whereas the low-priority ones may consider the encryption in Fig. 7.4
(b). For example, if we have an ordered scalable message M= [m 1 , m 2 , m 3 ] such that
Π(M) = [1, 2, 2], the symbol m 1 should be encrypted according to Fig. 7.4 (a), whereas
m 2 and m 3 may follow Fig. 7.4 (b), if the CWSL vector C= [1, 2] and the GSPT vector
P t = [ 1 , 1 ] are specified by the application layer. (See Definition 7.4.)
|F| |F|
7.7 Physical-layer Key Encoding in the Scalable Security
Framework
Scalable security can be realized in physical-layer key encoding by reducing the number
of 1-elements in (7.10) or (7.11) and shifting the groups of 1-elements to the left. This
will save the number of original key symbols needed. We will give an example problem
as follows.
Problem 7.1. The application layer requires that at most three data bits in the second