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) 105
Definition 7.3 A scalable message M= [m 1 , m 2 , ..., m ω ] classified by the vector Π(M) =
[π(m 1 ), π(m 2 ), ..., π(m ω )] is said to be ordered if and only if π(m v ) ≤ π(m u ), 1 ≤ v < u ≤
ω.
Definition 7.4 A combinatorial weak security limit (CWSL) vector C= [c 1 , c 2 , ..., c ψ ]
associated with a guessing success probability threshold (GSPT) vector P t = [p t1 , p t2 , ..., p tψ ]
of an ordered scalable message M= [m 1 , m 2 , ..., m ω ] is a vector whose each element c i
represent the maximum number of weakly secure symbols permitted by the application layer
for the priority class i, such that the ordered scalable message M is considered insecure
if, for some i, at least one of the following conditions holds.
7.4.1. The eavesdropper can decrypt at least one set of symbols N ci = {n 1 , n 2 , ..., n ci }
belonging to the i th priority with the guessing success probability p g , where p g
> p ti ≥
( 1
|F| )c i
.
7.4.2. The eavesdropper can decrypt at least one set of symbols N χi = {n 1 , n 2 , ..., n χi },
χ i > c i , belonging to the i th priority with the guessing success probability p g > p ti·( 1
|F| )χ i−c i
.
The first condition in Definition 7.4 states that, with a CWSL vector and a GSPT
vector given by the application layer, it is required that the encryption must not allow
the eavesdropper to decrypt c i symbols of priority i with the guessing success probability
exceeding p ti . Therefore, the GSPT vector specifies the limit of weakness in our security
scheme.
The second condition in Definition 7.4 states that, beyond the limit of c i symbols of
priority i, it is required that the guessing success probability is reduced by the factor 1
|F|
for each symbol beyond the limit. In other words, we only allow weak security for any
arbitrary set of up to c i symbols. Any more symbols added to the set must be regarded
as perfectly secure.
This scalable security framework is mainly designed for cryptosystems using Shamir’s
concept of secret sharing. We will discuss two such systems, which are weakly secure
network coding and our physical-layer key encoding in a one-time pad cryptosystem in
Sections 7.6 and 7.7, respectively.