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) 109
With the summation ∑ 3
i=1 Y i known by wiretapping, and K 1 ⊕ K 3 ⊕ K 5 known if
K 1 , K 3 , and K 5 are vulnerable symbols, the enemy knows the summation ∑ 3
i=1 X i. This
means, by guessing two bits out of three bits X i , i = 1, 2, 3, correctly, the enemy knows
all the three bits. The probability of guessing two bits correctly is 0.25, corresponding to
the requirement p t2 = 0.25. We can see that there is no way to derive any three bits with
more guessing success probability than 0.25. Therefore, the first condition is not met,
meaning that our encryption will be considered scalably secure if the second condition is
not met either.
By checking every possible way for the enemy to derive all four plaintext bits X i ,
i = 1, 2, 3, 4, we see that the maximum guessing success probability is 0.125.
Since
0.125 = p ti · ( 1
|F| )χ i−c i
= 0.25 · ( 1 2 )4−3 , the second condition is not met and our encryption
is therefore scalably secure according to the requirement.
One may observe that, by using the given scalable security requirement instead of the
perfect secrecy requirement, we reduce the number of original symbols I K needed by 40%.
7.8 Conclusion and Future Research
We have proposed physical-layer key encoding for the WPSG cryptosystem with onetime
pad encryptor. Four theorems indicate the required properties of the codes in order
to achieve perfect secrecy of the secret data. After that, we discuss a scalable security
framework specifying the degrees of security weakness that can be allowed in lower-priority
data. Our framework applies to weakly secure network coding as well as our physicallayer
key encoding in a WPSG cryptosystem and, indeed, any key encoding schemes for
one-time pad cryptosystems having vulnerable key bits. We show that the number of
original key bits needed can be significantly reduced if weak security is allowed.
In future, we hope that some algorithms are developed such that, given any scalable
security requirement, the code can be systematically designed.