1. Secure Nested Codes for Type II Wiretap Channels
Ruoheng Liu, Yingbin Liang, H. Vincent Poor, and Predrag Spasojevic (WINLAB)
Department of Electrical Engineering, Princeton University
Wiretap Channel
Approach I: Good Coarse Codes
Approach II: Dual Codes as Coarse Codes
consider a sequence of secure nested codes {C0(n), C1(n)}:
-- C0(n) = {0, 1}n (R0=1)
-- C1(n) is a good code of rate R1 and SNR threshold l*
for an AWGN-WT(l), {C0(n), C1(n)} can achieve the rate-equivocation pair
when using non-capacity-achieving C1(n), one cannot achieve perfect secrecy
construct an equivalent channel: BEC
choose e so that the channel f(z|z’) is feasible:
Type I: [Wyner, 1975]
degraded broadcast channel
confidential message W
Type II: [Ozarow & Wyner, 1984]
special wiretap channel
noiseless main channel
Wyner Codes and Secrecy Bins
secure nested codes {C0(n), C1(n)}
C0(n) = {0, 1}n C1(n) = C┴(n) (┴: the dual)
-- C(n) is a good code of rate Rc and erasure threshold d*
for an AWGN-WT(l), if
then {C0(n), C1(n)} can achieve (R, Re) = (Rc, Rc).
Example: LDPC Coarse Codes
Consider C0(n) = {0, 1}n C1(n) = CLDPC(n)
CLDPC(n): (4, 6) LDPC code
R1 = 1/3 and l* ≤ 0.302
{C0(n), CLDPC(n)} transmitted over AWGN-WT (l≤0.302),
(R, Re) =(2/3, 0.663) can be achieved.
here, the gap between R and Re is less than 0.004!
Achievable Rate-Equivocation Region
Approach I
Approach II
capacity−equivocation
region
achievable rate−equivocation
secure nested code sequence: {C0(n), C1(n)} :
C0(n) is a fine code of rate R0
C1(n) is a coarse code of rate R1
C1(n) ⊆ C0(n), R1 ≤ R0
information rate of this code sequence is R0 - R1
Equivocation Rate vs SNR
eavesdropper channel SNR -8 dB ≤ l ≤ 10 dB
transmission rate R
equivocation rate Re
Type II AWGN Wiretap Channel
Summary
eavesdropper channel is an BI-AWGN channel with SNR l
capacity-equivocation region contains all rate pairs
(R, Re): Re ≤ R ≤ 1
and 0 ≤ Re ≤ 1 - CBIAWGN(l)
BEC-WT(e)
BSC-WT(q)
AWGN-WT(l)
capacity e
h(q)
1-CBIAWGN(l)
code design
perfect secrecy
secrecy rate*
-log2(1-q), 2q
either loosen the perfect secrecy requirement
or reduce the transmission rate compared with the capacity