Secure Nested Codes for Type II Wiretap Channels

Secure Nested Codes for Type II Wiretap Channels Ruoheng Liu, Yingbin Liang, H. Vincent Poor, and Predrag Spasojevic Wiretap Channel (WINLAB) Department of Electrical Engineering, Princeton University Approach I: Good Coarse Codes Approach II: Dual Codes as Coarse Codes § consider a sequence of secure nested codes {C 0(n), C 1(n)}: -- C 0(n) = {0, 1}n (R 0=1) -- C 1(n) is a good code of rate R 1 and SNR threshold l* § for an AWGN-WT(l), {C 0(n), C 1(n)} can achieve the rateequivocation pair Type I: [Wyner, 1975] degraded broadcast channel confidential message W § construct an equivalent channel: BEC § choose e so that the channel f(z|z’) is feasible: Type II: [Ozarow & Wyner, 1984] special wiretap channel noiseless main channel Wyner Codes and Secrecy Bins § when using non-capacity-achieving C 1(n), one cannot achieve perfect secrecy Example: LDPC Coarse Codes Consider C 0(n) = {0, 1}n C 1(n) = CLDPC(n) § CLDPC(n): (4, 6) LDPC code R 1 = 1/3 and l* ≤ 0. 302 § {C 0(n), CLDPC(n)} transmitted over AWGN-WT (l≤ 0. 302), (R, Re) =(2/3, 0. 663) can be achieved. secure nested code sequence: {C 0(n), C 1(n)} : § C 0(n) is a fine code of rate R 0 § C 1(n) is a coarse code of rate R 1 § C 1(n) ⊆ C 0(n), R 1 ≤ R 0 § information rate of this code sequence is R 0 - R 1 Type II AWGN Wiretap Channel § secure nested codes {C 0(n), C 1(n)} C 0(n) = {0, 1}n C 1(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 {C 0(n), C 1(n)} can achieve (R, Re) = (Rc, Rc). Achievable Rate-Equivocation Region Approach I capacity−equivocation region § here, the gap between R and Re is less than 0. 004! Approach II Equivocation Rate vs SNR achievable rate−equivocation region § eavesdropper channel SNR -8 d. B ≤ l ≤ 10 d. B transmission rate R 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) • either loosen the perfect secrecy requirement • or reduce the transmission rate compared with the capacity equivocation rate Re capacity BEC-WT(e) BSC-WT(q) AWGN-WT(l) e h(q) 1 -CBIAWGN(l) code design perfect secrecy rate* -log 2(1 -q), 2 q