Nonagreement secret key generation based on spatial symmetric scrambling and secure polar coding

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Figure 1
SKG model. (a) Source-type; (b) channel-type
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Figure 2
SKG procedures based on SSS and SPC
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Figure 3
(Color online) DBER versus KOP $L_{\rm~K}=~128$
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Figure 6
SPC model

Table 1 Simulated parameters

Transmitted power at Alice	Received noise power at Bob	Received SNR threshold at Bob	Key length
$P~=~500$	$\sigma~_{\rm~B}^2~=~1$	$\rho~_{\rm~B}^\tau~=~10$ dB	${L_{\rm~K}}~=~128$
Received noise power at Eve	$P_{\rm~cop}$ threshold	Population number	SPC length
$\sigma~_{\rm~E}^2~=~0$	$P_{\rm~cop}^\tau~=~10^{-6}$	${N_{\rm~P}}~=~200$	$N~=~512$
Maximum generation	Crossover probability	Mutation probability
${G_{\rm~max}}~=~100$	${P_{\rm~crossover}}~=~0.6$	${P_{\rm~mutation}}~=~0.02$

Algorithm 1 GA$^2$SPCC algorithm

Require:$N,\;~{\rho~_{\rm~B}},\;~{\rho~_{\rm~E}},\;~{L_{\rm~K}},\;~{P_{\rm~kop}^\tau}$; ${N_{\rm~P}},\;~{G_{\rm~max}},\;~{P_{\rm~crossover}},\;~{P_{\rm~mutation}}$;
Output: $({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}})$; $\eta~,{L_{\rm~I}}$;
Compute $P_{\rm~e}^{\rm~AB}(~{W_N^{(i)}}~)$ and $P_{\rm~e}^{\rm~AE}(~{W_N^{(i)}}~)$ by Gaussian approximation and ${\rho~_{\rm~B}},~{\rho~_{\rm~E}}$;
Sort $P_{\rm~e}^{\rm~AB}(~{W_N^{(i)}}~)$ to screen the polarized sub-channels by (33);
Solve (32) by generic algorithm with the given parameters;
Return the SPC $({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}})$ and $\eta$, $L_{\rm~I}$;

Algorithm 2 SKG procedures at Alice

Require:${{\boldsymbol{x}}_{\rm~pilot}},{{\boldsymbol{y}}_{\rm~pilot}},{N_{\rm~A}},{N_{\rm~E}},P,\rho~_{\rm~B}^\tau~,P_{\rm~cop}^\tau~,P_{\rm~sop}^\tau~,{P_{\rm~kop}^\tau},{{\boldsymbol{u}}_{\rm~M}},~N,{L_{\rm~K}},{N_{\rm~P}},{G_{\rm~max}},{P_{\rm~crossover}},{P_{\rm~mutation}}$;
Output: $({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}})$; $\eta~,{L_{\rm~I}},~K$;
Estimate the legitimate channel ${~\boldsymbol{\hat~h}}$ using the received signal ${{\boldsymbol{y}}_{\rm~pilot}}$ and the public pilot ${{\boldsymbol{x}}_{\rm~pilot}}$;
Compute the SSS parameters $\phi,\alpha,{\rho~_{\rm~B}},{\rho~_{\rm~E}}$ by (19);
Construct the SPC $({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}})$ and obtain the parameter $\eta$, ${L_{\rm~I}}$ with GA$^2$SPCC algorithm;
Secure polar encoding ${{\boldsymbol{u}}_{\rm~M}}$ and then BPSK modulation to obtain ${v_1}(t)$;
Generate the signal-like noise ${v_i}(t),i~=~2,3,~\ldots~,{N_{\rm~A}}$ and send out with the desired signal together;
Obtain the secret keys $K$ by Hash function with the input length ${L_{\rm~I}}$.

Table 2 NIST test results

Frequency	Block frequency (128)	Cumulative sums (Fwd)	Cumulative sums (Rev)	Runs
0.472894	0.483268	0.479731	0.582617	0.318374
Longest run	Rank	Approximate entropy	Linear complexity	Serial
0.028386	0.526912	0.046022	0.953862	0.903275
FFT	Non overlapping template	Overlapping template
0.673452	0.927958	0.207544

Algorithm 3 SKG procedures at Bob

Require:$({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}}),{L_{\rm~I}},y_{\rm~B}$;
Output: $K$;
BPSK demodulate the received signal $y_{\rm~B}$;
Secure polar decode to get ${\boldsymbol{\hat~u}}_{\rm~M}^{\rm~B}$ by $({K_{\rm~M}},{{I}_{\rm~M}},{K_{\rm~R}},{{I}_{\rm~R}},{K_{\rm~F}},{{I}_{\rm~F}})$;
Obtain the secret keys $K$ by Hash function with the input length $L_{\rm~I}$.

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Nonagreement secret key generation based on spatial symmetric scrambling and secure polar coding

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