SCIENTIA SINICA Informationis, Volume 49 , Issue 4 : 486-502(2019) https://doi.org/10.1360/N112018-00119

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

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  • ReceivedMay 16, 2018
  • AcceptedDec 25, 2018
  • PublishedApr 3, 2019


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  • Figure 1

    SKG model. (a) Source-type; (b) channel-type

  • Figure 2

    SKG procedures based on SSS and SPC

  • Figure 3

    (Color online) DBER versus KOP $L_{\rm~K}=~128$

  • 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


    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


    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 $L_{\rm~I}$.