Secret Key Agreement Based on Secure Polar Code
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摘要: 针对密钥协商过程中的信息泄露问题，该文综合考虑信息协商和隐私放大提出了基于安全极化码(SPC)的密钥协商方法，打通了从量化误比特率(QBER)条件到密钥中断概率(SKOP)需求的桥梁。首先，将QBER建模为加性高斯白噪声(AWGN)信道的传输误比特率(TBER)，进而将QBER优势转化为AWGN信道优势；然后，利用高斯近似计算出各极化子信道的传输错误概率，并进一步推导出安全极化码的译码误比特率上下界；最后，根据密钥中断概率阈值利用遗传算法构造出合适的安全极化码实现密钥协商。仿真结果表明，该文所提的密钥协商方法能够满足设计的密钥中断概率需求，且具有比低密度奇偶校验(LDPC)码更高的密钥协商效率。Abstract: Focusing on the problem of information leakage in secret key agreement, combining information reconciliation and privacy amplification, a method based on Secure Polar Code (SPC) is proposed, which builds the bridge from the condition of Quantized Bit Error Rate (QBER) to the requirement of Secret Key Outage Probability (SKOP). Firstly, QBER is modeled as the Transmitted Bit Error Rate (TBER) of Additional White Gaussian Noise (AWGN) channel, so the advantage of QBER is converted to the advantage of AWGN channel; Then, the TBER of each polarized sub-channel is calculated by Gaussian approximation, and the upper and lower bounds of decoded bit error rate are also derived. Finally, the SPC is constructed based on generic algorithm and SKOP threshold. Simulation results show that the proposed method satisfies the requirement of SKOP and achieves higher secret key agreement efficiency, compared with Low Density Parity Check (LDPC)-based method.
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表 1 GA2SPCC算法
输入：$N,{\rho _{{\rm{AB}}}},\;{\rho _{{\rm{AE}}}},\;{L_{\rm{K}}},{\zeta _\tau },{G_{{m}}},{P_{{n}}},{P_{{c}}},{P_{{m}}}$ 输出：$({K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}}),\eta ,{L_{\rm{I}}}$
(1) 利用高斯近似法和${\rho _{{\rm{AB}}}}$, ${\rho _{{\rm{AE}}}}$分别计算$P_e^{{\rm{AB}}}\!\!\left(\! {W_N^{(i)}} \!\right)$和$P_e^{{\rm{AE}}}\!\!\left(\! {W_N^{(i)}} \!\right)$；
(2) 对$P_e^{{\rm{AB}}}\left( {W_N^{(i)}} \right)$进行排序并根据式(17)式初步筛选极化子信道；(3) 按照相应参数利用遗传算法求解式(16)； (4) 返回$({K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}}),\eta ,{L_{\rm{I}}}$。 
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表 2 Alice侧密钥协商算法
输入：$({K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}}),{L_{\rm{I}}},{S_{\rm{A}}}$ 输出：${H_{\rm{A}}},{K_{\rm{A}}}$ (1) 将${S_{\rm{A}}}$按照长度${K_{\rm{M}}}$分组； (2) 利用$({K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}})$进行EncoderA系统编码； (3) 取出校验比特${H_{\rm{A}}}$并打包由“公共无噪信道”发送至
Bob(Eve)；(4) 将${S_{\rm{A}}}$按照长度${L_{\rm{I}}}$分组，经过通用hash函数后生成密钥${K_{\rm{A}}}$。
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表 3 Bob(Eve)侧密钥协商算法
输入：$\left( {N,{K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}}} \right),{L_{\rm{I}}},{S_{\rm{B}}}({S_{\rm{E}}})$ 输出：${K_{\rm{B}}}({K_{\rm{E}}})$ (1) 将${S_{\rm{B}}}({S_{\rm{E}}})$按照长度${K_{\rm{M}}}$分组并与对应的${H_{\rm{A}}}$合并为新码字； (2) 利用$\left( {N,{K_{\rm{M}}},{I_{\rm{M}}},{K_{\rm{R}}},{I_{\rm{R}}},{K_{\rm{F}}},{I_{\rm{F}}}} \right)$进行系统译码得到${S'\!_{\rm{B}}}({S'\!_{\rm{E}}})$； (3) 将${S'\!_{\rm{B}}}({S'\!_{\rm{E}}})$按照长度${L_{\rm{I}}}$分组，经过通用hash函数后生成密钥
${K_{\rm{B}}}({K_{\rm{E}}})$。
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表 4 部分仿真参数
密钥长度 最大代数 种群个数 交叉概率 突变概率 ${L_{\rm{K}}} = 128$ ${G_{{m}}} = 100$ ${P_{{n}}} = 200$ ${P_{{c}}} = 0.6$ ${P_{{m}}} = 0.02$
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