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一种新的基于辛空间的密钥预分配方案

陈尚弟 张俊梅

陈尚弟, 张俊梅. 一种新的基于辛空间的密钥预分配方案[J]. 电子与信息学报. doi: 10.11999/JEIT211490
引用本文: 陈尚弟, 张俊梅. 一种新的基于辛空间的密钥预分配方案[J]. 电子与信息学报. doi: 10.11999/JEIT211490
CHEN Shangdi, ZHANG Junmei. A New Key Pre-distribution Scheme from Symplectic Spaces[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT211490
Citation: CHEN Shangdi, ZHANG Junmei. A New Key Pre-distribution Scheme from Symplectic Spaces[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT211490

一种新的基于辛空间的密钥预分配方案

doi: 10.11999/JEIT211490
基金项目: 中央高校基金(3122019192, 3122019152)
详细信息
    作者简介:

    陈尚弟:男,教授,研究方向为代数、图论、密码与编码

    张俊梅:女,硕士生,研究方向为密码

    通讯作者:

    陈尚弟 11csd@163.com

  • 中图分类号: TN918.4; TP212.9

A New Key Pre-distribution Scheme from Symplectic Spaces

Funds: The Fundamental Research Funds of the Central Universities of China (3122019192, 3122019152)
  • 摘要: 密钥预分配是无线传感器网络中最具挑战的安全问题之一。 该文基于有限域上辛空间中子空间之间的正交关系构造了一个新的组合设计,并基于该设计构造了一个密钥预分配方案。令V 是有限域上8维辛空间中的一个(4,2)型子空间,V 中每一个(1,0)型子空间看作密钥预分配方案中的一个节点,所有的(2,1)型子空间看作该方案的一个密钥池。将整个目标区域划分为若干个大小相同的小区,每个小区有普通节点和簇头两种类型的传感器节点。小区内的普通节点采用基于辛空间的密钥预分配方案分发密钥,不同小区内节点所用密钥池互不相同,因此不同小区内的节点需通过簇头建立间接通信,不同小区内簇头采用完全密钥预分配方式分发密钥。与其他方案相比,本方案的最大优势是网络中节点的抗捕获能力较强,且随着网络规模的不断扩大,网络的连通概率逐渐趋于1。
  • 图  1  $ \rho {\text{ = 1}} $$ {C_5} $的Lee球体区域

    图  2  损失概率图

    图  3  连通概率对比图

    图  4  损失概率对比图

    表  1  部分平衡$ 2 - ({q^4} + {q^2},{q^2};1,0) $设计到密钥预分配方案的对应关系

    部分平衡$ 2 - ({q^4} + {q^2},{q^2};1,0) $区组设计密钥预分配方案
    $ v = |X| = N(2,1;4) = {q^4} + {q^2} $密钥池大小
    $ b = |\mathcal{B}| = N(1,0;4) = {q^3} + {q^2} + q + 1 $方案所能支持的最大传感器节点数目
    $ k = |X(B)| = N(2,1;3,1;4) = {q^2} $密钥环大小
    $ r = |\mathcal{B}(x)| = N(1,0;2,1;4) = q + 1 $包含一给定密钥的节点数目
    $ \delta = 0 $或$ 1 $任意两个节点共享的密钥量
    下载: 导出CSV

    表  2  $ q = 4 $, $ q = 5 $, $ q = 7 $, $ q = 8 $时的局部损失概率

    $ q $$ N $$ b $$ k $$ S $${\rm{fail}}(1)$${\rm{fail}}(s)$
    440085165000.03610.0449
    440085166000.03610.0537
    59001562512000.02600.0345
    713004004918000.01510.0208
    816005856422000.01200.0165
    下载: 导出CSV

    表  3  各个方案的参数

    方案$ v $$ b $$ k $$ p $$ {\rm{fail}}(s) $
    RNC$ {q^2}{\text{ + }}q + 1 $$ {q^5} - {q^2} $$ q + 1 $$ {\mu _c}/(b - 1) $$ 1 - {(1 - {\rm{fail}}(1))^s} $
    NU-KP$ {q^3} + 1 $$ {q^4} - {q^3} + {q^2} $$ q + 1 $$ {(q + 1)^2}/({q^3} + q + 1) $$ 1 - C_{b - {q^2}}^s/C_b^s $
    2-UKP$ {q^3} + 1 $${b'}$$ 2(q + 1) $$ 1 - {(1 - {(q + 1)^2}/(v + q))^4} $$ \displaystyle\sum\nolimits_{i = 1}^4 {{{(1 - u)}^i}p(i)/p} $
    Kumar’s$ {q^2}{\text{ + }}q + 1 $$ {q^2}{\text{ + }}q + 1 $$ q + 1 $$ 1 $$ 1 - {(1 - (q - 1)/b)^s} $
    TKP$ {q^2} - q $$ 2{q^2} $$ 2(q+1) $$ ({k^2} - k)/(4{q^2} - 2) $$ 1 - (C_t^s + 4C_t^{s - 1}(q - 1))/C_b^s $
    Ex-w-BIBD$ 8{q^3} $$ 8{q^3} $$ 6q - 3 $$ 6q/(4{q^2} + 2q + 1) $$ p_1' r_1'(s){\text{ + }}p_2' r_2'(s) $
    本文$ {q^4}{\text{ + }}{q^2} $$ {q^3}{\text{ + }}{q^2}{\text{ + }}q + 1 $$ {q^2} $$ {q^3}/(b - 1) $$ 1 - {(1 - (q - 1)/(b - 2))^s} $
    注:(1)${\mu _c} = \sum\limits_{s = 1}^{t - 1} {C_k^s} \mu' _c(s)$, $\mu' _c(s) = {\lambda _s} - 1 - \sum\limits_{i = 1}^{t - 1 - s} {C_{k - s}^i} \mu _c'(s + i)$, ${\rm{fail}}(1) = (\sum\limits_{s = 1}^{t - 1} {C_k^s{\lambda _s}\mu '_c(s)} )/(b\sum\limits_{s = 1}^{t - 1} {C_k^s\mu '_c(s)} )$。
    (2)$ {q^4} - 2{q^3} + q \le 2{b'} \le{q^4} - {q^3} + {q^2} , u = C_{{q^4} - {q^3}}^{2s}/C_{{q^4} - {q^3} + {q^2}}^{2s} , p(i) = C_4^i{d^i} \times {(1 - d)^{4 - i}} , p{\text{ = }}1 - {(1 - d)^4} $。
    (3) ${p'_1} = 6q/(4{q^2} + 2q + 1) , r'_1(s) = 1 - 2{(1 - (6q - 5)/(8{q^3} - 2))^s} + {(1 - (12q - 10)/(8{q^3} - 2))^s} , {p_2}{'{ = 3/(4} }{ { {q} }^2}{ { + 2q + 1)} }$, ${r}'_{2}(s){ {=s} '_{22} }(s)/{C}_{s}^{8{q}^{2}-4{\rm{q}}} , t = 2{q^2} - 4q + 2 , { {s} }'_{22}(s){ {=C} }_{s}^{4q(2q-1)}-{C}_{1}^{2q-2}{C}_{s}^{2(2q-1)(2q-1)}+{C}_{2}^{2q-2}{C}_{s}^{2(2q-1)(2q-2)}+\cdots +{(-1)}^{\theta }{C}_{\theta }^{2q-2}{C}_{s}^{2(2q-1)(2q-\theta )}$。
    下载: 导出CSV

    表  4  符号表

    符号符号说明符号符号说明
    $ X $($ |X| = v $)点集(点集的大小/密钥池的大小)$ {C_i} $网络中第$ i $个小区
    $ \mathcal{B} $($ |\mathcal{B}| = b $)区组集(区组数目/方案所支持的最大传感器节点数目)${{\rm{CH}}_{ij} }$网络中第$ i $个小区中的$ j $类型簇头
    $ k $每个区组的大小(每个节点所含的密钥量)$ p $网络的连通概率
    $ r $包含一给定点的区组数(包含一给定密钥的节点数目)${\rm{fail}}(s)$$ s $个节点被捕获时网络的损失概率
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-12-13
  • 修回日期:  2022-05-27
  • 网络出版日期:  2022-06-10

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