一种改进的防窃听LT码

一种改进的防窃听LT码 An Improved LT Codes in Anti⁃Eavesdropping Method 1 first-author 牛芳琳，女，副教授，博士，现从事无线电通信技术及信息安全方面的研究。E⁃mail:dx_niufl@lnut.edu.cn 牛芳琳，女，副教授，博士，现从事无线电通信技术及信息安全方面的研究。E⁃mail:dx_niufl@lnut.edu.cn dx_niufl@lnut.edu.cn 1 1 1 1 辽宁工业大学 电子与信息工程学院，辽宁 锦州 121001 辽宁工业大学 电子与信息工程学院，辽宁 锦州 121001 School of Electronics and Information Engineering， Liaoning University of Technology， Jinzhou 121001， Liaoning, China School of Electronics and Information Engineering， Liaoning University of Technology， Jinzhou 121001， Liaoning, China 为了进一步提高LT（Luby transform）码的防窃听能力，提出将部分编码矩阵列依据度的大小由高到低重新排列，推迟发送度1符号，以延迟译码开始时间，阻止信息被窃听。实验结果表明，与传统的LT码相比，改进的LT码作为信道编码在保证接收者恢复信源信息的同时，迫使窃听者的误码率大幅度增加，当窃听信道删除概率高于主信道时候，误码率增幅更为明显。 In order to improve the anti⁃eavesdropping ability of LT（Luby transform）codes, this paper proposes the method that the part of the columns in the coding matrix is re⁃arranged from high to low according to the degree of LT codes, to delay the start time of decoding by delaying the symbols of degree 1. And the aim of preventing eavesdropping is realized. Experimental results show that compared with LT codes, the improved LT codes as channel codes force the eavesdropper s BER to increase greatly while guaranteeing the receiver to recover the source information, and the bit error rate increases more obviously when the deletion probability of the eavesdropping channel is higher than that of the main channel. 窃听信道 LT码 RSD度分布函数 BP译码 删除信道 eavesdropping channel LT（Luby transform） codes robust soliton distribution belief propagation decoder erasure channel 0　 引 言 1 无线通信技术使信息传输变得更为方便快捷，但其开放式的传输方式却给窃听者带来窃取信息的机会，致使无线通信存在较大的安全隐患。传统的防窃听技术是采用复杂的密钥对信息进行加密 [ 1 1 , 2 2 ] ，随着计算机运行速度的提升，这种密钥技术的安全性也逐渐变弱。研究表明 [ 3 3 ] 在物理层保护信息安全是可行的，因此物理层防窃听技术逐渐成为信息安全研究的重要方向。文献[ 4 4 , 5 5 , 6 6 ]提出将纠错码技术与加密体制相结合，在防窃听技术中获得了较好的效果。由于噪声具有干扰性且难以复制，文献[ 7 7 , 8 8 , 9 9 , 10 10 ]提出利用噪声进行信道干扰，可有效保护信息安全。在未知窃听者信道状态信息的情况下，文献[ 11 11 ]提出利用保密编码与人工噪声辅助信令一起对抗窃听者。 LT（Luby transform，LT）码 [ 12 12 ] 是适用于删除信道的纠错码，由于译码开销很小、编译码复杂度低，将其用作防窃听码不仅可以有效提高无线传输效率，而且在一定程度上可防止信息被窃听。文献[ 13 13 ]提出在LT编码中利用功率控制方法减少窃听者接收到正确编码符号数量；文献[ 14 14 ]提出从度2与度3编码符号中获取译码的关键数据包以防止窃听者恢复度1信息；文献[ 15 15 ]则利用增加人工噪声破坏窃听者LT码的置信传播（belief propagation，BP）译码顺序方法来增加窃听者的误码率，达到保证主信道通信安全的目的。 利用信道噪声与LT码编码技术相结合可以有效增加窃听者的误码率，本文提出一种改进的防窃听LT信道编码方法。这种方法在保证信息正常传输的同时，进一步提高窃听者接收信息的误码率，确保主信道安全通信。 1　 窃听信道及LT 码 1 1.1　窃听信道 2 窃听信道由信源、接收者和窃听者组成。信源与接收者称为主信道，信源与窃听者称为窃听信道。针对物理层编码方案，信源需要将信息符号进行信道编码，接收者对于接收到的信息需要进行信道译码，其结构如 图1 图1 所示。 1.2　LT码 2 ISD(ideal soliton distribution) [ 12 12 ] 为 <math display="block" id="M1"><mi>ρ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>/</mo><mi>k</mi><mtable><mtr><mtd><mo>,</mo></mtd><mtd/></mtr></mtable><mtable><mtr><mtd/><mtd/><mtd><mi>d</mi><mo>=</mo><mn mathvariant="normal">1</mn></mtd><mtd/></mtr></mtable></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac><mtable><mtr><mtd><mo>,</mo></mtd><mtd/><mtd><mi>d</mi><mo>=</mo><mn mathvariant="normal">2,3</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi></mtd><mtd/></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118013&type= 69.42666626 14.64733315 <math display="block" id="M2"><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>d</mi></mrow></munder><mi>ρ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118024&type= 19.72733307 6.60399961 增强因子 <math id="M3"><mi>τ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118039&type= 7.62000036 3.55599999 为 <math display="block" id="M4"><mi>τ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mtable><mtr><mtd/><mtd><mi>d</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>/</mo><mi>s</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd/></mtr></mtable></mtd></mtr><mtr><mtd><mfrac><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>s</mi><mo>/</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>,</mo><mtable><mtr><mtd/><mtd><mi>d</mi><mo>=</mo><mi>k</mi><mo>/</mo><mi>s</mi></mtd><mtd/><mtd/></mtr></mtable></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mtable><mtr><mtd/><mtd/><mtd/><mtd><mi>d</mi><mo>></mo><mi>k</mi><mo>/</mo><mi>s</mi></mtd><mtd/><mtd/><mtd/></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118050&type= 72.05133057 23.79133415 其中， <math id="M5"><mi>s</mi><mo>=</mo><mi>c</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>k</mi><mo>/</mo><mi>δ</mi><mo stretchy="false">)</mo><mroot><mrow><mi>k</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118060&type= 25.90799904 4.40266657 ， c <math id="M6"><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118073&type= 6.51933336 2.79399991 且为常数。 用 (2) 式对 <math id="M7"><mi>ρ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118085&type= 7.70466709 3.72533321 进行修正，并行归一化，得到RSD（robust soliton distribution）函数 <math display="block" id="M8"><mi>μ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>+</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mi>z</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118097&type= 33.61266708 7.70466709 其中， <math id="M9"><mi>z</mi><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>d</mi></mrow></munder><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>+</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118108&type= 34.96733475 6.60399961 。 依据 (3) 式的概率分布，可以得到每组信源原始长度 k 的LT编码矩阵 <math display="block" id="M10"><mi mathvariant="bold-italic">G</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>⋯</mo></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi>k</mi><mo>×</mo><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118119&type= 51.05400085 20.06599998 其中： k 表示信源原始符号数量， <math id="M11"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118130&type= 2.79399991 2.79399991 表示译码符号数量，由于LT码是无码率编码， m 数值难以确定，但 <math id="M12"><mi>m</mi><mo>≥</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118138&type= 10.07533360 3.30199981 。 LT码编码矩阵 G 中，每1列都对应1个LT编码符号，每列中含有1的数量即为编码度 <math id="M13"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118151&type= 2.20133328 2.79399991 。将信源消息符号与 G 相乘，即可得到LT码。 2　 改进的防窃听LT码编码方法 1 LT采用BP译码，该方法译码简单，译码符号数量与接收到编码符号顺序无关，只有接收到足够的编码符号数量即可恢复信源消息符号，但是需要接收到度1的编码符号才能开始译码。在删除信道中，用信道删除概率 <math id="M14"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118159&type= 4.23333359 3.72533321 表示接收符号发生错误的概率，传输过程中部分LT编码符号会发生错误，错误符号不参与BP译码。如果接收者需要 m 个正确编码符号恢复信源消息符号，则需要接收到编码符号数量约为 <math id="M15"><mi>m</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118167&type= 18.54199982 4.57200003 ，由此可知删除信道下的LT码编码矩阵 G 为 <math id="M16"><mi>k</mi><mo>×</mo><mi>m</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118175&type= 25.73866653 4.57200003 。 为了增加窃听者在窃听信道中的误码率，本文将LT码的部分编码矩阵重新排列，将度1符号延后接收，推迟BP译码开始的时间，达到LT码的防窃听能力，其实现方法如下： 首先，依据RSD函数设定 <math id="M17"><mi>k</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118184&type= 17.35666656 4.57200003 列编码矩阵 <math id="M18"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118194&type= 3.89466667 3.72533321 。将 <math id="M19"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118200&type= 3.89466667 3.72533321 中的列按照度 <math id="M20"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118209&type= 2.20133328 2.79399991 的数值由大到小重新进行排列，将得到的 <math id="M21"><mi>k</mi><mo>×</mo><mi>k</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118216&type= 24.55333328 4.57200003 编码矩阵设为 <math id="M22"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118223&type= 4.82599974 4.14866638 。 <math id="M23"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118231&type= 4.82599974 4.14866638 中将度1所对应的列排在后面，以实现推迟度1编码符号的时间； 然后，继续依据RSD函数获取编码矩阵 <math id="M24"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118238&type= 3.89466667 3.72533321 ，其列足够多，以保证产生足够多的LT码，使接收者完成译码。如果译码需要接收编码符号数量为 m ，则 <math id="M25"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118244&type= 3.89466667 3.72533321 的编码矩阵列为 <math id="M26"><mfrac><mrow><mi>m</mi><mo>-</mo><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118251&type= 11.68400002 8.46666718 。将 <math id="M27"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118258&type= 4.82599974 4.14866638 与 <math id="M28"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118265&type= 3.89466667 3.72533321 进行组合得到本文提出的改进的LT编码矩阵 <math id="M29"><mi mathvariant="bold-italic">G</mi><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118272&type= 3.47133350 3.30199981 <math display="block" id="M30"><mi mathvariant="bold-italic">G</mi><mi mathvariant="normal">'</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi><mo>×</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfrac></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118280&type= 32.93533325 8.04333305 其中，如果 <math id="M31"><mi>m</mi><mo>=</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118288&type= 9.31333351 2.79399991 ，则 <math id="M32"><msub><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118301&type= 10.83733368 3.72533321 。 依据 (5) 式进行编码可得到本文提出的改进的LT码。 由 图1 图1 窃听信道可知，信源将改进的LT码作为信道编码，将编码符号源源不断发送给接收者，直到接收者恢复信源消息符号，则开始发送下一组编码符号。 由于改进的LT码仅改变部分编码符号发生的顺序，其度分布依旧服从 (3) 式，度1符号出现的概率为 <math id="M33"><mi>μ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118316&type= 7.02733326 3.72533321 。删除概率条件下，前 <math id="M34"><mi>k</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118328&type= 17.35666656 4.57200003 个编码符号中，度1的符号个数为 <math id="M35"><mi>k</mi><mi>μ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118339&type= 24.89200020 4.82599974 。接收者译码第一次接收到度1符号时间 <math id="M36"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118350&type= 3.72533321 3.72533321 则满足 <math display="block" id="M37"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≥</mo><mfrac><mrow><mi>k</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>μ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118359&type= 36.91466904 8.63599968 在窃听信道中采用改进的LT码作为信道编码，无论是接收者还是窃听者，接收到度1符号以后，即满足 (6) 式，才能开始进行BP译码。 3　 改进的LT码在窃听信道中性能分析 1 3.1　改进的LT码防窃听原理分析 2 3.1.1　度1符号出现时间与BP译码关系 3 接收者接收到的LT编码符号，译码器首先需要寻找度为1的编码符号，再寻找与它相连接编码符号进行异或降低编码符号的度或恢复原始符号，因此接收度1编码符号的时间不同，译码开始时间也不同。设信源符号 k =200，在没有噪声干扰情况下，译码过程如 图2 图2 所示。 与误码率关系 图2 图2 中曲线①表示随机的一次译码过程，在接收到第52个符号度1符号首次出现，当接收到第253个编码符号时，由于信源原始符号已经累计到一定数量，BER迅速下降变为0形成 “瀑布区”，完成BP译码过程。将曲线①对应的LT编码矩阵中前122列中的 4个度为1的列向后随机移动，其他矩阵结构保持不变，则得到新的LT编码矩阵，将其用于LT编码，其译码过程结果见曲线②。新的编码矩阵中前117个列中不存在度1的列，不会产生度1符号，因此当接收端接收到117个编码符号无法开始译码，接收到度为1的第118个编码符号的时候，BP译码开始，随着接收译码符号增加，曲线②迅速下降，并逐渐靠近曲线①，最后与曲线①重合。 因此，延迟度1符号出现的时间可推迟BP译码时间但不会影响LT码的译码开销的大小。 3.1.2　删除信道中的BP译码 3 由信源广播发送LT码，选取2个不同位置的接收者接收符号并进行BP译码，继续选取 图2 图2 实验中曲线②的LT编码矩阵在信源进行LT编码，设信道删除概率相同均为0.3，随机选取二次测试结果如 图3 图3 所示。 与误码率关系 图3 图3 中曲线①与曲线②，均在第52个符号同时开始译码，受到信道删除概率影响，当信源发送符号数量大于180的时候，两条译码过程曲线完全分离，恢复信源原始符号需要的编码符号数量也不相同，曲线①为330，曲线②为304。 由于BP译码中，只有正确的符号才参与译码，不同的信道噪声所产生的错误图案位置不同，导致2个接收者收到正确符号顺序存在差异。这些差异使BP译码过程出现不同，因此同一信源发送编码符号，不同接收者译码过程曲线、译码符号数量不同。由此可知，LT码在不同信道相同信道删除概率，BP译码进入“瀑布区”存在时间不同的情况，译码所需要编码符号数量也出现不同。 如果将LT码用于窃听信道中（ 图1 图1 ），接收者与窃听者接收同一组LT码。由于信道不同，发生错误符号位置不同，接收者与窃听者接收到正确的LT码存在差异。当接收者结束译码后，由 图3 图3 实验结果可知窃听者的译码存在两种情况： 1） 曲线①为窃听者，曲线②为接收者，则窃听者早于接收者恢复信源信息，其译码率为100%，这种情况不会使窃听者产生误码率，但是与接收者相比，不会多接收到信源原始符号； 2） 曲线①为接收者，曲线②为窃听者，则窃听者滞后于接收者恢复信息，即接收者恢复信源消息的时候，窃听者仅恢复部分信源消息码字，其余则无法译出，对窃听者来说存在没有恢复的编码符号，因而产生误码率。 3.2　改进的LT码与传统LT码对比 2 3.2.1　窃听者误码率的比较 3 在窃听信道结构中，将本文提出的改进LT编码与传统的LT编码进行对比，观察窃听者接收信号的误码率与主信道删除概率关系曲线。 设 k =1 000，选取RSD度分布编码， <math id="M38"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">0.05</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118400&type= 13.46199894 2.79399991 ， c =0.03。主信道删除概率为 P AB ，窃听信道删除概率为 P AE 。根据文献[ 8 8 , 9 9 ]等方法加入人工噪声，使主信道优于窃听信道，即 <math id="M39"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118411&type= 24.29933357 3.81000018 ， <math id="M40"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118424&type= 3.55599999 3.72533321 表示在窃听信道与主信道的信道删除概率差值，分别选取 <math id="M41"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118433&type= 10.49866676 3.72533321 和 <math id="M42"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.01</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118443&type= 15.24000072 3.72533321 进行对比。采用Matlab进行仿真，仿真次数2 000，实验仿真结果如 图4 图4 所示。 由 图4 图4 结果可知，当 <math id="M43"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118470&type= 23.28333473 3.72533321 ， <math id="M44"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118480&type= 10.41399956 3.72533321 ，无论是改进的LT编码还是传统的LT编码，窃听者产生误码率均等于0，主要原因是信道无噪声干扰，窃听者与接收者接收到的编码符号均相同，BP译码过程也相同，两者同时恢复信源原始符号，因此这种情况LT码不适合作为防窃听码使用。 当 <math id="M45"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118489&type= 13.12333298 3.72533321 ， <math id="M46"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118500&type= 15.91733360 3.72533321 ， <math id="M47"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118508&type= 10.41399956 3.72533321 ，两种LT码都可以使窃听者误码率增加，但是改进的LT使窃听者产生的误码率大于传统的LT码。由于改进的LT编码方法推迟了第一次接收到度1编码符号时间，存在接收者已经恢复完信源消息、窃听者却还没有恢复信源的情况，使窃听者BER增加。 当窃听信道删除概率高于主信道，即 <math id="M48"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">0.01</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118521&type= 27.43200111 3.81000018 ，由 (6) 式可知，改进的LT码方法使窃听者接收到度1的时间与接收者相比显然滞后。当信源停止发送编码符号时候，窃听者有大量LT编码符号无法译码，因而增加了窃听者的误码率。 另外， 图4 图4 的实验结果中，传统的LT码作为防窃听码，当 <math id="M49"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118536&type= 5.41866684 3.72533321 较小时，窃听者产生的误码率随着 <math id="M50"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118549&type= 5.41866684 3.72533321 的增加而增加， <math id="M51"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118560&type= 5.41866684 3.72533321 达到一定值，窃听者的BER开始下降。而采用改进的LT码作为防窃听码，当 <math id="M52"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118568&type= 5.41866684 3.72533321 增加到一定值时，窃听者的BER虽然也下降，但是下降幅度较小，尤其是当 <math id="M53"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>></mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118579&type= 16.67933464 3.72533321 时，窃听者的BER几乎不会下降。产生这一现象的原因是： <math id="M54"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118586&type= 9.90599918 3.72533321 <math id="M55"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118594&type= 5.41866684 3.72533321 时，传统的LT码作为防窃听码，不同信道的信道删除概率会干扰接收者和窃听者原本相同的译码顺序， <math id="M56"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118603&type= 5.41866684 3.72533321 越大，接收者和窃听者接收到编码符号差异越多，窃听者的误码率随着 <math id="M57"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118611&type= 5.41866684 3.72533321 的增加而增加。但是当 <math id="M58"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118618&type= 5.41866684 3.72533321 增加到一定值，虽然接收者和窃听者接收到正确符号数量保持相同，但接收到不同正确符号数量也会增多，窃听者只要先于接收者接收到度1符号就可以按照自己接收到正确符号的顺序开始译码，如果符号数量足够则可以获取信源的原始符号，降低了窃听者的误码率，因此出现窃听者的误码率随着 <math id="M59"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118627&type= 5.41866684 3.72533321 增加而下降的现象。而采用改进的LT编码作防窃听码，由 (6) 式可知， <math id="M60"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118634&type= 5.41866684 3.72533321 越大，接收者与窃听者开始 BP译码的时间越迟，窃听者与接收者几乎是同时接收到度1符号，窃听者很难以较强的优势先于接收者进入“瀑布区”恢复信源信息，因此 图4 图4 中当 <math id="M61"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118641&type= 5.41866684 3.72533321 较大时，改进的LT码方法使窃听者的误码率随 <math id="M62"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118649&type= 5.41866684 3.72533321 的增加下降较缓慢。 3.2.2　接收者恢复信源符号过程对比 3 设信道删除概率 <math id="M63"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118656&type= 11.09133339 3.72533321 ，将改进的LT编码与传统LT编码译码过程相对比，比较接收到译码符号数量 m 与恢复信源符号数量变化情况，其他实验条件同上，实验结果如 图5 图5 所示。 图5 图5 中，选取传统的LT码作为防窃听码时，接收到度1编码符号是随机发生的，平均接收到第112个编码符号开始译码，而本文提出的方法接收到第988个编码符号才是度1的编码符号，BP译码开始时间比传统LT码推迟。接收到度1符号后，BP开始译码，被恢复的信源符号数量迅速增加。当 <math id="M64"><mi>m</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mn mathvariant="normal">000</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118680&type= 16.59466553 3.30199981 时，改进LT码与传统LT码的译码曲线几乎完全重合，直至译出所有信源原始符号。 因此，改进的LT码与传统的LT码译码所需要编码符号数量完全相同，只是延迟译码开始时间，不会增加LT码的译码开销。 4　 结 论 1 由以上实验结果和分析可知，在窃听信道中，改进LT码编码矩阵方法在译码开销不变的情况下，增加了窃听者接收符号的误码率，有效防止窃听者窃听到相关信息，提高了无线通信的安全性能。 参考文献 MAO W . 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DOI:10.14188/j.1671⁃8836.2019.02.008(Ch). 辽宁省自然科学基金指导计划项目(201602373)，辽宁省教育厅高校基本科研项目（JQL201715405） 2018-07-05 2019-04-24 2020-11-25 武汉大学学报（理学版） 2 65