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基于矩阵特征值的可验证无可信中心门限方案
可信计算 | 更新时间：2020-09-30
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基于矩阵特征值的可验证无可信中心门限方案
- Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue
- 武汉大学学报（理学版）
- 作者机构：
1.北京电子科技学院 密码科学与技术系，北京 100070
2.密码科学技术国家重点实验室，北京 100878
3.数据通信科学技术研究所，北京 100191
4.兴唐通信科技有限公司，北京 100191
- 作者简介：
张艳硕，男，副教授，博士，主要研究领域为密码理论及其应用。E-mail：zhang_yanshuo@163.com
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- 基金信息：国家重点研发计划(2018YFB1004101);中央高校基本科研业务费项目(328201902)
- DOI：10.14188/j.1671-8836.2019.0509 中图分类号： TP309.7
纸质出版日期：2020-04-24，
收稿日期：2019-09-01，
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引用本文
张艳硕,王泽豪,杜耀刚, 等.基于矩阵特征值的可验证无可信中心门限方案[J].武汉大学学报（理学版）,2020,66(2):135-140.
ZHANG Yanshuo,WANG Zehao,DU Yaogang,et al.Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue [J].J Wuhan Univ （Nat Sci Ed）,2020,66(2):135-140.
张艳硕,王泽豪,杜耀刚, 等.基于矩阵特征值的可验证无可信中心门限方案[J].武汉大学学报（理学版）,2020,66(2):135-140. DOI：10.14188/j.1671-8836.2019.0509.
ZHANG Yanshuo,WANG Zehao,DU Yaogang,et al.Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue [J].J Wuhan Univ （Nat Sci Ed）,2020,66(2):135-140. DOI：10.14188/j.1671-8836.2019.0509(Ch).
基于矩阵特征值的可验证无可信中心门限方案
基于矩阵特征值的可验证无可信中心门限方案 Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue 1 2 first-author 张艳硕，男，副教授，博士，主要研究领域为密码理论及其应用。E-mail：zhang_yanshuo@163.com 张艳硕，男，副教授，博士，主要研究领域为密码理论及其应用。E-mail：zhang_yanshuo@163.com zhang_yanshuo@163.com 3 4 1 1 北京电子科技学院 密码科学与技术系，北京 100070 北京电子科技学院 密码科学与技术系，北京 100070 Department of Cryptology Science and Technology,Beijing Electronic Science & Technology Institute, Beijing 100070, China Department of Cryptology Science and Technology,Beijing Electronic Science & Technology Institute, Beijing 100070, China 密码科学技术国家重点实验室，北京 100878 密码科学技术国家重点实验室，北京 100878 State Key Laboratory of Cryptology,Beijing 100878,China State Key Laboratory of Cryptology,Beijing 100878,China 数据通信科学技术研究所，北京 100191 数据通信科学技术研究所，北京 100191 Data Communication Science and Technology Research Institute, Beijing 100191, China Data Communication Science and Technology Research Institute, Beijing 100191, China 兴唐通信科技有限公司，北京 100191 兴唐通信科技有限公司，北京 100191 Xingtang Telecommunications Technology Co. Ltd., Beijing 100191, China Xingtang Telecommunications Technology Co. Ltd., Beijing 100191, China 传统的门限（秘密共享）方案大多存在一个可信中心，可信中心负责秘密份额的产生、分配以及秘密的恢复，这会影响系统的安全性、鲁棒性和可用性，本文利用矩阵特征值的特点，设计了一个可验证无可信中心门限方案。在本文方案中，所有参与者提供相同秘密份额的值给黑盒子，构成一个2 n 维的可逆方阵 P ，该可逆方阵 P 和对角矩阵 L 生成一个矩阵 A ，将矩阵 A 的特征向量标准正交化，作为子密钥分发给各参与者。每个参与者各分得两个子密钥，这两个子密钥满足两个条件：正交和所对应的特征值相同。在子密钥生成和主密钥恢复的过程中，均可以利用这两个条件保证方案安全实施。分析表明，该方案是正确且安全的，信息率为1/2。通过实例说明了方案的可行性。 Mostly,there is a trusted center in traditional secret sharing （threshold） schemes，and the trusted center is responsible for the generation and distribution of secret shares and the recovery of secrets, which may affect the security, robustness and usability of the system. In this paper, we use the eigenvalues of matrices to design a verifiable untrusted central threshold scheme. In this scheme, all participants provide the same value of secret share equally to the black box, forming a 2 n -dimensional matrix P of invertible matrix. A matrix A is generated for the reversible matrix P and the diagonal matrix L , and the standard orthogonalization of the eigenvector of the matrix A is distributed as a sub-key to each participant. Each participant is divided into two subkeys, which satisfies two conditions: orthogonality and the corresponding eigenvalues are the same. In the process of subkey generation and master key recovery, these two conditions can be used to ensure the security implementation of the scheme. The analysis results show that the scheme is correct and safe, and the information rate is 1/2. At the end of the article, examples are used to illustrate the feasibility of the implementation of the program. 门限 秘密共享 可信 可验证 黑盒子 特征向量 threshold secret sharing credible verifiable black box eigenvector 0 引 言 1 门限(秘密共享)方案常被作为密码学协议中的一种工具。Shamir [ 1 1 ] 和Blakley [ 2 2 ] 最先提出了两种( t , n )门限方案。随后，各类门限方案便被应用于不同场景，如机器间通信技术（machine-to-machine ,M2M）中应用的效率扩展( t , n )门限方案 [ 3 3 ] ，应用于公共信息频繁更新场景的基于格和Ajtai的单向函数的门限方案 [ 4 4 ] 、多秘密共享 [ 5 5 ] 、多级多秘密共享 [ 6 6 ] 的门限方案，以及基于量子加密的多方量子秘密共享方案 [ 7 7 ] 。上所述方案都需要一个秘密分发方，即可信中心的存在方能成立。 然而，可信中心的存在会导致“权威欺骗”问题，成为系统单点失败源，其任何安全失误或故障会暴露秘密或使系统崩溃。因此，“无可信中心”这一概念被提出 [ 8 8 , 9 9 ] ，并在此基础上发展出不同的应用。例如基于无可信中心的全分布式密钥管理方案 [ 10 10 ] ；无可信中心的动态门限签名方案 [ 11 11 ] ；无共享分布式中心（SDC）的秘密共享方案 [ 12 12 ] ；成员可协同交互的无可信中心秘密共享方案 [ 13 13 ] 。 2018年，文献[ 14 14 ]首次借助于特征值提出一种无可信中心的门限秘密共享方案,是无可信中心概念下的又一次创新性研究。但该方案没有提供验证功能。因此，本文从矩阵特征值的角度设计了一种可验证的无可信中心的门限方案。所有的参与者均提供相同的秘密份额即列向量给黑盒子，协商生成子密钥，每个参与者均得到两个子密钥，且子密钥满足以下特征：子密钥的范数均为1；任意两个子密钥都是正交的；同一个参与者得到的子密钥所对应的特征值是相同的。根据以上特征，利用黑盒子可以进行欺诈检测。 1 准备知识 1 1.1 数学相关基础知识 2 1.1.1 矩阵的特征值和特征向量 3 定义1 [ 15 15 ] 设 A 是 n 阶矩阵，如果数 <math id="M1"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413655&type= 1.77800000 2.79399991 和 n 维非零列向量 x 使关系式 <math display="block" id="M2"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">x</mi><mo>=</mo><mi>λ</mi><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413663&type= 13.54666710 2.79399991 成立，那么，这样的数 <math id="M3"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413676&type= 1.77800000 2.79399991 称为矩阵 A 的特征值，非零向量 x 称为 A 的对应于特征值 <math id="M4"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413683&type= 1.77800000 2.79399991 的特征向量。 1.1.2 相似矩阵 3 定义2 [ 15 15 ] 设矩阵 A 、 B 都是 n 阶矩阵，若有可逆矩阵 P ，使 <math display="block" id="M5"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413694&type= 18.88066673 3.04799986 则称 B 是 A 的相似矩阵，可逆矩阵 P 称为把 A 变成 B 的相似变换矩阵。 推论1 若 n 阶矩阵 A 与对角矩阵 <math id="M6"><mi mathvariant="bold-italic">Λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413706&type= 2.70933342 2.79399991 相似，则存在可逆矩阵 P ，使 <math display="block" id="M7"><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">Λ</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413715&type= 50.71533203 18.96533394 <math id="M8"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413721&type= 18.28800011 4.14866638 即是 A 的 n 个特征值。 P 用其列向量表示为 <math id="M9"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413728&type= 29.29466629 4.65666676 ，于是有 <math display="block" id="M10"><mi mathvariant="bold-italic">A</mi><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>⋯</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413736&type= 34.71333313 4.57200003 P 的列向量 <math id="M11"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413743&type= 2.70933342 4.57200003 就是 A 的对应于 <math id="M12"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413754&type= 2.53999996 3.72533321 的特征向量。 定理1 n 阶矩阵 A 与对角阵 <math id="M13"><mi mathvariant="bold-italic">Λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413763&type= 2.70933342 2.79399991 相似（即 A 能对角化）的充分必要条件是 A 有 n 个线性无关的特征向量。 定理2 若 n 阶矩阵 A 与 B 相似，则 A 与 B 的特征值相同。 1.1.3 向量组的线性相关性 3 定义3 [ 15 15 ] 给定向量组 <math id="M14"><mi mathvariant="bold-italic">A</mi><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413771&type= 23.36800003 4.14866638 ，如果存在不全为零的数 <math id="M15"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413778&type= 18.03400040 4.14866638 ，使 <math display="block" id="M16"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413782&type= 44.28066635 3.81000018 则称向量组 A 是线性相关的，否则称它线性无关。 1.1.4 标准正交化 3 定义4 设 <math id="M17"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413789&type= 18.71133423 4.14866638 是向量空间 V 的一个基，求 <math id="M18"><mi mathvariant="bold-italic">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413800&type= 2.96333337 2.96333337 的一个标准正交基，即要找一组两两正交的单位向量 <math id="M19"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413810&type= 17.69533348 4.14866638 ，使得 <math id="M20"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413823&type= 17.69533348 4.14866638 与 <math id="M21"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413832&type= 18.71133423 4.14866638 等价。这一过程称为将基 <math id="M22"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413842&type= 18.71133423 4.14866638 标准正交化。 标准正交化包括两个步骤，即施密特正交化和单位化。施密特正交化过程了如下 <math display="block" id="M23"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413849&type= 11.09133339 3.72533321 <math display="block" id="M24"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">]</mo></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413858&type= 31.07266808 8.63599968 <math display="block" id="M25"><mo>⋮</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413868&type= 3.72533321 3.30199981 <math display="block" id="M26"><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">]</mo></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mo>⋯</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413876&type= 63.58466721 8.72066689 <math display="block" id="M27"><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">]</mo></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413884&type= 25.39999962 8.72066689 容易验证 <math id="M28"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413892&type= 18.71133423 4.14866638 两两正交，且 <math id="M29"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413898&type= 9.05933285 4.14866638 <math id="M30"><mo>⋯</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413906&type= 8.80533314 4.14866638 与 <math id="M31"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413913&type= 18.71133423 4.14866638 等价。 单位化步骤如下 <math display="block" id="M32"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413919&type= 19.30399895 9.56733322 ， <math display="block" id="M33"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413927&type= 19.30399895 9.56733322 ， <math display="block" id="M34"><mo>⋮</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413935&type= 3.72533321 3.30199981 <math display="block" id="M35"><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfenced></mrow></mfrac><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413945&type= 19.30399895 9.65199947 <math id="M36"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413953&type= 17.69533348 4.14866638 就是 V 的一个标准正交基。 1.2 黑盒子 2 1.2.1 黑盒子的定义 3 定义5 所谓“黑盒子”，是指对于一个器件或产品，用户并不知其内部构造和原理，而只知它的功能及如何使用这些功能 [ 16 16 ] 。 1.2.2 黑盒子的功能 3 本方案中的黑盒子需要满足以下功能： 1) 输入一个列向量，黑盒子可以判断该列向量和存储在黑盒子中向量组的线性相关性。本功能的实现根据公式(4)设计的，即 <math display="block" id="M37"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413962&type= 46.05867004 4.57200003 其中， <math id="M38"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413971&type= 19.30399895 4.57200003 是每次参与者输入的列向量。当新输入的列向量和存储在黑盒子中的向量组线性相关时，黑盒子要求参与者继续输入列向量，直到满足线性无关的条件为止。 2) 输入一个线性无关的向量组，可以将该向量组构成一个可逆的矩阵 P ，根据(2)式求出对角矩阵 <math id="M39"><mi mathvariant="bold-italic">Λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413980&type= 2.70933342 2.79399991 的相似矩阵 A 。求出矩阵 A 的特征向量，并将其标准正交化，得到 n 个正交的单位向量，作为子密钥分发给参与者。 3) 输入一个特征向量，黑盒子输出该特征向量对应的特征值。 本功能的实现是根据(3)式设计的，即 <math display="block" id="M40"><mi mathvariant="bold-italic">A</mi><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413989&type= 17.27199936 4.57200003 其中， <math id="M41"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3413999&type= 3.30199981 4.57200003 是参与者输入的子密钥，输出的是该子密钥所应用的特征值。 2 本文方案 1 本方案可分为三个步骤：参数选取、秘密生成以及秘密恢复。 2.1 参数选取 2 本门限系统包含 n 个参与者。 n 个参与者构成集合 B ，将 B 划分为几个不相交的集合， <math display="block" id="M42"><mi>B</mi><mo>=</mo><mover><mrow><munder><mrow><mo>∪</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>t</mi></mrow></mover><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414006&type= 15.57866669 8.63599968 其中: <math id="M43"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414015&type= 66.12466431 4.91066647 。 首先，每个集合的参与者通过Diffie-Hellman密钥交换协议获得会话密钥 <math id="M44"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414022&type= 2.53999996 3.72533321 值( <math id="M45"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414030&type= 13.54666710 3.30199981 )，并将 <math id="M46"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414039&type= 2.53999996 3.72533321 值发送给黑盒子。黑盒子根据得到的 <math id="M47"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414046&type= 2.53999996 3.72533321 值，生成一个对角矩阵 <math display="block" id="M48"><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414054&type= 64.51599884 43.09533310 <math display="block" id="M49"><mtable columnalign="left"><mtr><mtd><mfenced open="" close="}" separators="|"><mrow><mtable columnalign="left"><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable></mrow></mfenced><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mfenced open="" close="}" separators="|"><mrow><mtable columnalign="left"><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable></mrow></mfenced><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mfenced open="" close="}" separators="|"><mrow><mtable columnalign="left"><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable></mrow></mfenced><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414062&type= 7.11199999 42.67199707 其中， <math id="M50"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414071&type= 2.87866688 3.72533321 有 <math id="M51"><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414081&type= 4.91066647 3.72533321 个， <math id="M52"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414090&type= 2.87866688 3.72533321 有 <math id="M53"><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414100&type= 4.91066647 3.72533321 个，…， <math id="M54"><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414111&type= 2.45533323 3.72533321 有 <math id="M55"><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414119&type= 4.48733330 3.72533321 个。本文方案中，各集合的参与者利用了密钥交换协议得到会话密钥，而不知道会话密钥的内容。 然后，选定一个素数 p ，选定 t 个小于 p 的两两不相同整数 <math id="M56"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414129&type= 18.71133423 4.14866638 (如选择1,2,…, t )和参与者提供的初始信息 <math id="M57"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414142&type= 2.53999996 3.72533321 组合成 t 个数对（ <math id="M58"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414148&type= 6.77333355 4.14866638 ）。假设主秘密为 <math id="M59"><mi>k</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414158&type= 9.56733322 3.72533321 ，选取 <math id="M60"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414168&type= 10.07533360 3.72533321 组成秘密数对（ <math id="M61"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414180&type= 6.26533318 4.14866638 ）。（ <math id="M62"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414191&type= 6.26533318 4.14866638 ）和 t 个数对（ <math id="M63"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414203&type= 6.77333355 4.14866638 ）经过Lagrange Interpolation 公式生成一个 t 次多项式 <math display="block" id="M64"><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>x</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414216&type= 57.40399933 4.91066647 此时的主秘密为 <math id="M65"><mi>k</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414228&type= 9.56733322 3.72533321 。 2.2 秘密生成 2 首先，集合 B 1 中的参与者 B 11 生成一个2 n 维的列向量 <math id="M66"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414241&type= 4.06400013 4.57200003 给黑盒子，黑盒子将其存储在 P 中，此时 <math id="M67"><mi mathvariant="bold-italic">P</mi><mo>:</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414248&type= 8.21266651 4.57200003 ，其构成的矩阵 <math id="M68"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414256&type= 20.23533440 4.65666676 。 接着，参与者 B 11 又生成一个2 n 维的列向量 <math id="M69"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414264&type= 4.06400013 4.57200003 给黑盒子，黑盒子验证 <math id="M70"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414273&type= 4.06400013 4.57200003 与 P 是否线性相关。若相关，参与者 B 11 继续提供列向量给黑盒子，否则，将该列向量存储在 P 中，此时 <math id="M71"><mi mathvariant="bold-italic">P</mi><mo>:</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414282&type= 14.22399998 4.57200003 ，其构成的矩阵 <math display="block" id="M72"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi><mo>×</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414290&type= 26.16199875 4.65666676 以此类推，集合 B 1 中的每个参与者均提供两个列向量协商生成子密钥，得到 <math display="block" id="M73"><mi mathvariant="bold-italic">P</mi><mo>:</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414298&type= 14.90133286 4.57200003 <math display="block" id="M74"><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414307&type= 5.75733376 3.21733332 <math display="block" id="M75"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414315&type= 11.76866722 4.06400013 <math display="block" id="M76"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414323&type= 6.77333355 4.06400013 ， 其构成的矩阵 <math display="block" id="M77"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414331&type= 14.98599911 4.06400013 <math display="block" id="M78"><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414339&type= 40.04733276 5.07999992 同理，其他集合的参与者也按照集合 B 1 中参与者的方式提供2 n 维的列向量给黑盒子。最后，所有参与者提供的 <math display="block" id="M79"><mi mathvariant="bold-italic">P</mi><mo>:</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414348&type= 3.47133350 2.79399991 <math display="block" id="M80"><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414357&type= 23.62199974 4.99533367 <math display="block" id="M81"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414363&type= 6.18066692 4.06400013 <math display="block" id="M82"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414369&type= 19.55799866 4.06400013 其构成的矩阵 <math display="block" id="M83"><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi><mo>×</mo><mn mathvariant="normal">2</mn><mi>n</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414376&type= 13.46199894 3.72533321 <math display="block" id="M84"><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414382&type= 44.53466797 5.07999992 <math display="block" id="M85"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414390&type= 37.76133347 4.23333359 然后，黑盒子利用推论1计算出 <math display="block" id="M86"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">Λ</mi><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414397&type= 61.55266953 5.33399963 · <math display="block" id="M87"><mtable columnalign="left"><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd/><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd/><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><mo>⋱</mo></mtd><mtd/></mtr><mtr><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd/><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414403&type= 66.88667297 57.23466492 · 求出该相似矩阵的特征向量，将其标准正交化，得到 <math display="block" id="M88"><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi><mo>×</mo><mn mathvariant="normal">2</mn><mi>n</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414411&type= 13.71600056 3.72533321 <math display="block" id="M89"><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1,2</mn><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414417&type= 53.93266296 5.07999992 <math display="block" id="M90"><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414425&type= 5.75733376 3.21733332 <math display="block" id="M91"><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414431&type= 20.06599998 4.23333359 将其作为子密钥分发给各参与者。 最后，黑盒子将( <math id="M92"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414437&type= 7.11199999 4.14866638 )和( <math id="M93"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>j</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414444&type= 10.66799927 4.14866638 ) <math id="M94"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414450&type= 34.79800034 4.82599974 作为子密钥分发给各集合的参与者，同时也将 <math id="M95"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414458&type= 20.40466690 4.48733330 发送给每个参与者，其中 <math id="M96"><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414463&type= 12.69999981 4.06400013 ，且与 <math id="M97"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414469&type= 18.71133423 4.14866638 均不相同。 2.3 秘密恢复 2 由于门限的特殊性，要求恢复秘密的参与者数目不少于门限值 t ，也就是每个参与者集合必须至少出一个人(不失一般性)。 首先，假设集合 B 1 中参与者 B 11 提供两个子密钥对 <math id="M98"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414476&type= 11.76866722 4.48733330 和 <math id="M99"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414482&type= 11.76866722 4.48733330 ，集合 B 2 中参与者 B 21 提供两个子密钥对 <math id="M100"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414489&type= 11.76866722 4.48733330 和 <math id="M101"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414496&type= 11.76866722 4.48733330 ，…，集合 <math id="M102"><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414502&type= 3.21733332 3.72533321 中参与者 <math id="M103"><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414508&type= 4.23333359 3.72533321 提供两个子密钥 <math id="M104"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414514&type= 10.92199993 4.48733330 和 <math id="M105"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414520&type= 10.92199993 4.48733330 。在恢复密钥之前，他们需要将自己手中两个子密钥输入黑盒子进行验证： ① <math id="M106"><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414526&type= 3.30199981 3.72533321 和 <math id="M107"><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414532&type= 3.30199981 3.72533321 正交； ② <math id="M108"><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414538&type= 3.30199981 3.72533321 和 <math id="M109"><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414544&type= 3.30199981 3.72533321 对应的特征值相等； 只有同时满足以上两个条件才进行下一步操作，否则，停止密钥恢复工作。 然后，他们均将自己的子密钥输入到黑盒子中去，根据(2)式，可以得到 <math id="M110"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414550&type= 17.86466599 4.14866638 。 最后，将 <math id="M111"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414556&type= 15.07066631 4.48733330 和 t 个数对 <math id="M112"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414562&type= 10.24466705 4.48733330 一起代入Lagrange Interpolation 公式，得到共享主密钥 k 。 3 方案分析 1 3.1 参数选取 2 命题1 在参与者诚实而正确地执行协议的情况下，任意授权子集都可以恢复秘密 k 。 证 设 Q 是一个最小授权子集， <math display="block" id="M113"><mi>Q</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>⋃</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>⋃</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>⋃</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414568&type= 53.50933075 4.65666676 在黑盒子中输入子密钥 <math id="M114"><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414574&type= 2.87866688 3.72533321 后即可得到集合 <math id="M115"><mi>H</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414580&type= 29.04066658 4.48733330 。将 <math id="M116"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414586&type= 15.07066631 4.48733330 和 t 个数对 <math id="M117"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414592&type= 10.24466705 4.48733330 代入(7)式，从而得到方程组 <math display="block" id="M118"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⋅</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⋅</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⋅</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414596&type= 65.87066650 21.25133324 则系数矩阵为 <math display="block" id="M119"><mi mathvariant="bold-italic">N</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>t</mi></mrow></msubsup></mtd></mtr><mtr><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><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414602&type= 41.06333542 19.04999924 N 可以看成一个的 <math id="M120"><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414608&type= 25.31533432 3.55599999 范德蒙矩阵，那么它的行列式 <math id="M121"><mi>D</mi><mo>=</mo><munder><mrow><mo movablelimits="false">∏</mo></mrow><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>≥</mo><mi>m</mi><mo>></mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></mrow></munder><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414616&type= 40.63999939 6.94266701 。因为 <math id="M122"><msub><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414622&type= 25.82333374 4.14866638 互不相等，所以 <math id="M123"><mi>D</mi><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414629&type= 9.56733322 3.04799986 。由线性方程组的卡莱姆法则，知方程组有唯一解，从而可以求出 <math id="M124"><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414634&type= 7.11199999 3.55599999 。证毕。 3.2 安全性 2 为了防止系统内部不诚信成员的欺骗和系统外部伪成员的攻击，本方案可以通过(3)~(6)式对成员的子秘密份额进行信息认证，具体确认方法如下。 在子密钥生成阶段，分发给各参与者的子密钥均满足以下特点： 1) 子密钥的范数为1； 2) 同一集合参与者得到的任意两个子密钥都是正交的； 3) 同一集合参与者所得到的子密钥对应的特征值是相同的。 当所有集合的参与者得到的子密钥满足以上所有条件时，说明子密钥生成阶段是诚实的；否则，说明存在欺诈行为。 在主密钥恢复阶段，参与主密钥恢复的参与者的子密钥均满足以下特点： 1) 子密钥的范数为1； 2) 任意两个不同集合参与者得到的子密钥都是正交的； 3) 参与主密钥恢复的参与者的两个子密钥所对应的特征值是相同的。 当参与秘密恢复的所有参与者满足以上所有条件时，说明参与者是诚实的；否则，说明参与秘密恢复的参与者中存在伪成员。 通过以上分析，说明该方案是正确的、安全的，同时还具有防欺诈检测的功能。门限方案的参与者每人需提供2个随机2 n 维列向量参与秘密生成，由此得出，该方案的信息率为1/2。在预防欺诈方面是无条件安全的。 3.3 创新性 2 本方案首次从特征值的角度，设计了一个门限方案。该门限方案无需可信中心，从而避免了可信中心的欺诈。所有参与者均提供两个 <math id="M125"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414640&type= 1.86266661 2.79399991 维列向量作为子密钥生成的原始信息，体现了参与者的公平性。在子密钥生成和主密钥恢复的过程中，都可以利用两个子密钥的特征，检测活动的真实性，有效避免了可信中心欺诈和参与成员的诈骗行为。 4 方案的具体实例 1 下面用一个例子说明方案的可行性。 设有 B 1 、 B 2 两个集合，集合 B 1 中有1个参与者 B 11 ，集合 B 2 中有1个参与者 B 21 。试为这2个集合的2个参与者分配密钥，并分析重构密钥 k 的过程。 4.1 安全性参数选取 2 首先，利用密钥协商协议，集合 B 1 和集合 B 2 中的参与者分别得到协商的结果 <math id="M126"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414645&type= 9.82133293 3.72533321 和 <math id="M127"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414651&type= 9.82133293 3.72533321 。将 <math id="M128"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414658&type= 2.87866688 3.72533321 和 <math id="M129"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414664&type= 2.87866688 3.72533321 加密传输到黑盒中去，生成对角矩阵 <math display="block" id="M130"><mi mathvariant="bold-italic">Λ</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414670&type= 29.71799850 18.45733452 假设共享秘密 <math id="M131"><mi>k</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414677&type= 8.38199997 2.79399991 。然后，选取 <math id="M132"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,4</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414683&type= 23.19866753 4.48733330 ， <math id="M133"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1,2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414688&type= 23.87599945 4.48733330 和 <math id="M134"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2,1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414695&type= 23.87599945 4.48733330 ，经过Lagrange Interpolation，得到 <math display="block" id="M135"><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">3</mn><mi>x</mi><mo>+</mo><mn mathvariant="normal">4</mn><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414700&type= 46.65133286 3.81000018 此时的共享主密钥是 <math id="M136"><mi>k</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414707&type= 8.38199997 2.79399991 。 4.2 秘密生成 2 首先，集合 B 1 中的参与者生成一个4维的列向量 <math id="M137"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414713&type= 36.83000183 3.89466667 给黑盒子，黑盒子将其存储到 P 中，此时 <math id="M138"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414719&type= 48.76799774 4.48733330 。接着，参与者又生成了一个4维的列向量 <math id="M139"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414725&type= 4.06400013 4.57200003 给黑盒子，并验证 <math id="M140"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414731&type= 4.06400013 4.57200003 和向量组 P 之间的线性相关性。当二者线性相关时，该参与者继续提供列向量给黑盒子，否则，将 <math id="M141"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414737&type= 4.06400013 4.57200003 存储到向量组 P 中，此时的 <math id="M142"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414743&type= 33.02000046 3.89466667 。将其加入到向量组 P 中，得到 <math id="M143"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414749&type= 52.66266632 9.22866726 。 集合 B 1 中的参与者也按同样的方式提供两个线性无关的列向量 <math id="M144"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414755&type= 36.83000183 3.89466667 , <math id="M145"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414761&type= 36.83000183 3.89466667 给黑盒子，验证通过后分别存储到向量组 P 中，得到: <math id="M146"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mtd></mtr></mtable><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mtable columnalign="right"><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414767&type= 75.09933472 19.21933365 。 由推论1可得矩阵 <math display="block" id="M147"><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">Λ</mi><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414774&type= 45.55066681 18.45733452 求出矩阵 <math id="M148"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414780&type= 2.79399991 2.79399991 的特征向量并将其标准正交化，得到向量组 <math id="M149"><mi mathvariant="bold-italic">E</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414786&type= 30.56466866 4.48733330 ，其所构成的矩阵为 <math display="block" id="M150"><mi mathvariant="bold-italic">E</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414791&type= 45.12733459 4.48733330 <math display="block" id="M151"><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">3</mn></mtd><mtd><mn mathvariant="normal">4</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">3</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">5</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">7</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">5</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">4</mn></mtd><mtd><mn mathvariant="normal">6</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414795&type= 26.75466537 18.54199982 黑盒子将 E 中的列向量作为子密钥分发给参与者，得到两组密钥： <math id="M152"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋃</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414799&type= 34.62866592 4.65666676 ； <math id="M153"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋃</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414803&type= 34.62866592 4.65666676 。将 <math id="M154"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414807&type= 2.79399991 3.72533321 和 <math id="M155"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414811&type= 2.79399991 3.72533321 分发给集合 B 1 中参与者和集合 B 2 中的参与者，同时将( <math id="M156"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414815&type= 3.72533321 2.79399991 1，7)发送给所有参与者。 4.3 秘密恢复 2 首先，集合 B 1 中参与者和集合 B 2 中的参与者将提供的正确的子密钥输入到黑盒中，分别得到 <math id="M157"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414819&type= 9.82133293 3.72533321 ， <math id="M158"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414823&type= 9.82133293 3.72533321 。 然后，将( <math id="M159"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414827&type= 3.72533321 2.79399991 1，7)、(1，2)和(2，1)代入Lagrange Interpolation 公式得到 <math display="block" id="M160"><mn mathvariant="normal">7</mn><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414831&type= 52.83199692 8.38199997 <math display="block" id="M161"><mn mathvariant="normal">14</mn><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414836&type= 28.53266716 3.55599999 <math display="block" id="M162"><mn mathvariant="normal">2</mn><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414841&type= 53.25532913 8.38199997 <math display="block" id="M163"><mn mathvariant="normal">10</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414846&type= 28.44799995 3.55599999 <math display="block" id="M164"><mn mathvariant="normal">1</mn><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414850&type= 53.08599854 8.38199997 <math display="block" id="M165"><mn mathvariant="normal">4</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414854&type= 26.58533287 3.55599999 所以 <math display="block" id="M166"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">14</mn><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">10</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">4</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mtext> </mtext><mn mathvariant="normal">11</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414860&type= 122.00466919 3.81000018 <math display="block" id="M167"><mo>≡</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">3</mn><mi>x</mi><mo>+</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414863&type= 24.72266769 3.13266659 从而获得共享密钥 <math id="M168"><mi>k</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3414867&type= 8.38199997 2.79399991 。 5 结 语 1 利用矩阵特征值的特性，设计了一个可验证无可信中心的门限方案。所有的参与者均提供相同秘密份额的值给黑盒子，协助子密钥的生成，每个参与者得到两个子密钥。通过分析参与者提供的子密钥的数值特征，可以进行防欺诈检测。该方案的信息率为1/2。 参考文献 SHAMIR A . How to share a secret [J]. Communications of the ACM , 1979 , 22 ( 11 ): 612 - 613 . DOI: 10.1145/359168.359176 http://dx.doi.org/10.1145/359168.359176 . How to share a secret BLAKLEY G R . Safeguarding Cryptographic Keys [DB/OL].[ 2019-01-30 ]. https://www.computer.org/csdl/proceedings https://www.computer.org/csdl/proceedings ⁃article/afips/ 1979/50870313/12 OmNCeK 2 a 1 . Safeguarding Cryptographic Keys SHEN J , ZHOU T Q , LIU X G , et al . A novel Latin⁃ square-based secret sharing for M2M communications [J]. IEEE Transactions on Industrial Informatics , 2018 , 14 ( 8 ): 3659 - 3668 . DOI： 10.1109/TII.2018.2810840 http://dx.doi.org/10.1109/TII.2018.2810840 . A novel Latin⁃ square-based secret sharing for M2M communications PILARAM H , EGHLIDOS T . An efficient lattice based multi-stage secret sharing scheme [J]. IEEE Transactions on Dependable and Secure Computing , 2015 , 14 ( 1 ): 2 - 8 . DOI： 10.1109/TDSC.2015.2432800 http://dx.doi.org/10.1109/TDSC.2015.2432800 . An efficient lattice based multi-stage secret sharing scheme HSU C F , HARN L , CUI G H . An ideal multi-secret sharing scheme based on connectivity of graphs [J]. Wireless Personal Communications , 2014 , 77 ( 1 ): 383 - 394 . DOI： 10.1007/s11277-013-1511-3 http://dx.doi.org/10.1007/s11277-013-1511-3 . An ideal multi-secret sharing scheme based on connectivity of graphs SINGH N , TENTU A N , BASIT A , et al . Sequential Secret Sharing Scheme Based on Chinese Remainder Theorem [DB/OL]. [ 2019-02-20 ]. https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7919678 https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7919678 . Sequential Secret Sharing Scheme Based on Chinese Remainder Theorem AIN N U . A Novel Approach for Secure Multi-party Secret Sharing Scheme via Quantum Cryptography [DB/OL] [ 2019-02-22 ]. https://ieeexplore.ieee.org/document/7918912 https://ieeexplore.ieee.org/document/7918912 . A Novel Approach for Secure Multi-party Secret Sharing Scheme via Quantum Cryptography PEDERSEN T P . A threshold cryptosystem without a trusted party [C]// Advances in Cryptology—EUROCRYPT’91 . Berlin : Springer-Verlag , 1991 : 522 - 526 . DOI: 10.1007/3-540-46416-6_47 http://dx.doi.org/10.1007/3-540-46416-6_47 . A threshold cryptosystem without a trusted party LAIH C S , HARN L . Generalized Threshold Cryptosystems [DB/OL]. [ 2019-02-21 ]. https://link.springer.com/content/pdf/10.1007%2F3-540-57332-1_13.pdf https://link.springer.com/content/pdf/10.1007%2F3-540-57332-1_13.pdf . Generalized Threshold Cryptosystems LUO H , KONG J , ZERFOS P , et al . URSA: Ubiquitous and robust access control for mobile ad hoc networks [J]. IEEE/ACM Transactions on Networking , 2004 , 12 ( 6 ): 1049 - 1063 . DOI: 10.1109/TNET.2004.838598 http://dx.doi.org/10.1109/TNET.2004.838598 . URSA: Ubiquitous and robust access control for mobile ad hoc networks 张毅 , 侯整风 , 胡东辉 . 一种动态的无可信中心( t , n )门限签名认证方案 [J]. 合肥工业大学学报(自然科学版) , 2011 , 34 ( 9 ): 1341 - 1344 . DOI: 1003-5060(2011)09-1341-04 http://dx.doi.org/1003-5060(2011)09-1341-04 . 一种动态的无可信中心(t,n)门限签名认证方案 ZHANG Y , HOU Z F , HU D H . A dynamic ( t , n ) threshold signature authentication scheme without a trusted party [J]. Journal of Hefei University of Technology(Natural Science) , 2011 , 34 ( 9 ): 1341 - 1344 . DOI:1003-5060(2011)09-1341-04 (Ch) . A dynamic (t,n) threshold signature authentication scheme without a trusted party XUE Y S , WU S L , CHEN H D . A Blakley Secret Sharing Scheme without Trusted Share Distributed Center [DB/OL]. [ 2019-02-21 ]. https://ieeexplore.ieee.org/document/6128196 https://ieeexplore.ieee.org/document/6128196 . A Blakley Secret Sharing Scheme without Trusted Share Distributed Center 何二庆 , 侯整风 , 朱晓玲 . 一种无可信中心动态秘密共享方案 [J]. 计算机应用研究 , 2013 , 30 ( 2 ): 491 - 493 . 一种无可信中心动态秘密共享方案 HE E Q , HOU Z F , ZHU X L . Proactive secret sharing scheme without trusted party [J]. Application Research of Computers , 2013 , 30 ( 2 ): 491 - 493 (Ch) . Proactive secret sharing scheme without trusted party 张艳硕 , 李文敬 , 陈雷 , 等 . 基于特征值的可验证特殊门限秘密共享方案 [J]. 通信学报 , 2018 , 39 ( 8 ): 169 - 175 . DOI: 10.11959/j.issn.1000-436x.2018143 http://dx.doi.org/10.11959/j.issn.1000-436x.2018143 . 基于特征值的可验证特殊门限秘密共享方案 ZHANG Y S , LI W J , CHEN L , et al . Verifiable special threshold secret sharing scheme based on eigenvalue [J]. Journal on Communications , 2018 , 39 ( 8 ): 169 - 175 . DOI:10.11959/j.issn.1000-436x.2018143 (Ch) . Verifiable special threshold secret sharing scheme based on eigenvalue 同济大学数学系编 . 工程数学线性代数 [M]. 北京 ： 高等教育出版社 , 2014 . 工程数学线性代数 Department of Mathematics of Tongji University . Linear Algebra of Engineering Mathematics [M]. Beijing : Heigher Education Press , 2014（Ch) . Linear Algebra of Engineering Mathematics 曹尔强 , 张沂 , 曹晔 , 等 . “软件黑盒子”文件加锁和加密的一个方法 [J]. 长春邮电学院学报 , 1991 , 9 ( 3 ): 11 - 14 . “软件黑盒子”文件加锁和加密的一个方法 CAO E Q , ZHANG Y , CAO Y , et al . A technique of locking a disk and secreting a whole disk [J]. Journal of Changchun Post and Telecommunication Institute , 1991 , 9 ( 3 ): 11 - 14 （Ch) . A technique of locking a disk and secreting a whole disk 10.14188/j.1671-8836.2019.0509 TP309.7 A 1671-8836（2020）02-0135-06 ZHANG Yanshuo,WANG Zehao,DU Yaogang, et al .Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue [J]. J Wuhan Univ （ Nat Sci Ed ）,2020,66(2):135-140. DOI:10.14188/j.1671-8836.2019.0509(Ch). ZHANG Yanshuo,WANG Zehao,DU Yaogang,et al.Verifiable Threshold Scheme without Trusted Center Based on Matrix Eigenvalue [J].J Wuhan Univ（Nat Sci Ed）,2020,66(2):135-140. DOI:10.14188/j.1671-8836.2019.0509(Ch). 张艳硕,王泽豪,杜耀刚, 等.基于矩阵特征值的可验证无可信中心门限方案[J].武汉大学学报（理学版）,2020,66(2):135-140. DOI:10.14188/j.1671-8836.2019.0509. 张艳硕,王泽豪,杜耀刚, 等.基于矩阵特征值的可验证无可信中心门限方案[J].武汉大学学报（理学版）,2020,66(2):135-140. DOI:10.14188/j.1671-8836.2019.0509. 国家重点研发计划(2018YFB1004101),中央高校基本科研业务费项目(328201902) 2019-09-01 2020-04-24 2020-09-30 武汉大学学报（理学版） 2 66
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