搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

基于DNA编码与交替量子随机行走的彩色图像加密算法

王一诺 宋昭阳 马玉林 华南 马鸿洋

引用本文:
Citation:

基于DNA编码与交替量子随机行走的彩色图像加密算法

王一诺, 宋昭阳, 马玉林, 华南, 马鸿洋

Color image encryption algorithm based on DNA code and alternating quantum random walk

Wang Yi-Nuo, Song Zhao-Yang, Ma Yu-Lin, Hua Nan, Ma Hong-Yang
PDF
HTML
导出引用
  • 近年来, 图像加密技术备受关注. 随着人们对通信隐私及网络安全重视程度的提高, 对信息加密技术的要求更加严格, 图像作为信息的载体之一, 因携带信息的有效性和生动性而受到重视. 本文提出一种基于DNA编码与交替量子随机行走的彩色图像加密算法. 量子随机行走作为出色的密码学工具参与算法流程中各个部分, DNA编码作为核心加密方式完成算法. 本文详细描述加密、解密流程, 并对所提出算法进行仿真实验验证与结果分析. 仿真阶段设计模拟密钥参数, 编码进行彩色图像加密、解密实验, 并进行了相关分析. 实验结果表明, 本文提出的彩色图像加密算法能够进行安全有效的彩色图像加密, 且相关分析表示其加密后图像直方图平稳、像素相关性系数趋近于0、密钥空间${2^{ 128}} $, 三通道信息熵达到7.997以上, 能够抵御统计攻击、穷举攻击等攻击手段. 此外, DNA编码除新颖的编码及运算方式之外还有其独特的生物学特性, 为密码学的研究提供了新的思路与方向.
    In recent years, image encryption technology has attracted much attention. As people pay more attention to communication privacy and network security, the requirements for information encryption technology are more stringent. As one of the information carriers, images are valuable for carrying the effectiveness and vividness of the information. This paper proposes a color image encryption algorithm based on DNA encoding and alternating quantum random walk. Quantum random walk is an excellent cryptographic tool that participates in all parts of the algorithm process, and DNA encoding is used as the core encryption method to complete the algorithm. This article describes the encryption and decryption process in detail, and conducts simulation experiments to verify and analyze the results of the proposed algorithm. In the simulation stage, we design the simulation key parameters, encode the color image encryption and decryption experiments, and carry out related analysis. The experimental results show that the color image encryption algorithm proposed in this paper can perform safe and effective color image encryption. The correlation analysis shows that the image histogram after encryption is stable, the pixel correlation coefficient approaches 0, and the key space is ${2^{ 128 }} $, the three-channel information entropy reaches more than 7.997, which can resist statistical attacks, brute force attacks and other attack methods. In addition, DNA coding has unique biological characteristics in addition to the novel coding and calculation methods, which provide new ideas and directions for cryptographic research.
      通信作者: 马鸿洋, hongyang_ma@aliyun.com
    • 基金项目: 国家自然科学基金(批准号: 11975132, 61772295)、山东省自然科学基金(批准号: ZR2019YQ01)和山东省高等教育科技计划(批准号: J18KZ012)资助的课题.
      Corresponding author: Ma Hong-Yang, hongyang_ma@aliyun.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11975132, 61772295), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2019YQ01), and the Higher Educational Science and Technology Program of Shandong Province, China(Grant No. J18KZ012).
    [1]

    Zheng R H, Xiao Y, Su S L, Chen Y H, Shi Z C, Song J, Xia Y, Zheng S B 2021 Phys. Rev. A 103 052402Google Scholar

    [2]

    Kang Y H, Shi Z C, Huang B H, Song J, Xia Y 2020 Phys. Rev. A 101 032322Google Scholar

    [3]

    Long G L 2001 Phys. Rev. A 64 022307Google Scholar

    [4]

    Long G L, Li X, Sun Y 2002 Phys. Lett. A 294 143Google Scholar

    [5]

    Li T, Zhang S, Fu X Q, Wang X, Wang Y, Lin J, Bao W S 2019 Chin. Phys. B 28 120301Google Scholar

    [6]

    Liu F, Zhang X, Xu P A, He Z X, Ma H Y 2020 Int. J. Theor. Phys. 59 3491Google Scholar

    [7]

    Zhou N R, Li J F, Yu Z B, Gong L H, A Farouk 2017 Quantum Inf. Process. 16 1Google Scholar

    [8]

    Gong L H, Li J F, Zhou N R 2018 Laser Phys. Lett. 15 105204Google Scholar

    [9]

    Hu X M, Huang C X, Sheng Y B, Zhou L, Liu B H, Guo Y, Zhang C, Xing W B, Huang Y F, Li C F, Guo G C 2021 Phys. Rev. Lett. 126 010503Google Scholar

    [10]

    Zhou L, Sheng Y B, Long G L 2020 Sci. Bull. 65 12Google Scholar

    [11]

    Yan Z H, Qin J L, Qin Z Z, Su X L, Jia X J, Xie C D, Peng K C 2021 Fundam. Res. 1 43Google Scholar

    [12]

    Zhao J B, Zhang W B, Ma Y L, Zhang X H, Ma H Y 2020 Appl. Sci. 10 1935Google Scholar

    [13]

    Qiu T H, Li H, Xie M, Liu Q, Ma H Y 2019 Opt. Express 27 27477Google Scholar

    [14]

    Ma H Y, Xu P A, Shao C H, Chen L B, Li J X, Pan Q 2019 Int. J. Theor. Phys. 58 4241Google Scholar

    [15]

    Li H S, Fan P, Xia H Y, Peng H L, Long G L 2020 Sci. China Phys. Mech. 63 1

    [16]

    Zheng R H, Kang Y H, Su S L, Song J, Xia Y 2020 Phys. Rev. A 102 012609Google Scholar

    [17]

    Ma H Y, He Z X, Xu P A, Dong Y M, Fan X K 2020 Quantum Inf. Process 19 1

    [18]

    Xu P A, He Z X, Qiu T H, Ma H Y 2020 Opt. Express 28 12508Google Scholar

    [19]

    Zhou N R, Huang L X, Gong L H, Zheng Q W 2020 Quantum Inf. Process. 19 1

    [20]

    Li H S, Fan P, Xia H Y, Peng H L, Long G L 2020 Sci. China. Phys. Mech. 63 1

    [21]

    Castagnoli G 2016 Found Phys. 46 360Google Scholar

    [22]

    Castagnoli G 2016 Quanta 5 34Google Scholar

    [23]

    Gong L H, Song H C, He C S, Liu Y, Zhou N R 2014 Phys. Scr. 89 035101Google Scholar

    [24]

    Li H H, Gong L H, Zhou N R 2020 Chin. Phys. B 29 110304Google Scholar

    [25]

    Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 1687Google Scholar

    [26]

    Farhi E, Gutmann S 1998 Phys. Rev. A 58 915Google Scholar

    [27]

    Watrous J 2001 Comput. Syst. Sci. 62 376Google Scholar

    [28]

    Abd El-Latif A A, Abd-El-Atty B, Venegas-Andraca S E, Elwahsh H, Piran M J, Bashir A K, Song O, Mazurczyk W 2020 IEEE Access 8 92687Google Scholar

    [29]

    Abd-El-Atty B, Iliyasu A M, Alaskar H, Abd El-Latif A A 2020 Sensors 20 3108Google Scholar

    [30]

    Abd EL-Latif A A, Abd-El-Atty B, Venegas-Andraca S E 2020 Physica A 547 123869Google Scholar

    [31]

    Abd El-Latif A A, Abd-El-Atty B, Elseuofi S, Khalifa H S, Alghamdi A S, Polat K, Amin M 2020 Physica A 541 123687Google Scholar

    [32]

    Abd-El-Atty B, Amin M, Iliyasu A M 2020 Sci. Rep. 10

    [33]

    Godsil C, Zhan H 2019 J. Comb. Theory Ser. A 167 181Google Scholar

    [34]

    Abd-El-Atty B, Iliyasu A M, Alanezi A, Abd EL-Latif A A 2021 Opt. Lasers Eng. 138 106403Google Scholar

    [35]

    Adleman L M 1994 Science 266 1021Google Scholar

    [36]

    Leier A, Richter C, Banzhaf W, Rauhe H 2000 Biosystems 57 13Google Scholar

    [37]

    Chen J 2003 Proceedings of the 2003 International Symposium on Circuits and Systems ISCAS'03. IEEE 2003 3 III-III.

    [38]

    Chang W L, Guo M, Ho M S H 2005 IEEE Trans Nanobiosci. 4 149Google Scholar

    [39]

    Lu M X, Lai X J, Xiao G Z, Qin L 2007 Sci. China Inf. Sci. 50 324Google Scholar

    [40]

    Lai X J, Lu M X, Qin L, Han J S, Fang X W 2010 Sci. China Inf. Sci. 53 506Google Scholar

    [41]

    Wei X, Guo L, Zhang Q, Zhang J, Lian S G 2012 J. Syst. Softw. 85 290Google Scholar

    [42]

    Niu Y, Zhang X, Han F 2017 Comput. Intell. Neurosci. 2017

    [43]

    Kalsi S, Kaur H, Chang V 2018 J. Med. Syst. 42 1Google Scholar

    [44]

    Basu S, Karuppiah M, Nasipuri M, Halder A K, Radhakrishnan 2019 J. Syst. Archit. 94 24Google Scholar

    [45]

    Biswas M R, Alam K M R, Tamura S, Morimoto Y 2019 J. Syst. Archit. 48 102363

    [46]

    Huang L, Wang S, Xiang J, Sun Y 2020 Math. Prob. Eng. 2020

    [47]

    Alghafis A, Firdousi F, Khan M, Batool A L, Amin M 2020 Math. Comput. Simul. 177 441Google Scholar

    [48]

    Eswaran P, Shankar K 2017 Int. J. Pure Appl. Math. 118 393

    [49]

    Lu Q, Zhu C, Deng X 2020 IEEE Access 8 25664Google Scholar

    [50]

    Wang Y, Ye S C, Wang Y 2020 Microelectron. Comput. 37 71

  • 图 1  双方向格点上交替量子随机行走

    Fig. 1.  Alternating quantum random walking on the bidirectional grid.

    图 2  算法流程图

    Fig. 2.  Algorithm flowchart

    图 3  加密算法仿真效果图

    Fig. 3.  Encryption algorithm simulation renderings.

    图 4  Lena图像加密前后相关性分析

    Fig. 4.  Correlation analysis of images before and after Lena encryption.

    图 6  小猫图像加密前后相关性分析

    Fig. 6.  Correlation analysis of images before and after cat encryption.

    图 5  蛋糕图像加密前后相关性分析

    Fig. 5.  Correlation analysis of images before and after cake encryption.

    图 7  Lena图像加密前后三通道直方图分析

    Fig. 7.  Analysis of R, G, B three-channel histogram before and after lena image encryption.

    表 1  8种DNA编码方案

    Table 1.  Eight DNA coding schemes.

    1 2 3 4 5 6 7 8
    A 00 00 01 01 10 10 11 11
    G 01 10 00 11 00 11 01 10
    C 10 01 11 00 11 00 10 01
    T 11 11 10 10 01 01 00 00
    注: 满足DNA的生物学特性A-T互补、G-C互补.
    下载: 导出CSV

    表 2  DNA编码方案1对应的加法方案1

    Table 2.  Addition plan 1 corresponding to DNA coding plan 1.

    + A G C T
    A A G C T
    G G C T A
    C C T A G
    T T A G C
    下载: 导出CSV

    表 3  DNA编码方案1对应的减法方案1

    Table 3.  DNA coding scheme 1 corresponding to subtraction scheme 1.

    A G C T
    A A G C T
    G T A G C
    C C T A G
    T G C T A
    下载: 导出CSV

    表 4  像素相关性分析数据

    Table 4.  Pixel correlation analysis data.

    图像 方向 数值
    原始图像 加密图像
    Lena 水平 0.9275 0.0017
    垂直 0.9603 –0.0005
    对角 0.8941 0.0007
    小猫 水平 0.9429 0.0026
    垂直 0.9377 0.0009
    对角 0.9097 –0.0012
    蛋糕 水平 0.9229 0.0076
    垂直 0.9056 0.0017
    对角 0.8647 0.0017
    下载: 导出CSV

    表 5  相关性分析数据对比

    Table 5.  Correlation analysis data comparison.

    算法 通道 平均相关性
    本文 Red 水平: 0.0040
    Green 垂直: 0.0010
    Blue 对角: 0.0012
    文献[42] Red 水平: 0.0042
    Green 垂直: 0.0033
    Blue 对角: 0.0024
    文献[49] Red 水平: 0.0025
    Green 垂直: 0.0010
    Blue 对角: 0.0022
    文献[50] Red 水平: 0.0012
    Green 垂直: 0.0011
    Blue 对角: 0.0032
    下载: 导出CSV

    表 6  密钥敏感性分析数据

    Table 6.  Key sensitivity analysis data.

    图像 通道 NPCR UACI
    Lena Red 99.6040% 33.4005%
    Green 99.6195% 33.2311%
    Blue 99.5826% 33.5902%
    小猫 Red 99.6326% 33.6785%
    Green 99.6124% 33.4455%
    Blue 99.6147% 33.5784%
    蛋糕 Red 99.6124% 33.4461%
    Green 99.6528% 33.5315%
    Blue 99.7004% 33.4499%
    下载: 导出CSV

    表 7  NPCR, UACI数据对比

    Table 7.  Comparison of NPCR and UACI data.

    算法 NPCR平均 UACI平均
    本文 99.6257% 33.4835%
    文献[42] 99.5506% 33.4055%
    文献[49] 99.6554% 33.4675%
    文献[50] 99.6150% 33.3900%
    下载: 导出CSV
  • [1]

    Zheng R H, Xiao Y, Su S L, Chen Y H, Shi Z C, Song J, Xia Y, Zheng S B 2021 Phys. Rev. A 103 052402Google Scholar

    [2]

    Kang Y H, Shi Z C, Huang B H, Song J, Xia Y 2020 Phys. Rev. A 101 032322Google Scholar

    [3]

    Long G L 2001 Phys. Rev. A 64 022307Google Scholar

    [4]

    Long G L, Li X, Sun Y 2002 Phys. Lett. A 294 143Google Scholar

    [5]

    Li T, Zhang S, Fu X Q, Wang X, Wang Y, Lin J, Bao W S 2019 Chin. Phys. B 28 120301Google Scholar

    [6]

    Liu F, Zhang X, Xu P A, He Z X, Ma H Y 2020 Int. J. Theor. Phys. 59 3491Google Scholar

    [7]

    Zhou N R, Li J F, Yu Z B, Gong L H, A Farouk 2017 Quantum Inf. Process. 16 1Google Scholar

    [8]

    Gong L H, Li J F, Zhou N R 2018 Laser Phys. Lett. 15 105204Google Scholar

    [9]

    Hu X M, Huang C X, Sheng Y B, Zhou L, Liu B H, Guo Y, Zhang C, Xing W B, Huang Y F, Li C F, Guo G C 2021 Phys. Rev. Lett. 126 010503Google Scholar

    [10]

    Zhou L, Sheng Y B, Long G L 2020 Sci. Bull. 65 12Google Scholar

    [11]

    Yan Z H, Qin J L, Qin Z Z, Su X L, Jia X J, Xie C D, Peng K C 2021 Fundam. Res. 1 43Google Scholar

    [12]

    Zhao J B, Zhang W B, Ma Y L, Zhang X H, Ma H Y 2020 Appl. Sci. 10 1935Google Scholar

    [13]

    Qiu T H, Li H, Xie M, Liu Q, Ma H Y 2019 Opt. Express 27 27477Google Scholar

    [14]

    Ma H Y, Xu P A, Shao C H, Chen L B, Li J X, Pan Q 2019 Int. J. Theor. Phys. 58 4241Google Scholar

    [15]

    Li H S, Fan P, Xia H Y, Peng H L, Long G L 2020 Sci. China Phys. Mech. 63 1

    [16]

    Zheng R H, Kang Y H, Su S L, Song J, Xia Y 2020 Phys. Rev. A 102 012609Google Scholar

    [17]

    Ma H Y, He Z X, Xu P A, Dong Y M, Fan X K 2020 Quantum Inf. Process 19 1

    [18]

    Xu P A, He Z X, Qiu T H, Ma H Y 2020 Opt. Express 28 12508Google Scholar

    [19]

    Zhou N R, Huang L X, Gong L H, Zheng Q W 2020 Quantum Inf. Process. 19 1

    [20]

    Li H S, Fan P, Xia H Y, Peng H L, Long G L 2020 Sci. China. Phys. Mech. 63 1

    [21]

    Castagnoli G 2016 Found Phys. 46 360Google Scholar

    [22]

    Castagnoli G 2016 Quanta 5 34Google Scholar

    [23]

    Gong L H, Song H C, He C S, Liu Y, Zhou N R 2014 Phys. Scr. 89 035101Google Scholar

    [24]

    Li H H, Gong L H, Zhou N R 2020 Chin. Phys. B 29 110304Google Scholar

    [25]

    Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 1687Google Scholar

    [26]

    Farhi E, Gutmann S 1998 Phys. Rev. A 58 915Google Scholar

    [27]

    Watrous J 2001 Comput. Syst. Sci. 62 376Google Scholar

    [28]

    Abd El-Latif A A, Abd-El-Atty B, Venegas-Andraca S E, Elwahsh H, Piran M J, Bashir A K, Song O, Mazurczyk W 2020 IEEE Access 8 92687Google Scholar

    [29]

    Abd-El-Atty B, Iliyasu A M, Alaskar H, Abd El-Latif A A 2020 Sensors 20 3108Google Scholar

    [30]

    Abd EL-Latif A A, Abd-El-Atty B, Venegas-Andraca S E 2020 Physica A 547 123869Google Scholar

    [31]

    Abd El-Latif A A, Abd-El-Atty B, Elseuofi S, Khalifa H S, Alghamdi A S, Polat K, Amin M 2020 Physica A 541 123687Google Scholar

    [32]

    Abd-El-Atty B, Amin M, Iliyasu A M 2020 Sci. Rep. 10

    [33]

    Godsil C, Zhan H 2019 J. Comb. Theory Ser. A 167 181Google Scholar

    [34]

    Abd-El-Atty B, Iliyasu A M, Alanezi A, Abd EL-Latif A A 2021 Opt. Lasers Eng. 138 106403Google Scholar

    [35]

    Adleman L M 1994 Science 266 1021Google Scholar

    [36]

    Leier A, Richter C, Banzhaf W, Rauhe H 2000 Biosystems 57 13Google Scholar

    [37]

    Chen J 2003 Proceedings of the 2003 International Symposium on Circuits and Systems ISCAS'03. IEEE 2003 3 III-III.

    [38]

    Chang W L, Guo M, Ho M S H 2005 IEEE Trans Nanobiosci. 4 149Google Scholar

    [39]

    Lu M X, Lai X J, Xiao G Z, Qin L 2007 Sci. China Inf. Sci. 50 324Google Scholar

    [40]

    Lai X J, Lu M X, Qin L, Han J S, Fang X W 2010 Sci. China Inf. Sci. 53 506Google Scholar

    [41]

    Wei X, Guo L, Zhang Q, Zhang J, Lian S G 2012 J. Syst. Softw. 85 290Google Scholar

    [42]

    Niu Y, Zhang X, Han F 2017 Comput. Intell. Neurosci. 2017

    [43]

    Kalsi S, Kaur H, Chang V 2018 J. Med. Syst. 42 1Google Scholar

    [44]

    Basu S, Karuppiah M, Nasipuri M, Halder A K, Radhakrishnan 2019 J. Syst. Archit. 94 24Google Scholar

    [45]

    Biswas M R, Alam K M R, Tamura S, Morimoto Y 2019 J. Syst. Archit. 48 102363

    [46]

    Huang L, Wang S, Xiang J, Sun Y 2020 Math. Prob. Eng. 2020

    [47]

    Alghafis A, Firdousi F, Khan M, Batool A L, Amin M 2020 Math. Comput. Simul. 177 441Google Scholar

    [48]

    Eswaran P, Shankar K 2017 Int. J. Pure Appl. Math. 118 393

    [49]

    Lu Q, Zhu C, Deng X 2020 IEEE Access 8 25664Google Scholar

    [50]

    Wang Y, Ye S C, Wang Y 2020 Microelectron. Comput. 37 71

  • [1] 郭牧城, 汪福东, 胡肇高, 任苗苗, 孙伟业, 肖婉婷, 刘书萍, 钟满金. 微纳尺度稀土掺杂晶体的量子相干性能及其应用研究进展. 物理学报, 2023, 72(12): 120302. doi: 10.7498/aps.72.20222166
    [2] 刘瀚扬, 华南, 王一诺, 梁俊卿, 马鸿洋. 基于量子随机行走和多维混沌的三维图像加密算法. 物理学报, 2022, 71(17): 170303. doi: 10.7498/aps.71.20220466
    [3] 方洁, 姜明浩, 安小宇, 孙军伟. 基于混沌加密和DNA编码的“一图一密”图像加密算法. 物理学报, 2021, 70(7): 070501. doi: 10.7498/aps.70.20201642
    [4] 李保民, 胡明亮, 范桁. 量子相干. 物理学报, 2019, 68(3): 030304. doi: 10.7498/aps.68.20181779
    [5] 史保森, 丁冬生, 张伟, 李恩泽. 基于拉曼协议的量子存储. 物理学报, 2019, 68(3): 034203. doi: 10.7498/aps.68.20182215
    [6] 窦建鹏, 李航, 庞晓玲, 张超妮, 杨天怀, 金贤敏. 量子存储研究进展. 物理学报, 2019, 68(3): 030307. doi: 10.7498/aps.68.20190039
    [7] 李明, 陈阳, 郭光灿, 任希锋. 表面等离激元量子信息应用研究进展. 物理学报, 2017, 66(14): 144202. doi: 10.7498/aps.66.144202
    [8] 李卓, 邢莉娟. 差错基、量子码与群代数. 物理学报, 2013, 62(13): 130306. doi: 10.7498/aps.62.130306
    [9] 王云江, 白宝明, 彭进业, 王新梅. 针对X-Z型Pauli信道的量子稀疏图码的反馈式和积译码算法. 物理学报, 2011, 60(3): 030306. doi: 10.7498/aps.60.030306
    [10] 邢莉娟, 李卓, 张武军. 加强的量子汉明限. 物理学报, 2011, 60(5): 050304. doi: 10.7498/aps.60.050304
    [11] 王云江, 白宝明, 王新梅. 量子稀疏图码的反馈式迭代译码. 物理学报, 2010, 59(11): 7591-7595. doi: 10.7498/aps.59.7591
    [12] 姜福仕, 赵翠兰. 量子环中量子比特的声子效应. 物理学报, 2009, 58(10): 6786-6790. doi: 10.7498/aps.58.6786
    [13] 尹辑文, 肖景林, 于毅夫, 王子武. 库仑势对抛物量子点量子比特消相干的影响. 物理学报, 2008, 57(5): 2695-2698. doi: 10.7498/aps.57.2695
    [14] 邢莉娟, 李 卓, 白宝明, 王新梅. 量子卷积码的编译码方法. 物理学报, 2008, 57(8): 4695-4699. doi: 10.7498/aps.57.4695
    [15] 李 卓, 邢莉娟. 量子Generalized Reed-Solomon码. 物理学报, 2008, 57(1): 28-30. doi: 10.7498/aps.57.28
    [16] 王子武, 肖景林. 抛物线性限制势量子点量子比特及其光学声子效应. 物理学报, 2007, 56(2): 678-682. doi: 10.7498/aps.56.678
    [17] 李 卓, 邢莉娟. 一类基于级联结构的量子好码. 物理学报, 2007, 56(10): 5602-5606. doi: 10.7498/aps.56.5602
    [18] 张权, 张尔扬. 非对称二状态量子密钥分配协议最优参量研究. 物理学报, 2002, 51(8): 1684-1689. doi: 10.7498/aps.51.1684
    [19] 张权, 唐朝京, 张森强. B92量子密钥分配协议的变形及其无条件安全性证明. 物理学报, 2002, 51(7): 1439-1447. doi: 10.7498/aps.51.1439
    [20] 张权, 唐朝京, 高峰. 量子Turbo码. 物理学报, 2002, 51(1): 15-20. doi: 10.7498/aps.51.15
计量
  • 文章访问数:  5020
  • PDF下载量:  210
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-07-05
  • 修回日期:  2021-08-04
  • 上网日期:  2021-08-20
  • 刊出日期:  2021-12-05

/

返回文章
返回