# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021050
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## A multivariate identity-based broadcast encryption with applications to the internet of things

 1 Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur-831014, India 2 Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA 3 SAG Lab, Defense Research & Development Organization, Delhi-110054, India

* Corresponding author: sdebnath.math@nitjsr.ac.in

Received  April 2021 Revised  September 2021 Early access November 2021

Fund Project: The work is supported by DRDO, India (ERIP/ER/202005001/M/01/1775)

When Kevin Ashton proposed the catchword 'Internet of Things' in 1999, little did he know that technology will become an indispensable part of human lives in just two decades. In short, the Internet of Things (IoT), is a catch-all terminology used to describe devices connected to the internet. These devices can share and receive data as well as provide instructions over a network. By design itself, the IoT system requires multicasting data and information to a set of designated devices, securely. Taking everything into account, Broadcast Encryption (BE) seems to be the natural choice to address the problem. BE allows an originator to broadcast ciphertexts to a big group of receivers in a well-organized and competent way, while ensuring that only designated people can decrypt the data. In this work, we put forward the first Identity-Based Broadcast Encryption scheme based on multivariate polynomials that achieves post-quantum security. Multivariate public key cryptosystems (MPKC), touted as one of the most promising post-quantum cryptography candidates, forms the foundation on which our scheme relies upon, which allows it to be very cost-effective and faster when implemented. In addition, it also provides resistance to collusion attack, and as a consequence our scheme can be utilized to form an efficient and robust IoT system.

Citation: Vikas Srivastava, Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Saibal Kumar Pal. A multivariate identity-based broadcast encryption with applications to the internet of things. Advances in Mathematics of Communications, doi: 10.3934/amc.2021050
##### References:
 [1] L. Bettale, J.-C. Faugëre and L. Perret, Hybrid approach for solving multivariate systems over finite fields, J. Math. Cryptology, 3 (2009), 177-197.  doi: 10.1515/JMC.2009.009. [2] A. Bogdanov, T. Eisenbarth, A. Rupp and C. Wolf, Time-area optimized public-key engines: MQ-cryptosystems as replacement for elliptic curves?, Cryptographic Hardware and Embedded Systems-CHES 2008, 5154 (2008), 45-61.  doi: 10.1007/978-3-540-85053-3_4. [3] D. Boneh, C. Gentry and B. Waters, Collusion resistant broadcast encryption with short ciphertexts and private keys, Advances in Cryptology–CRYPTO 2005, 3621 (2005), 258-275.  doi: 10.1007/11535218_16. [4] R. Canetti, J. Garay, G. Itkis, D. Micciancio, M. Naor and B. Pinkas, Multicast security: A taxonomy and some efficient constructions, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), IEEE, 1999. doi: 10.1109/INFCOM.1999.751457. [5] A. I.-T. Chen, M.-S. Chen, T.-R. Chen, C.-M. Cheng, J. Ding, E. L.-H. Kuo, F. Y.-S. Lee and B.-Y. Yang, SSE implementation of multivariate PKCs on modern s86 CPUs, Cryptographic Hardware and Embedded Systems - CHES 2009, (2009), 33–48. doi: 10.1007/978-3-642-04138-9_3. [6] N. T. Courtois, Efficient zero-knowledge authentication based on a linear algebra problem MinRank, Advances in Cryptology–ASIACRYPT 2001, 2248 (2001), 402-421.  doi: 10.1007/3-540-45682-1_24. [7] N. T. Courtois, A. Klimov, J. Patarin and A. Shamir, Efficient algorithms for solving overdefined systems of multivariate polynomial equations, Advances in Cryptology–EUROCRYPT 2000, 1807 (2000), 392-407.  doi: 10.1007/3-540-45539-6_27. [8] C. Delerablée, Identity-based broadcast encryption with constant size ciphertexts and private keys, Advances in Cryptology–ASIACRYPT 2007, 4833 (2007), 200-215.  doi: 10.1007/978-3-540-76900-2_12. [9] C. Delerablée, Identity-based broadcast encryption with constant size ciphertexts and private keys, Advances in Cryptology–ASIACRYPT 2007, 4833 (2007), 200-215.  doi: 10.1007/978-3-540-76900-2_12. [10] J. Ding, L. Hu, X. Nie, J. Li and J. Wagner, High order linearization equation hole attack on multivariate public key cryptosystems, Public Key Cryptography – PKC 2007, 4450 (2007), 233-248.  doi: 10.1007/978-3-540-71677-8_16. [11] J. Ding, A. Petzoldt and D. S. Schmidt, Multivariate Public Key Cryptosystems, 2$^nd$ edition, Advances in Information Security, 80. Springer, New York, 2020. doi: 10.1007/978-1-0716-0987-3. [12] Y. Dodis and N. Fazio, Public key broadcast encryption for stateless receivers, Digital Rights Management, 2696 (2002), 61-80.  doi: 10.1007/978-3-540-44993-5_5. [13] J. C. Faugére, A new efficient algorithm for computing Gröbner bases without reduction to zero ($F_5$), Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, (2002), 75–83. [14] J.-C. Faugére, A new efficient algorithm for computing Gröbner bases ($F_4$), J. Pure Appl. Algebra, 139 (1999), 61-88.  doi: 10.1016/S0022-4049(99)00005-5. [15] A. Fiat and M. Naor, Broadcast encryption, Advances in Cryptology–CRYPTO' 93, 773 (1993), 480-491.  doi: 10.1007/3-540-48329-2_40. [16] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, A Series of Books in the Mathematical Sciences, 1979. [17] M. T. Goodrich, J. Z. Sun and R. Tamassia, Efficient tree-based revocation in groups of low-state devices, Advances in Cryptology–CRYPTO 2004, 3152 (2004), 511-527.  doi: 10.1007/978-3-540-28628-8_31. [18] L. Goubin and N. T. Courtois, Cryptanalysis of the TTM cryptosystem, Advances in Cryptology–ASIACRYPT 2000, 1976 (2000), 44-57.  doi: 10.1007/3-540-44448-3_4. [19] D. Halevy and A. Shamir, The LSD broadcast encryption scheme, Advances in Cryptology–CRYPTO 2002, 2442 (2002), 47-60.  doi: 10.1007/3-540-45708-9_4. [20] K. He, J. Weng, J.-N. Liu, J. K. Liu, W. Liu and R. H. Deng, Anonymous identity-based broadcast encryption with chosen-ciphertext security, In Proceedings of the 11th ACM on Asia Conference on Computer and Communications Security, (2016), 247–255. [21] J. Kim, S. Camtepe, W. Susilo, S. Nepal and J. Baek, Identity-based broadcast encryption with outsourced partial decryption for hybrid security models in edge computing, Proceedings of the 2019 ACM Asia Conference on Computer and Communications Security, (2019), 55–66. [22] D. Naor, M. Naor and J. Lotspiech, Revocation and tracing schemes for stateless receivers, Advances in Cryptology–CRYPTO 2001, 2139 (2001), 41-62.  doi: 10.1007/3-540-44647-8_3. [23] J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt'88, Advances in Cryptology–CRYPT0' 95, 963 (1995), 248-261.  doi: 10.1007/3-540-44750-4_20. [24] R. Sakai and J. Furukawa, Identity-based broadcast encryption, IACR Cryptol. ePrint Arch., 20072/17, URL http://eprint.iacr.org/2007/217. [25] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev, 41 (1999), 303-332.  doi: 10.1137/S0036144598347011. [26] C. Tao, H. Xiang, A. Petzoldt and J. Ding, Simple matrix–a multivariate public key cryptosystem (MPKC) for encryption, Finite Fields Appl., 35 (2015), 352-368.  doi: 10.1016/j.ffa.2015.06.001. [27] B.-Y. Yang, C.-M. Cheng, B.-R. Chen and J.-M. Chen, Implementing minimized multivariate PKC on low-resource embedded systems,, Security in Pervasive Computing, Springer Berlin Heidelberg, 3934 (2006), 73–88. doi: 10.1007/11734666_7. [28] T. Yasuda, X. Dahan, Y.-J. Huang, T. Takagi and K. Sakurai, MQ Challenge: Hardness Evaluation of Solving Multivariate Quadratic Problems, Cryptology ePrint Archive, Report, 2015/275, 2015, https://eprint.iacr.org/2015/275. [29] Z. Zhao, F. Guo, J. Lai, W. Susilo, B. Wang and Y. Hu, Accountable authority identity-based broadcast encryption with constant-size private keys and ciphertexts, Theoret. Comput. Sci., 809 (2020), 73-87.  doi: 10.1016/j.tcs.2019.11.035. [30] X. Zhao and F. Zhang, Fully CCA2 secure identity-based broadcast encryption with black-box accountable authority, Journal of Systems and Software, 85 (2012), 708-716.

show all references

##### References:
 [1] L. Bettale, J.-C. Faugëre and L. Perret, Hybrid approach for solving multivariate systems over finite fields, J. Math. Cryptology, 3 (2009), 177-197.  doi: 10.1515/JMC.2009.009. [2] A. Bogdanov, T. Eisenbarth, A. Rupp and C. Wolf, Time-area optimized public-key engines: MQ-cryptosystems as replacement for elliptic curves?, Cryptographic Hardware and Embedded Systems-CHES 2008, 5154 (2008), 45-61.  doi: 10.1007/978-3-540-85053-3_4. [3] D. Boneh, C. Gentry and B. Waters, Collusion resistant broadcast encryption with short ciphertexts and private keys, Advances in Cryptology–CRYPTO 2005, 3621 (2005), 258-275.  doi: 10.1007/11535218_16. [4] R. Canetti, J. Garay, G. Itkis, D. Micciancio, M. Naor and B. Pinkas, Multicast security: A taxonomy and some efficient constructions, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), IEEE, 1999. doi: 10.1109/INFCOM.1999.751457. [5] A. I.-T. Chen, M.-S. Chen, T.-R. Chen, C.-M. Cheng, J. Ding, E. L.-H. Kuo, F. Y.-S. Lee and B.-Y. Yang, SSE implementation of multivariate PKCs on modern s86 CPUs, Cryptographic Hardware and Embedded Systems - CHES 2009, (2009), 33–48. doi: 10.1007/978-3-642-04138-9_3. [6] N. T. Courtois, Efficient zero-knowledge authentication based on a linear algebra problem MinRank, Advances in Cryptology–ASIACRYPT 2001, 2248 (2001), 402-421.  doi: 10.1007/3-540-45682-1_24. [7] N. T. Courtois, A. Klimov, J. Patarin and A. Shamir, Efficient algorithms for solving overdefined systems of multivariate polynomial equations, Advances in Cryptology–EUROCRYPT 2000, 1807 (2000), 392-407.  doi: 10.1007/3-540-45539-6_27. [8] C. Delerablée, Identity-based broadcast encryption with constant size ciphertexts and private keys, Advances in Cryptology–ASIACRYPT 2007, 4833 (2007), 200-215.  doi: 10.1007/978-3-540-76900-2_12. [9] C. Delerablée, Identity-based broadcast encryption with constant size ciphertexts and private keys, Advances in Cryptology–ASIACRYPT 2007, 4833 (2007), 200-215.  doi: 10.1007/978-3-540-76900-2_12. [10] J. Ding, L. Hu, X. Nie, J. Li and J. Wagner, High order linearization equation hole attack on multivariate public key cryptosystems, Public Key Cryptography – PKC 2007, 4450 (2007), 233-248.  doi: 10.1007/978-3-540-71677-8_16. [11] J. Ding, A. Petzoldt and D. S. Schmidt, Multivariate Public Key Cryptosystems, 2$^nd$ edition, Advances in Information Security, 80. Springer, New York, 2020. doi: 10.1007/978-1-0716-0987-3. [12] Y. Dodis and N. Fazio, Public key broadcast encryption for stateless receivers, Digital Rights Management, 2696 (2002), 61-80.  doi: 10.1007/978-3-540-44993-5_5. [13] J. C. Faugére, A new efficient algorithm for computing Gröbner bases without reduction to zero ($F_5$), Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, (2002), 75–83. [14] J.-C. Faugére, A new efficient algorithm for computing Gröbner bases ($F_4$), J. Pure Appl. Algebra, 139 (1999), 61-88.  doi: 10.1016/S0022-4049(99)00005-5. [15] A. Fiat and M. Naor, Broadcast encryption, Advances in Cryptology–CRYPTO' 93, 773 (1993), 480-491.  doi: 10.1007/3-540-48329-2_40. [16] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, A Series of Books in the Mathematical Sciences, 1979. [17] M. T. Goodrich, J. Z. Sun and R. Tamassia, Efficient tree-based revocation in groups of low-state devices, Advances in Cryptology–CRYPTO 2004, 3152 (2004), 511-527.  doi: 10.1007/978-3-540-28628-8_31. [18] L. Goubin and N. T. Courtois, Cryptanalysis of the TTM cryptosystem, Advances in Cryptology–ASIACRYPT 2000, 1976 (2000), 44-57.  doi: 10.1007/3-540-44448-3_4. [19] D. Halevy and A. Shamir, The LSD broadcast encryption scheme, Advances in Cryptology–CRYPTO 2002, 2442 (2002), 47-60.  doi: 10.1007/3-540-45708-9_4. [20] K. He, J. Weng, J.-N. Liu, J. K. Liu, W. Liu and R. H. Deng, Anonymous identity-based broadcast encryption with chosen-ciphertext security, In Proceedings of the 11th ACM on Asia Conference on Computer and Communications Security, (2016), 247–255. [21] J. Kim, S. Camtepe, W. Susilo, S. Nepal and J. Baek, Identity-based broadcast encryption with outsourced partial decryption for hybrid security models in edge computing, Proceedings of the 2019 ACM Asia Conference on Computer and Communications Security, (2019), 55–66. [22] D. Naor, M. Naor and J. Lotspiech, Revocation and tracing schemes for stateless receivers, Advances in Cryptology–CRYPTO 2001, 2139 (2001), 41-62.  doi: 10.1007/3-540-44647-8_3. [23] J. Patarin, Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt'88, Advances in Cryptology–CRYPT0' 95, 963 (1995), 248-261.  doi: 10.1007/3-540-44750-4_20. [24] R. Sakai and J. Furukawa, Identity-based broadcast encryption, IACR Cryptol. ePrint Arch., 20072/17, URL http://eprint.iacr.org/2007/217. [25] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev, 41 (1999), 303-332.  doi: 10.1137/S0036144598347011. [26] C. Tao, H. Xiang, A. Petzoldt and J. Ding, Simple matrix–a multivariate public key cryptosystem (MPKC) for encryption, Finite Fields Appl., 35 (2015), 352-368.  doi: 10.1016/j.ffa.2015.06.001. [27] B.-Y. Yang, C.-M. Cheng, B.-R. Chen and J.-M. Chen, Implementing minimized multivariate PKC on low-resource embedded systems,, Security in Pervasive Computing, Springer Berlin Heidelberg, 3934 (2006), 73–88. doi: 10.1007/11734666_7. [28] T. Yasuda, X. Dahan, Y.-J. Huang, T. Takagi and K. Sakurai, MQ Challenge: Hardness Evaluation of Solving Multivariate Quadratic Problems, Cryptology ePrint Archive, Report, 2015/275, 2015, https://eprint.iacr.org/2015/275. [29] Z. Zhao, F. Guo, J. Lai, W. Susilo, B. Wang and Y. Hu, Accountable authority identity-based broadcast encryption with constant-size private keys and ciphertexts, Theoret. Comput. Sci., 809 (2020), 73-87.  doi: 10.1016/j.tcs.2019.11.035. [30] X. Zhao and F. Zhang, Fully CCA2 secure identity-based broadcast encryption with black-box accountable authority, Journal of Systems and Software, 85 (2012), 708-716.
Proposed practical parameters for ${\sf MulIB-BE}$ [26]
 Level of Security (in bit) Field ($\mathbb{F}_q$) Number of equations ($m$) Number of variables ($n$) 80 $\mathbb{F}_{2^{32}}$ 112 56 $\mathbb{F}_{2^{16}}$ 200 100 $\mathbb{F}_{2^{8}}$ 264 128 90 $\mathbb{F}_{2^{32}}$ 144 72 $\mathbb{F}_{2^{16}}$ 242 121 $\mathbb{F}_{2^{8}}$ 312 153 100 $\mathbb{F}_{2^{32}}$ 180 90 $\mathbb{F}_{2^{16}}$ 288 144 $\mathbb{F}_{2^{8}}$ 364 180
 Level of Security (in bit) Field ($\mathbb{F}_q$) Number of equations ($m$) Number of variables ($n$) 80 $\mathbb{F}_{2^{32}}$ 112 56 $\mathbb{F}_{2^{16}}$ 200 100 $\mathbb{F}_{2^{8}}$ 264 128 90 $\mathbb{F}_{2^{32}}$ 144 72 $\mathbb{F}_{2^{16}}$ 242 121 $\mathbb{F}_{2^{8}}$ 312 153 100 $\mathbb{F}_{2^{32}}$ 180 90 $\mathbb{F}_{2^{16}}$ 288 144 $\mathbb{F}_{2^{8}}$ 364 180
Communication and Storage Overheads of ${\sf MulIB-BE}$
 MPK Size $m\binom{n+2}{2}\binom{N+8}{8}$ field $(\mathbb{F}_q)$ elements Ciphertext Size $m\binom{N+9}{9}+1$ field $(\mathbb{F}_q)$ elements MSK Size $[m(m+1)+ n(n+1)+m\binom{n+2}{2}]\binom{N+2}{2}$ field ($\mathbb{F}_q$) elements SK Size $[m(m+1)+ n(n+1)+m\binom{n+2}{2}]$ field ($\mathbb{F}_q$) elements
 MPK Size $m\binom{n+2}{2}\binom{N+8}{8}$ field $(\mathbb{F}_q)$ elements Ciphertext Size $m\binom{N+9}{9}+1$ field $(\mathbb{F}_q)$ elements MSK Size $[m(m+1)+ n(n+1)+m\binom{n+2}{2}]\binom{N+2}{2}$ field ($\mathbb{F}_q$) elements SK Size $[m(m+1)+ n(n+1)+m\binom{n+2}{2}]$ field ($\mathbb{F}_q$) elements
Time complexity of ${\sf MulIB-BE}$ for 80-bit security level over $GF(256)$
 Time (in seconds) Setup 11.91 Key Extraction 0.56 Encryption 2.17 Decryption 1.25
 Time (in seconds) Setup 11.91 Key Extraction 0.56 Encryption 2.17 Decryption 1.25
Comparison with existing schemes for $100$-bit security level
 Scheme Secret key size (in kb) Ciphertext size (in kb) Post-quantum secure ZhanoZhang-IB-BE [30] 0.375 1.25 $\times$ A-IBBE [29] 0.05 0.875 $\times$ Delerablée-IB-BE [9] 0.06 0.5 $\times$ Kim, Jongkil et al. [21] 0.06 0.5 $\times$ He, Kai et al. [20] 0.06 0.28 $\times$ ${\sf MulIB-BE}$ 21.36 7.09 $\checkmark$
 Scheme Secret key size (in kb) Ciphertext size (in kb) Post-quantum secure ZhanoZhang-IB-BE [30] 0.375 1.25 $\times$ A-IBBE [29] 0.05 0.875 $\times$ Delerablée-IB-BE [9] 0.06 0.5 $\times$ Kim, Jongkil et al. [21] 0.06 0.5 $\times$ He, Kai et al. [20] 0.06 0.28 $\times$ ${\sf MulIB-BE}$ 21.36 7.09 $\checkmark$
 [1] Ramprasad Sarkar, Mriganka Mandal, Sourav Mukhopadhyay. Quantum-safe identity-based broadcast encryption with provable security from multivariate cryptography. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022026 [2] Jintai Ding, Sihem Mesnager, Lih-Chung Wang. Letters for post-quantum cryptography standard evaluation. Advances in Mathematics of Communications, 2020, 14 (1) : i-i. doi: 10.3934/amc.2020012 [3] Gérard Maze, Chris Monico, Joachim Rosenthal. Public key cryptography based on semigroup actions. Advances in Mathematics of Communications, 2007, 1 (4) : 489-507. doi: 10.3934/amc.2007.1.489 [4] Javier de la Cruz, Ricardo Villanueva-Polanco. Public key cryptography based on twisted dihedral group algebras. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022031 [5] Gerhard Frey. Relations between arithmetic geometry and public key cryptography. Advances in Mathematics of Communications, 2010, 4 (2) : 281-305. doi: 10.3934/amc.2010.4.281 [6] Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281 [7] Lidong Chen, Dustin Moody. New mission and opportunity for mathematics researchers: Cryptography in the quantum era. Advances in Mathematics of Communications, 2020, 14 (1) : 161-169. doi: 10.3934/amc.2020013 [8] Diego F. Aranha, Ricardo Dahab, Julio López, Leonardo B. Oliveira. Efficient implementation of elliptic curve cryptography in wireless sensors. Advances in Mathematics of Communications, 2010, 4 (2) : 169-187. doi: 10.3934/amc.2010.4.169 [9] Andreas Klein. How to say yes, no and maybe with visual cryptography. Advances in Mathematics of Communications, 2008, 2 (3) : 249-259. doi: 10.3934/amc.2008.2.249 [10] Anna-Lena Horlemann-Trautmann, Violetta Weger. Information set decoding in the Lee metric with applications to cryptography. Advances in Mathematics of Communications, 2021, 15 (4) : 677-699. doi: 10.3934/amc.2020089 [11] Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046 [12] Felipe Cabarcas, Daniel Cabarcas, John Baena. Efficient public-key operation in multivariate schemes. Advances in Mathematics of Communications, 2019, 13 (2) : 343-371. doi: 10.3934/amc.2019023 [13] Yu-Chi Chen. Security analysis of public key encryption with filtered equality test. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021053 [14] Sikhar Patranabis, Debdeep Mukhopadhyay. Identity-based key aggregate cryptosystem from multilinear maps. Advances in Mathematics of Communications, 2019, 13 (4) : 759-778. doi: 10.3934/amc.2019044 [15] Yang Lu, Jiguo Li. Forward-secure identity-based encryption with direct chosen-ciphertext security in the standard model. Advances in Mathematics of Communications, 2017, 11 (1) : 161-177. doi: 10.3934/amc.2017010 [16] Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215 [17] Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046 [18] Zhiping Zhou, Xinbao Liu, Jun Pei, Panos M. Pardalos, Hao Cheng. Competition of pricing and service investment between iot-based and traditional manufacturers. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1203-1218. doi: 10.3934/jimo.2018006 [19] Rainer Steinwandt, Adriana Suárez Corona. Attribute-based group key establishment. Advances in Mathematics of Communications, 2010, 4 (3) : 381-398. doi: 10.3934/amc.2010.4.381 [20] Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053

2020 Impact Factor: 0.935