doi: 10.3934/amc.2021050
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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:
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L. BettaleJ.-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.  Google Scholar

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A. BogdanovT. EisenbarthA. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[10]

J. DingL. HuX. NieJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. T. GoodrichJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

D. NaorM. 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.  Google Scholar

[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.  Google Scholar

[24]

R. Sakai and J. Furukawa, Identity-based broadcast encryption, IACR Cryptol. ePrint Arch., 20072/17, URL http://eprint.iacr.org/2007/217. Google Scholar

[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.  Google Scholar

[26]

C. TaoH. XiangA. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

Z. ZhaoF. GuoJ. LaiW. SusiloB. 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.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

L. BettaleJ.-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.  Google Scholar

[2]

A. BogdanovT. EisenbarthA. 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.  Google Scholar

[3]

D. BonehC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. T. CourtoisA. KlimovJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. DingL. HuX. NieJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. T. GoodrichJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

D. NaorM. 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.  Google Scholar

[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.  Google Scholar

[24]

R. Sakai and J. Furukawa, Identity-based broadcast encryption, IACR Cryptol. ePrint Arch., 20072/17, URL http://eprint.iacr.org/2007/217. Google Scholar

[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.  Google Scholar

[26]

C. TaoH. XiangA. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

Z. ZhaoF. GuoJ. LaiW. SusiloB. 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.  Google Scholar

[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.   Google Scholar

Table 1.  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
Table 2.  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
Table 3.  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
Table 4.  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 $\checkmark$
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