An NTRU-like Message Recoverable Signature Algorithm
Abstract
:1. Introduction
2. Research Content
- Design and improve a group-based NTRU-like signature algorithm based on the GTRU public key cryptosystem studied by Shuai Li [28] et al. It is also ensured that this signature algorithm can statute to the SVP, guaranteeing that it can withstand lattice-based attacks.
- The signature scheme proposed by Nyberg and Rueppel [13] is adapted to a group-based NTRU-like signature algorithm to make it message-recovery friendly and resistant to some well-known forgery attacks and other related lattice attacks.
- The efficiency and security of the signature scheme are verified and analyzed.
3. Mathematical Background
3.1. NTRU
3.1.1. Key Generation
- First choose two random polynomials , and have to satisfy:In some cases, f has no corresponding , in which case another polynomial must be re-selected to compute .
- Compute the public key h using the chosen :
- The public key is () and the private key is ().
3.1.2. Encrypt
- Select the plaintext to be encrypted , after randomly selecting .
- Calculate c:
3.1.3. Decrypt
- To recover the plaintext m, you need to multiply c by the private key f, and then take the modulus of q after calculating the result:
- Finally, using a and , the plaintext m is computed:
3.2.
3.3. Poly- Group
4. An NTRU-like Message Recoverable Signature Algorithm
4.1. Parameters
4.2. Key Generation
- Choose and and there exist , , and satisfying theIn some cases, has no corresponding , in which case the other polynomials and must be re-selected.
- Choose and , let (where E is the identity matrix), and satisfy
- Calculates h and l
- The private key of A is and the public key of A is h; The private key of B is u and the public key of B is l.
4.3. Signature
- A randomly selected .
- A calculates
- A randomly selected .
- A calculates and
- A calculates
- A calculates s and t
- A sends to B.
4.4. Verification
- B calculates , verifies equality, and rejects otherwise.
- B calculates
- B verifies that , otherwise reject.
- B restores message m
5. Performance Analysis
5.1. Parameter
5.2. Key Generation
5.3. Signature and Verification
5.4. Analysis
6. Security Analysis
6.1. Brute Force Attacks
6.2. Lattice-Based Attacks
6.3. Forgery and Key Recovery Attacks
7. Advantages and Challenges
- Against quantum attacks: The traditional signature algorithm is primarily implemented by using algorithms such as RSA, DSA and finite field cryptography. However, all these schemes are practically inadequate against quantum attacks [38]. The NR-GTRU signature algorithm is implemented on top of the GTRU encryption algorithm, and the security of GTRU is associated with the difficult problem of SVP for lattice cryptography, so it is reasonable to believe that the NR-GTRU signature algorithm can be well protected against quantum attacks.
- Faster execution time: Although NR-GTRU loses some of its efficiency when compared to the traditional NTRU signature algorithm, it is still more efficient than the traditional RSA signature algorithm and the Elliptic Curve Digital Signature Algorithm (ECDSA).
- Computation cost: Due to the large number of matrix operations used in NR-GTRU and the resource constrained IoT devices, which usually have low processing power, limited memory and low computational power, this definitely increases the computational cost in IoT devices.
- Appropriate parameters selection: NR-GTRU has certain parameter requirements and it is a difficult task to select the appropriate parameters and mathematical functions for a particular application.
- Communication cost: NR-GTRU is a lattice-based signature algorithm that is more efficient and secure than traditional encryption and signature algorithms, but this also results in more communication bits to be processed.
- Side channel attacks: Digital signature algorithms based on lattice cryptography theory can operate securely in a quantum environment, but in practice, they still face the threat of side-channel attacks such as energy analysis attacks and timing attacks.
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1. Key Generation
- Choose and and there exist , , and
- Choose and and let
- Calculate h and l
- The private key of A is and the public key of A is h; The private key of B is u and the public key of B is l.
Appendix A.2. Signature
- A randomly selected
- A uses the public key l of B to calculate c
- A calculates
- A calculates
- A randomly selected
- A calculates and
- A calculates a
- A calculates s and t
- A sends to B.
Appendix A.3. Verification
- B calculates , verifies equality, and rejects otherwise.
- B calculates:
- B verifies that , otherwise reject.
- B restores message m
Appendix B
Abbreviation | Annotation |
---|---|
DDoS | Distributed Denial of Service Attack |
DSA | Digital Signature Algorithm |
ECC | Elliptic Curve Cipher |
ECDLP | Elliptic Curve Discrete Logarithm Problem |
ECDSA | Elliptic Curve Digital Signature Algorithm |
GTRU | A Group-Based NTRU-Like Public-Key Cryptosystem |
IoT | Internet of Things |
NR | Nyberg-Rueppel |
NR-GTRU | An NTRU-Like Message Recoverable Signature Algorithm |
NR-NTRU | An Efficient Nyberg-Rueppel Type of NTRU Digital Signature Algorithm |
NTRU | Number Theory Research Unit |
SIS | Short Integer Solution problem |
SVP | Shortest Vector Problem |
References
- Ling, Z.; Luo, J.; Xu, Y.; Gao, C.; Wu, K.; Fu, X. Security vulnerabilities of internet of things: A case study of the smart plug system. IEEE Internet Things J. 2017, 4, 1899–1909. [Google Scholar] [CrossRef]
- Das, A.K.; Zeadally, S.; He, D. Taxonomy and analysis of security protocols for internet of things. Future Gener. Comput. Syst. 2018, 89, 110–125. [Google Scholar] [CrossRef]
- Rivest, R.L.; Shamir, A.; Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 1978, 21, 120–126. [Google Scholar] [CrossRef]
- Shor, P.W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 1999, 41, 303–332. [Google Scholar] [CrossRef]
- Stergiou, C.; Psannis, K.E.; Kim, B.-G.; Gupta, B. Secure integration of IoT and cloud computing. Future Gener. Comput. Syst. 2018, 78, 964–975. [Google Scholar] [CrossRef]
- Jose, D.V.; Vijyalakshmi, A. An overview of security in internet of things. Procedia Comput. Sci. 2018, 143, 744–748. [Google Scholar] [CrossRef]
- Koblitz, N. Elliptic curve cryptosystems. Math. Comput. 1987, 48, 203–209. [Google Scholar] [CrossRef]
- Liu, T.; Ramachandran, G.; Jurdak, R. Post-Quantum Cryptography for Internet of Things: A Survey on Performance and Optimization. arXiv 2024, arXiv:2401.17538. [Google Scholar]
- IBM. Ibm Unveils Breakthrough 127-Qubit Quantum Processor. Nov 2021. Available online: https://newsroom.ibm.com/2021-11-16-IBM-Unveils-Breakthrough-127-Qubit-Quantum-Processor (accessed on 24 March 2023).
- IBM. Ibm’s Roadmap for Scaling Quantum Technology. 2020. Available online: https://www.ibm.com/quantum/roadmap (accessed on 24 March 2023).
- Mosca, M. Cybersecurity in an era with quantum computers: Will we be ready? IEEE Secur. Priv. 2018, 16, 38–41. [Google Scholar] [CrossRef]
- Lu, X.; Wen, Q.; Yin, W.; Liang, K.; Jin, Z.; Panaousis, E.; Chen, J. Quantum-resistant identity-based signature with message recovery and proxy delegation. Symmetry 2019, 11, 272. [Google Scholar] [CrossRef]
- Nyberg, K.; Rueppel, R.A. A new signature scheme based on the DSA giving message recovery. In Proceedings of the 1st ACM Conference on Computer and Communications Security, Fairfax, VA, USA, 3–5 November 1993; pp. 58–61. [Google Scholar]
- Hoffstein, J.; Pipher, J.; Silverman, J.H. NTRU: A ring-based public key cryptosystem. In Proceedings of the International Algorithmic Number Theory Symposium, Portland, OR, USA, 21–25 June 1998; pp. 267–288. [Google Scholar]
- Kamal, A.A.; Youssef, A.M. An FPGA implementation of the NTRUEncrypt cryptosystem. In Proceedings of the 2009 International Conference on Microelectronics-ICM, Marrakech, Morocco, 19–22 December 2009; pp. 209–212. [Google Scholar]
- Shen, X.; Du, Z.; Chen, R. Research on NTRU algorithm for mobile java security. In Proceedings of the 2009 International Conference on Scalable Computing and Communications; Eighth International Conference on Embedded Computing, Dalian, China, 25–27 September 2009; pp. 366–369. [Google Scholar]
- Howgrave-Graham, N.; Silverman, J.H.; Whyte, W. Choosing parameter sets for NTRUEncrypt with NAEP and SVES-3. In Proceedings of the Cryptographers’ Track at the RSA Conference, San Francisco, CA, USA, 14–18 February 2005; pp. 118–135. [Google Scholar]
- Hoffstein, J.; Howgrave-Graham, N.; Pipher, J.; Silverman, J.H.; Whyte, W. NTRUSIGN: Digital signatures using the NTRU lattice. In Proceedings of the Cryptographers’ Track at the RSA Conference, San Francisco, CA, USA, 13–17 April 2003; pp. 122–140. [Google Scholar]
- Gentry, C.; Peikert, C.; Vaikuntanathan, V. Trapdoors for hard lattices and new cryptographic constructions. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, Victoria, BC, Canada, 17–20 May 2008; pp. 197–206. [Google Scholar]
- IEEE Std 1363.1-2008; IEEE Standard Specification for Public Key Cryptographic Techniques Based on Hard Problems over Lattices. IEEE: New York, NY, USA, 2009; pp. 1–81. [CrossRef]
- ANSI X9.98-2010; Lattice-Based Polynomial Public Key Establishment Algorithm for the Financial Services Industry. Accredited Standards Committee, Inc.: Annapolis, MD, USA, 2010.
- Lyubashevsky, V. Lattice signatures without trapdoors. In Proceedings of the Annual International Conference on the Theory and Applications of Cryptographic Techniques, Cambridge, UK, 15–19 April 2012; pp. 738–755. [Google Scholar]
- Tian, M.; Huang, L. Lattice-based message recovery signature schemes. Int. J. Electron. Secur. Digit. Forensics 2013, 5, 257–269. [Google Scholar] [CrossRef]
- NIST. Post-Quantum Cryptography: Proposed Requirements and Evaluation Criteria; NIST: Gaithersburg, MD, USA, 2016.
- ElGamal, T. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory 1985, 31, 469–472. [Google Scholar] [CrossRef]
- Fouque, P.-A.; Hoffstein, J.; Kirchner, P.; Lyubashevsky, V.; Pornin, T.; Prest, T.; Ricosset, T.; Seiler, G.; Whyte, W.; Zhang, Z. Falcon: Fast-Fourier lattice-based compact signatures over NTRU. Submiss. Nist’S-Post-Quantum Cryptogr. Stand. Process. 2018, 36, 1–75. [Google Scholar]
- Wu, F.; Yao, W.; Zhang, X.; Zheng, Z. An Efficient Lattice-Based Proxy Signature with Message Recovery. In Proceedings of the International Conference on Security, Privacy and Anonymity in Computation, Communication and Storage, Guangzhou, China, 12–15 December 2017; Volume 10656, pp. 321–331. [Google Scholar]
- Shuai, L.; Xu, H.; Miao, L.; Zhou, X. A Group-based NTRU-like Public-key Cryptosystem for IoT. IEEE Access 2019, 7, 75732–75740. [Google Scholar] [CrossRef]
- Singh, S.; Padhye, S. Generalisations of NTRU cryptosystem. Secur. Commun. Netw. 2016, 9, 6315–6334. [Google Scholar] [CrossRef]
- Segal, D. Polycyclic Groups; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Gebhardt, V. Efficient collection in infinite polycyclic groups. J. Symb. Comput. 2002, 34, 213–228. [Google Scholar] [CrossRef]
- Cavallo, B. Algorithmic Properties of Poly-Z Groups and Secret Sharing Using Non-Commutative Groups; City University of New York: New York, NY, USA, 2015. [Google Scholar]
- Hall, B.C.; Hall, B.C. Lie Groups, Lie Algebras, and Representations; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Elverdi, F.; Akleylek, S.; Kirlar, B.B. Efficient Nyberg-Rueppel type of NTRU digital signature algorithm. Turk. J. Math. 2022, 46, 59–70. [Google Scholar]
- Lenstra, A.K.; Lenstra, H.W.; Lovász, L. Factoring polynomials with rational coefficients. Math. Ann. 1982, 261, 515–534. [Google Scholar] [CrossRef]
- Schanck, J. Practical Lattice Cryptosystems: NTRUEncrypt and NTRUMLS. Master’s Thesis, University of Waterloo, Waterloo, ON, Canada, 2015. [Google Scholar]
- Hoffstein, J.; Pipher, J.; Schanck, J.M.; Silverman, J.H.; Whyte, W. Transcript secure signatures based on modular lattices. In Proceedings of the Post-Quantum Cryptography: 6th International Workshop, PQCrypto 2014, Waterloo, ON, Canada, 1–3 October 2014; Proceedings 6. Springer: Cham, Switzerland, 2014; pp. 142–159. [Google Scholar]
- Perlner, R.A.; Cooper, D.A. Quantum resistant public key cryptography: A survey. In Proceedings of the 8th Symposium on Identity and Trust on the Internet, Gaithersburg, MD, USA, 14–16 April 2009; ACM: New York, NY, USA, 2009; pp. 85–93. [Google Scholar]
NR-GTRU | NR-NTRU | |
---|---|---|
Parameter N | 17 | 293 |
Parameter p | 3 | 3 |
Parameter q | 1009 | 1009 |
key (bit) | ||
Key Generation (ms) | 58.3568 | 37.9395 |
Signature (ms) | 10.3836 | 9.4576 |
Verification (ms) | 7.5134 | 6.4936 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Shen, T.; Miao, L.; Hua, B.; Li, S. An NTRU-like Message Recoverable Signature Algorithm. Mathematics 2024, 12, 2051. https://doi.org/10.3390/math12132051
Shen T, Miao L, Hua B, Li S. An NTRU-like Message Recoverable Signature Algorithm. Mathematics. 2024; 12(13):2051. https://doi.org/10.3390/math12132051
Chicago/Turabian StyleShen, Tingle, Li Miao, Bin Hua, and Shuai Li. 2024. "An NTRU-like Message Recoverable Signature Algorithm" Mathematics 12, no. 13: 2051. https://doi.org/10.3390/math12132051