Lightweight Computational Complexity Stepping Up the NTRU Post-Quantum Algorithm Using Parallel Computing

The Nth-degree Truncated polynomial Ring Unit (NTRU) is one of the famous post-quantum cryptographic algorithms. Researchers consider NTRU to be the most important parameterized family of lattice-based public key cryptosystems that has been established to the IEEE P1363 standards. Lattice-based prot...

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Veröffentlicht in:Symmetry (Basel) Jg. 16; H. 1; S. 12
Hauptverfasser: Elkabbany, Ghada Farouk, Ahmed, Hassan I. Sayed, Aslan, Heba K., Cho, Young-Im, Abdallah, Mohamed S.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Basel MDPI AG 01.01.2024
Schlagworte:
Algorithms
Computer systems
Computers
Cryptography
Digital signatures
Investment analysis
Multiplication
Multiplication & division
Polynomials
Quantum computing
Rings (mathematics)
Standardization
California
ISSN:2073-8994, 2073-8994
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Abstract The Nth-degree Truncated polynomial Ring Unit (NTRU) is one of the famous post-quantum cryptographic algorithms. Researchers consider NTRU to be the most important parameterized family of lattice-based public key cryptosystems that has been established to the IEEE P1363 standards. Lattice-based protocols necessitate operations on large vectors, which makes parallel computing one of the appropriate solutions to speed it up. NTRUEncrypt operations contain a large amount of data that requires many repetitive arithmetic operations. These operations make it a strong candidate to take advantage of the high degree of parallelism. The main costly operation that is repeated in all NTRU algorithm steps is polynomial multiplication. In this work, a Parallel Post-Quantum NTRUEncrypt algorithm called PPQNTRUEncrypt is proposed. This algorithm exploits the capabilities of parallel computing to accelerate the NTRUEncrypt algorithm. Both analytical and Apache Spark simulation models are used. The proposed algorithm enhanced the NTRUEncrypt algorithm by approximately 49.5%, 74.5%, 87.6%, 92.5%, 93.4%, and 94.5%, assuming that the number of processing elements is 2, 4, 8, 12, 16, and 20 respectively.
AbstractList The Nth-degree Truncated polynomial Ring Unit (NTRU) is one of the famous post-quantum cryptographic algorithms. Researchers consider NTRU to be the most important parameterized family of lattice-based public key cryptosystems that has been established to the IEEE P1363 standards. Lattice-based protocols necessitate operations on large vectors, which makes parallel computing one of the appropriate solutions to speed it up. NTRUEncrypt operations contain a large amount of data that requires many repetitive arithmetic operations. These operations make it a strong candidate to take advantage of the high degree of parallelism. The main costly operation that is repeated in all NTRU algorithm steps is polynomial multiplication. In this work, a Parallel Post-Quantum NTRUEncrypt algorithm called PPQNTRUEncrypt is proposed. This algorithm exploits the capabilities of parallel computing to accelerate the NTRUEncrypt algorithm. Both analytical and Apache Spark simulation models are used. The proposed algorithm enhanced the NTRUEncrypt algorithm by approximately 49.5%, 74.5%, 87.6%, 92.5%, 93.4%, and 94.5%, assuming that the number of processing elements is 2, 4, 8, 12, 16, and 20 respectively.
Audience Academic
Author Cho, Young-Im
Elkabbany, Ghada Farouk
Aslan, Heba K.
Abdallah, Mohamed S.
Ahmed, Hassan I. Sayed
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Cites_doi 10.1007/978-3-030-60890-3
10.3390/cryptography7030040
10.1016/j.array.2022.100242
10.1088/1757-899X/981/2/022017
10.1145/237814.237866
10.1109/ICACCI.2014.6968468
10.1109/EuroSP.2018.00032
10.1109/MSP.2017.3151334
10.1109/MSP.2017.3151326
10.20944/preprints202104.0734.v1
10.1007/978-3-319-72565-9_12
10.23939/istcmtm2021.02.005
10.1007/3-540-68339-9_4
10.20944/preprints202007.0450.v1
10.1007/978-3-642-40041-4_3
10.20944/preprints202305.0515.v1
10.3390/s22020489
10.1109/HPCSim.2016.7568376
10.1007/978-3-662-53018-4_21
10.3390/en15030714
10.1155/2018/7012056
10.1145/1568318.1568324
10.1109/MSP.2017.3151328
10.1002/spy2.200
10.1038/s41534-021-00400-7
10.1109/PDGC.2010.5679887
10.13052/jsn2445-9739.2017.10
10.1007/978-3-319-89339-6_16
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References ref_50
Wang (ref_2) 2021; 7
ref_58
ref_57
ref_12
ref_56
ref_11
Karbasi (ref_46) 2018; 7
Tallapally (ref_49) 2020; 981
ref_55
ref_10
ref_54
McEliece (ref_9) 1978; 4244
Jao (ref_14) 2011; 7071
ref_53
ref_52
ref_51
ref_19
ref_17
ref_15
ref_60
Ding (ref_21) 2017; 15
ref_25
ref_24
ref_68
ref_23
Park (ref_61) 2018; 2018
ref_67
ref_22
ref_66
ref_20
ref_63
Regev (ref_13) 2009; 56
ref_29
ref_28
ref_27
ref_26
Kumar (ref_43) 2022; 5
Deshai (ref_64) 2019; 8
Butin (ref_18) 2007; 15
ref_36
ref_35
ref_34
ref_33
ref_32
ref_31
ref_30
ref_39
ref_38
ref_37
Jangla (ref_62) 2015; 4
Kumar (ref_40) 2022; 15
Rasslan (ref_59) 2018; 20
Azize (ref_16) 2021; 10
ref_47
ref_45
ref_44
ref_42
ref_41
Buchmann (ref_6) 2017; 15
ref_1
ref_48
Kumar (ref_65) 2014; 2
ref_8
ref_5
ref_4
Nielsen (ref_3) 2002; 54
ref_7
References_xml – ident: ref_15
  doi: 10.1007/978-3-030-60890-3
– ident: ref_32
– ident: ref_41
  doi: 10.3390/cryptography7030040
– ident: ref_55
– ident: ref_26
– volume: 15
  start-page: 100242
  year: 2022
  ident: ref_40
  article-title: Post-quantum cryptography Algorithm’s standardization and performance analysis
  publication-title: Array
  doi: 10.1016/j.array.2022.100242
– volume: 981
  start-page: 022017
  year: 2020
  ident: ref_49
  article-title: Suitable encrypting algorithms in Parallel Processing for improved efficiency
  publication-title: IOP Conf. Ser. Mater. Sci. Eng.
  doi: 10.1088/1757-899X/981/2/022017
– ident: ref_5
  doi: 10.1145/237814.237866
– ident: ref_45
  doi: 10.1109/ICACCI.2014.6968468
– ident: ref_28
  doi: 10.1109/EuroSP.2018.00032
– volume: 15
  start-page: 37
  year: 2007
  ident: ref_18
  article-title: Hash-based signatures: State of play
  publication-title: IEEE Secur. Priv.
  doi: 10.1109/MSP.2017.3151334
– ident: ref_23
– ident: ref_58
– volume: 4
  start-page: 229
  year: 2015
  ident: ref_62
  article-title: Development of an Intrusion Detection System based on Big Data for Detecting Unknown Attacks
  publication-title: Int. J. Adv. Res. Comput. Commun. Eng.
– volume: 15
  start-page: 12
  year: 2017
  ident: ref_6
  article-title: Postquantum Cryptography-State of the Art
  publication-title: IEEE Secur. Priv.
  doi: 10.1109/MSP.2017.3151326
– ident: ref_8
– ident: ref_4
– ident: ref_56
– volume: 4244
  start-page: 114
  year: 1978
  ident: ref_9
  article-title: A Public-Key Cryptosystem Based on Algebraic
  publication-title: Coding Thv
– ident: ref_48
– ident: ref_10
– ident: ref_1
  doi: 10.20944/preprints202104.0734.v1
– ident: ref_31
  doi: 10.1007/978-3-319-72565-9_12
– volume: 7
  start-page: 11
  year: 2018
  ident: ref_46
  article-title: PairTRU: Pairwise Non-commutative Extension of The NTRU Public key Cryptosystem
  publication-title: Int. J. Inf. Secur. Sci.
– ident: ref_68
  doi: 10.23939/istcmtm2021.02.005
– ident: ref_66
– volume: 54
  start-page: 60
  year: 2002
  ident: ref_3
  article-title: Quantum Computation and Quantum Information
  publication-title: Phys. Today
– ident: ref_38
– ident: ref_17
– ident: ref_11
  doi: 10.1007/3-540-68339-9_4
– volume: 2
  start-page: 11
  year: 2014
  ident: ref_65
  article-title: Evaluation Metrics for Intrusion Detection Systems—A Study
  publication-title: Int. J. Comput. Sci. Mob. Appl.
– ident: ref_20
– ident: ref_7
– ident: ref_24
– ident: ref_34
– ident: ref_60
  doi: 10.20944/preprints202007.0450.v1
– ident: ref_47
– ident: ref_27
  doi: 10.1007/978-3-642-40041-4_3
– ident: ref_42
  doi: 10.20944/preprints202305.0515.v1
– ident: ref_44
  doi: 10.3390/s22020489
– ident: ref_67
– ident: ref_37
– ident: ref_63
– ident: ref_52
  doi: 10.1109/HPCSim.2016.7568376
– volume: 10
  start-page: 725
  year: 2021
  ident: ref_16
  article-title: Boosted Performances of NTRUencrypt Post-Quantum Cryptosystem
  publication-title: J. Cyber Secur. Mobil.
– ident: ref_35
  doi: 10.1007/978-3-662-53018-4_21
– ident: ref_25
– ident: ref_50
– ident: ref_39
  doi: 10.3390/en15030714
– ident: ref_29
– ident: ref_33
– ident: ref_54
– volume: 2018
  start-page: 7012056
  year: 2018
  ident: ref_61
  article-title: Efficient Parallel Implementation of Matrix Multiplication for Lattice-Based Cryptography on Modern ARM Processor
  publication-title: Secur. Commun. Netw.
  doi: 10.1155/2018/7012056
– volume: 7071
  start-page: 19
  year: 2011
  ident: ref_14
  article-title: Towards Quantum-Resistant Cryptosystems from Supersingular Elliptic Curve Isogenies
  publication-title: PQCrypto
– ident: ref_12
– volume: 56
  start-page: 34
  year: 2009
  ident: ref_13
  article-title: On Lattices, Learning with Errors, Random Linear Codes, and Cryptography
  publication-title: J. ACM (JACM)
  doi: 10.1145/1568318.1568324
– volume: 20
  start-page: 371
  year: 2018
  ident: ref_59
  article-title: New Generic Design to Expedite Asymmetric Cryptosystems using Three-level Parallelism
  publication-title: Int. J. Netw. Secur. (IJNS)
– volume: 15
  start-page: 28
  year: 2017
  ident: ref_21
  article-title: Current state of Multivariate Cryptography
  publication-title: IEEE Secur. Priv.
  doi: 10.1109/MSP.2017.3151328
– volume: 5
  start-page: e200
  year: 2022
  ident: ref_43
  article-title: Securing the future internet of things with post-quantum cryptography
  publication-title: Secur. Priv.
  doi: 10.1002/spy2.200
– volume: 7
  start-page: 67
  year: 2021
  ident: ref_2
  article-title: Experimental Authentication of Quantum Key Distribution with Post-Quantum Cryptography
  publication-title: Npj Quantum Inf.
  doi: 10.1038/s41534-021-00400-7
– ident: ref_51
  doi: 10.1109/PDGC.2010.5679887
– ident: ref_36
– ident: ref_19
– ident: ref_22
– volume: 8
  start-page: 45
  year: 2019
  ident: ref_64
  article-title: Mllib: Machine learning in apache spark
  publication-title: Int. J. Recent Technol. Eng.
– ident: ref_57
– ident: ref_53
  doi: 10.13052/jsn2445-9739.2017.10
– ident: ref_30
  doi: 10.1007/978-3-319-89339-6_16
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Snippet The Nth-degree Truncated polynomial Ring Unit (NTRU) is one of the famous post-quantum cryptographic algorithms. Researchers consider NTRU to be the most...
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SubjectTerms Algorithms
Computer systems
Computers
Cryptography
Digital signatures
Investment analysis
Multiplication
Multiplication & division
Polynomials
Quantum computing
Rings (mathematics)
Standardization
Title Lightweight Computational Complexity Stepping Up the NTRU Post-Quantum Algorithm Using Parallel Computing
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