Efficient Large-Width Montgomery Modular Multiplier Design Based on Toom–Cook-5
Toom–Cook-n multiplication is an efficient large-width multiplication algorithm based on a divide-and-conquer strategy, widely used in modular multiplication operations for cryptographic algorithms. Theoretically, as the degree n increases, Toom–Cook-n can split the multiplicands into more sub-terms...
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| Published in: | Electronics (Basel) Vol. 14; no. 7; p. 1402 |
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| Abstract | Toom–Cook-n multiplication is an efficient large-width multiplication algorithm based on a divide-and-conquer strategy, widely used in modular multiplication operations for cryptographic algorithms. Theoretically, as the degree n increases, Toom–Cook-n can split the multiplicands into more sub-terms to further enhance the performance of the multiplier. However, constrained by the computational burden brought by the growing size of the interpolation matrix as the degree increases, current research predominantly focuses on Toom–Cook-4 and Toom–Cook-3. This paper proposes a Montgomery modular multiplication design based on Toom–Cook-5, which alleviates the computational difficulty of the interpolation step by introducing an interpolation matrix pre-simplification strategy. Additionally, the design incorporates and optimizes carry–save adder and Karatsuba multiplication, enabling Toom–Cook-5 multiplication to be applied in practical and efficient hardware implementation. This paper presents the ASIC implementation results of the hardware architecture under a 90nm process, demonstrating superior performance compared to previous works. |
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| AbstractList | Toom–Cook-n multiplication is an efficient large-width multiplication algorithm based on a divide-and-conquer strategy, widely used in modular multiplication operations for cryptographic algorithms. Theoretically, as the degree n increases, Toom–Cook-n can split the multiplicands into more sub-terms to further enhance the performance of the multiplier. However, constrained by the computational burden brought by the growing size of the interpolation matrix as the degree increases, current research predominantly focuses on Toom–Cook-4 and Toom–Cook-3. This paper proposes a Montgomery modular multiplication design based on Toom–Cook-5, which alleviates the computational difficulty of the interpolation step by introducing an interpolation matrix pre-simplification strategy. Additionally, the design incorporates and optimizes carry–save adder and Karatsuba multiplication, enabling Toom–Cook-5 multiplication to be applied in practical and efficient hardware implementation. This paper presents the ASIC implementation results of the hardware architecture under a 90nm process, demonstrating superior performance compared to previous works. |
| Audience | Academic |
| Author | Hao, Yue Wang, Xiaohua Liu, Kuanhao Wang, Weijiang Zhang, Jingqi |
| Author_xml | – sequence: 1 givenname: Kuanhao surname: Liu fullname: Liu, Kuanhao – sequence: 2 givenname: Xiaohua surname: Wang fullname: Wang, Xiaohua – sequence: 3 givenname: Yue orcidid: 0000-0002-6084-1544 surname: Hao fullname: Hao, Yue – sequence: 4 givenname: Jingqi orcidid: 0000-0003-4140-7029 surname: Zhang fullname: Zhang, Jingqi – sequence: 5 givenname: Weijiang surname: Wang fullname: Wang, Weijiang |
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| Cites_doi | 10.1109/TCSI.2018.2878598 10.17487/rfc5639 10.1109/TVLSI.2012.2227846 10.1109/TVLSI.2021.3058509 10.1109/TCSI.2024.3435473 10.1109/IRETELC.1962.5407919 10.1090/S0025-5718-1985-0777282-X 10.3390/electronics11223712 10.1109/TVLSI.2010.2049037 10.1145/359340.359342 10.1109/ACCESS.2023.3252504 10.3390/electronics11071131 10.23919/DATE.2019.8715287 10.1109/TC.2010.93 10.1109/TCSI.2023.3332880 10.1090/S0002-9947-1969-0249212-8 10.6028/NIST.SP.800-57p3 10.1109/TVLSI.2015.2409113 10.1109/MM.2005.24 10.1103/PhysRevA.98.012311 10.6028/NIST.SP.800-57p1r3 10.1109/TVLSI.2023.3277865 10.1109/CISE.2010.5676912 |
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| References | Dutta (ref_16) 2018; 98 Bedrij (ref_30) 1962; EC-11 ref_35 ref_12 Kuang (ref_34) 2015; 24 ref_10 ref_32 Rivest (ref_1) 1978; 21 Hao (ref_22) 2023; 71 Karatsuba (ref_6) 1963; 7 Choe (ref_5) 2020; 24 ref_17 Gu (ref_21) 2019; 66 ref_38 ref_15 Knezevic (ref_37) 2010; 59 ref_25 ref_24 ref_23 Heidarpur (ref_7) 2021; 29 Toom (ref_9) 1963; 3 Ding (ref_20) 2018; 66 Putranto (ref_19) 2023; 11 Montgomery (ref_29) 1985; 44 ref_3 ref_2 Miyamoto (ref_36) 2010; 19 ref_28 ref_27 Wang (ref_18) 2023; 71 ref_26 Cook (ref_11) 1969; 142 Ramkumar (ref_31) 2010; 42 ref_8 Das (ref_14) 2024; 72 Wang (ref_13) 2023; 31 Kuang (ref_33) 2012; 21 Eberle (ref_4) 2005; 25 |
| References_xml | – ident: ref_28 – volume: 66 start-page: 1003 year: 2018 ident: ref_20 article-title: High-Speed ECC Processor Over NIST Prime Fields Applied With Toom–Cook Multiplication publication-title: IEEE Trans. Circuits Syst. I Regul. Pap. doi: 10.1109/TCSI.2018.2878598 – ident: ref_25 doi: 10.17487/rfc5639 – volume: 42 start-page: 53 year: 2010 ident: ref_31 article-title: ASIC implementation of modified faster carry save adder publication-title: Eur. J. Entific Res. – volume: 21 start-page: 1999 year: 2012 ident: ref_33 article-title: Energy-efficient high-throughput Montgomery modular multipliers for RSA cryptosystems publication-title: IEEE Trans. Very Large Scale Integr. (VLSI) Syst. doi: 10.1109/TVLSI.2012.2227846 – ident: ref_3 – volume: 29 start-page: 667 year: 2021 ident: ref_7 article-title: An Efficient and High-Speed Overlap-Free Karatsuba-Based Finite-Field Multiplier for FGPA Implementation publication-title: IEEE Trans. Very Large Scale Integr. (VLSI) Syst. doi: 10.1109/TVLSI.2021.3058509 – volume: 72 start-page: 600 year: 2024 ident: ref_14 article-title: Area and Delay Trade-Offs in Three-Way Toom-Cook Large Integer Multipliers Implemented on FPGAs publication-title: IEEE Trans. Circuits Syst. I Regul. Pap. doi: 10.1109/TCSI.2024.3435473 – volume: 24 start-page: 961 year: 2020 ident: ref_5 article-title: A High Performance Modular Multiplier for ECC publication-title: J. IKEEE – volume: EC-11 start-page: 340 year: 1962 ident: ref_30 article-title: Carry-Select Adder publication-title: IRE Trans. Electron. Comput. doi: 10.1109/IRETELC.1962.5407919 – volume: 44 start-page: 519 year: 1985 ident: ref_29 article-title: Modular multiplication without trial division publication-title: Math. Comp doi: 10.1090/S0025-5718-1985-0777282-X – ident: ref_35 doi: 10.3390/electronics11223712 – volume: 66 start-page: 1401 year: 2019 ident: ref_21 article-title: A Division-Free Toom–Cook Multiplication-Based Montgomery Modular Multiplication publication-title: IEEE Trans. Circuits Syst. Part II Express Briefs – volume: 19 start-page: 1136 year: 2010 ident: ref_36 article-title: Systematic design of RSA processors based on high-radix Montgomery multipliers publication-title: IEEE Trans. Very Large Scale Integr. (VLSI) Syst. doi: 10.1109/TVLSI.2010.2049037 – volume: 21 start-page: 120 year: 1978 ident: ref_1 article-title: A method for obtaining digital signatures and public-key cryptosystems publication-title: Commun. ACM doi: 10.1145/359340.359342 – volume: 11 start-page: 21848 year: 2023 ident: ref_19 article-title: Space and Time-Efficient Quantum Multiplier in Post Quantum Cryptography Era publication-title: IEEE Access doi: 10.1109/ACCESS.2023.3252504 – volume: 3 start-page: 498 year: 1963 ident: ref_9 article-title: The complexity of a scheme of functional elements realizing the multiplication of integers publication-title: Soviet Math – ident: ref_17 doi: 10.3390/electronics11071131 – ident: ref_26 doi: 10.23919/DATE.2019.8715287 – volume: 7 start-page: 595 year: 1963 ident: ref_6 article-title: Multiplication of multidigit numbers on automata publication-title: Sov. Phys. Dokl. – ident: ref_8 – volume: 59 start-page: 1715 year: 2010 ident: ref_37 article-title: Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods publication-title: IEEE Trans. Comput. doi: 10.1109/TC.2010.93 – ident: ref_27 – ident: ref_2 – volume: 71 start-page: 359 year: 2023 ident: ref_18 article-title: A High-Throughput Toom-Cook-4 Polynomial Multiplier for Lattice-Based Cryptography Using a Novel Winograd-Schoolbook Algorithm publication-title: IEEE Trans. Circuits Syst. I Regul. Pap. doi: 10.1109/TCSI.2023.3332880 – ident: ref_12 – volume: 71 start-page: 862 year: 2023 ident: ref_22 article-title: Efficient Barrett Modular Multiplication Based on Toom–Cook Multiplication publication-title: IEEE Trans. Circuits Syst. II Express Briefs – ident: ref_10 – volume: 142 start-page: 291 year: 1969 ident: ref_11 article-title: On the minimum computation time of functions publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1969-0249212-8 – ident: ref_24 doi: 10.6028/NIST.SP.800-57p3 – volume: 24 start-page: 434 year: 2015 ident: ref_34 article-title: Low-cost high-performance VLSI architecture for Montgomery modular multiplication publication-title: IEEE Trans. Very Large Scale Integr. (VLSI) Syst. doi: 10.1109/TVLSI.2015.2409113 – volume: 25 start-page: 52 year: 2005 ident: ref_4 article-title: Accelerating next-generation public-key cryptosystems on general-purpose CPUs publication-title: IEEE Micro doi: 10.1109/MM.2005.24 – ident: ref_15 – volume: 98 start-page: 012311 year: 2018 ident: ref_16 article-title: Quantum circuits for Toom-Cook multiplication publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.98.012311 – ident: ref_23 doi: 10.6028/NIST.SP.800-57p1r3 – volume: 31 start-page: 1153 year: 2023 ident: ref_13 article-title: TCPM: A Reconfigurable and Efficient Toom-Cook-Based Polynomial Multiplier Over Rings Using a Novel Compressed Postprocessing Algorithm publication-title: IEEE Trans. Very Large Scale Integr. (VLSI) Syst. doi: 10.1109/TVLSI.2023.3277865 – ident: ref_38 – ident: ref_32 doi: 10.1109/CISE.2010.5676912 |
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| SubjectTerms | Algorithms Design Efficiency Hardware Interpolation Multipliers Optimization Performance enhancement |
| Title | Efficient Large-Width Montgomery Modular Multiplier Design Based on Toom–Cook-5 |
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